Loop group actions on categories and Whittaker invariants
Dario Beraldo
We develop some aspects of the theory of $\fD$-modules on ind-schemes of pro-finite type. These notions are used to define $\fD$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. %We also extend the Fourier-Deligne transform to Tate vector spaces. \\ Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $\chi: N\ppart \to \GG_a$ and a category $\C$ acted upon by $N \ppart$, we define the category $\C^{N \ppart, \chi}$ of $(N \ppart, \chi)$-invariant objects, along with the coinvariant category $\C_{N \ppart, \chi}$. These are the Whittaker categories of $\C$, which are in general not equivalent. However, there is always a family of functors $\Theta_k: \C_{N \ppart, \chi} \to \C^{N \ppart, \chi}$, parametrized by $k \in \ZZ$. We conjecture that each $\Theta_k$ is an equivalence, provided that the $N \ppart$-action on $\C$ extends to a $G \ppart$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained by taking invariants of $\C$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G \ppart$. %
1310.5127.tex
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\nc\tstar{\wt{\star}}
\nc{\alp}{\alpha}

\nc{\ZZ}{{\mathbb Z}}
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\nc{\DD}{{\mathbb D}}
\nc{\GG}{{\mathbb G}}
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\nc{\Fqb}{\ol{{\mathbb F}_q}}
\nc{\Ql}{\ol{{\mathbb Q}_\ell}}
\nc{\id}{\text{id}}
\nc\X{\mathcal X}

\nc{\Hom}{\on{Hom}}
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\nc{\Pic}{\on{Pic}}
\nc{\Bun}{\on{Bun}}
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\nc{\rk}{\on{rk}}
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\nc{\Conv}{\on{Conv}}
\nc{\Sph}{\on{Sph}}
\nc{\Sym}{\on{Sym}}
\renewcommand{\dim}{\on{dim}}
%\nc{\deg}{\on{deg}}
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\nc{\BunNb}{\overline{\Bun}_N}
\nc{\BunTb}{\overline{\Bun}_T}
\nc{\BunBbm}{\overline{\Bun}_{B^-}}
\nc{\BunBbel}{\overline{\Bun}_{B,el}}
\nc{\BunBbmel}{\overline{\Bun}_{B^-,el}}
\nc{\Buno}{\overset{o}{\Bun}}
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\nc{\BunBM}{\Bun_{B(M)}}
\nc{\BunBMb}{\overline{\Bun}_{B(M)}}
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\nc{\BunBP}{\widetilde{\Bun}_{B,P}}
\nc{\GUb}{\overline{G/U}}
\nc{\GUPb}{\overline{G/U(P)}}


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\nc\mul{\ol{\mu}}
\nc\Sum\Sigma
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\nc{\Y}{{\mathcal Y}}
\nc{\G}{{\mathcal G}}
\nc{\E}{{\mathcal E}}
\nc{\CalC}{{\mathcal C}}
%\renc{\Sb}{\overline{S}}
\nc\Dh{\widehat{\D}}
\renewcommand{\O}{{\mathcal O}}
\nc{\C}{{\mathcal C}}
\nc{\K}{{\mathcal K}}
\renewcommand{\H}{{\mathcal H}}
\renewcommand{\S}{{\mathcal S}}
\nc{\T}{{\mathcal T}}
\nc{\V}{{\mathcal V}}
\renc{\P}{{\mathcal P}}
\nc{\A}{{\mathcal A}}
\nc{\B}{{\mathcal B}}
\nc{\U}{{\mathcal U}}
\renewcommand{\L}{{\mathcal L}}
\nc{\Gr}{{\on{Gr}}}

%\nc{\fA}{{\mathfrak A}}
%\nc{\fP}{{\mathfrak P}}
\nc{\frn}{{\check{\mathfrak u}(P)}}

\nc{\fC}{\mathfrak C}
\nc{\p}{\mathfrak p}
\nc{\q}{\mathfrak q}
\nc\f{{\mathfrak f}}
\renewcommand\k{{\mathfrak k}}
\nc{\qo}{{\mathfrak q}}
\nc{\po}{{\mathfrak p}}
\nc{\s}{{\mathfrak s}}
\nc\w{\text{w}}
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\renewcommand{\mod}{{\on{-mod}}}
\newcommand{\comod}{{\on{-comod}}}

\nc\mathi\iota
\nc\Spec{\on{Spec}}
\nc\Mod{\on{Mod}}
\nc{\tw}{\widetilde{\mathfrak t}}
\nc{\pw}{\widetilde{\mathfrak p}}
\nc{\qw}{\widetilde{\mathfrak q}}
\nc{\jw}{\widetilde j}

\nc{\grb}{\overline{\Gr_{X^{\fset}}}}
\nc{\I}{\mathcal I}
\renewcommand{\i}{\mathfrak i}
\renewcommand{\j}{\mathfrak j}

%\nc{\lambdach}{{\check\lambda}}
%\nc{\Lambdach}{{\check\Lambda}{}}
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\nc{\rhoch}{{\check\rho}}
\nc{\ch}{{\check\fh}}

\nc{\Hb}{\overline{\H}}

%%%%%%%%%%%%%%%%%%%%%%%%%%


\emergencystretch=2cm

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\nc{\BC}{{\mathbb{C}}}
\nc{\BE}{{\mathbb{E}}}
\nc{\BG}{{\mathbb{G}}}
\nc{\BM}{{\mathbb{M}}}
\nc{\BO}{{\mathbb{O}}}
\nc{\BD}{{\mathbb{D}}}
\nc{\BN}{{\mathbb{N}}}
\nc{\BP}{{\mathbb{P}}}
\nc{\BR}{{\mathbb{R}}}
\nc{\BZ}{{\mathbb{Z}}}
\nc{\BS}{{\mathbb{S}}}
\nc{\BV}{{\mathbb{V}}}

\nc{\CA}{{\mathcal{A}}}
\nc{\CB}{{\mathcal{B}}}

\nc{\CE}{{\mathcal{E}}}
\nc{\CF}{{\mathcal{F}}}
\nc{\CH}{{\mathcal{H}}}

\nc{\CL}{{\mathcal{L}}}
%\nc{\CC}{{\mathcal{C}}}
\nc{\CG}{{\mathcal{G}}}
\nc{\CM}{{\mathcal{M}}}
\nc{\CN}{{\mathcal{N}}}
\nc{\CK}{{\mathcal{K}}}
\nc{\CO}{{\mathcal{O}}}
\nc{\CP}{{\mathcal{P}}}
\nc{\CQ}{{\mathcal{Q}}}
\nc{\CR}{{\mathcal{R}}}
\nc{\CS}{{\mathcal{S}}}
\nc{\CT}{{\mathcal{T}}}
\nc{\CU}{{\mathcal{U}}}
\nc{\CV}{{\mathcal{V}}}
\nc{\CW}{{\mathcal{W}}}
\nc{\CX}{{\mathcal{X}}}
\nc{\CY}{{\mathcal{Y}}}
\nc{\CZ}{{\mathcal{Z}}}
\nc{\CI}{{\mathcal{I}}}

%\nc{\cM}{{\check{\mathcal M}}{}}
\nc{\csM}{{\check{\mathcal A}}{}}
\nc{\oM}{{\overset{\circ}{\mathcal M}}{}}
\nc{\obM}{{\overset{\circ}{\mathbf M}}{}}
\nc{\oCA}{{\overset{\circ}{\mathcal A}}{}}
\nc{\obA}{{\overset{\circ}{\mathbf A}}{}}
\nc{\ooM}{{\overset{\circ}{M}}{}}
\nc{\osM}{{\overset{\circ}{\mathsf M}}{}}
\nc{\vM}{{\overset{\bullet}{\mathcal M}}{}}
\nc{\nM}{{\underset{\bullet}{\mathcal M}}{}}
\nc{\oD}{{\overset{\circ}{\mathcal D}}{}}
\nc{\obC}{{\overset{\circ}{\mathbf C}}{}}
\nc{\obD}{{\overset{\circ}{\mathbf D}}{}}
\nc{\oA}{{\overset{\circ}{\mathbb A}}{}}
\nc{\op}{{\overset{\bullet}{\mathbf p}}{}}
%\nc{\cp}{{\overset{\circ}{\mathbf p}}{}}
\nc{\oU}{{\overset{\bullet}{\mathcal U}}{}}
\nc{\oZ}{{\overset{\circ}{\mathcal Z}}{}}
\nc{\ofZ}{{\overset{\circ}{\mathfrak Z}}{}}
\nc{\oF}{{\overset{\circ}{\fF}}}

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\nc{\fb}{{\mathfrak{b}}}
\nc{\fc}{{\mathfrak{c}}}
\nc{\fd}{{\mathfrak{d}}}
\nc{\ff}{{\mathfrak{f}}}
\nc{\fg}{{\mathfrak{g}}}
\nc{\fgl}{{\mathfrak{gl}}}
\nc{\fh}{{\mathfrak{h}}}
\nc{\fj}{{\mathfrak{j}}}
\nc{\fl}{{\mathfrak{l}}}
\nc{\fm}{{\mathfrak{m}}}
\nc{\fn}{{\mathfrak{n}}}
\nc{\fu}{{\mathfrak{u}}}
\nc{\fp}{{\mathfrak{p}}}
\nc{\fr}{{\mathfrak{r}}}
\nc{\fs}{{\mathfrak{s}}}
%\nc{\ft}{{\mathfrak{t}}}
\nc{\fz}{{\mathfrak{z}}}
\nc{\fsl}{{\mathfrak{sl}}}
\nc{\hsl}{{\widehat{\mathfrak{sl}}}}
\nc{\hgl}{{\widehat{\mathfrak{gl}}}}
\nc{\hg}{{\widehat{\mathfrak{g}}}}
\nc{\chg}{{\widehat{\mathfrak{g}}}{}^\vee}
\nc{\hn}{{\widehat{\mathfrak{n}}}}
\nc{\chn}{{\widehat{\mathfrak{n}}}{}^\vee}

\nc{\fA}{{\mathfrak{A}}}
\nc{\fB}{{\mathfrak{B}}}
\nc{\fD}{{\mathfrak{D}}}
\nc{\fE}{{\mathfrak{E}}}
\nc{\fF}{{\mathfrak{F}}}
\nc{\fG}{{\mathfrak{G}}}
\nc{\fK}{{\mathfrak{K}}}
\nc{\fL}{{\mathfrak{L}}}
\nc{\fM}{{\mathfrak{M}}}
\nc{\fN}{{\mathfrak{N}}}
\nc{\fP}{{\mathfrak{P}}}
\nc{\fU}{{\mathfrak{U}}}
\nc{\fV}{{\mathfrak{V}}}
\nc{\fZ}{{\mathfrak{Z}}}

\nc{\bb}{{\mathbf{b}}}
\nc{\bc}{{\mathbf{c}}}
\nc{\bd}{{\mathbf{d}}}
\nc{\bbf}{{\mathbf{f}}}
\nc{\be}{{\mathbf{e}}}
\nc{\bg}{{\mathbf{g}}}
\nc{\bi}{{\mathbf{i}}}
\nc{\bj}{{\mathbf{j}}}
\nc{\bn}{{\mathbf{n}}}
\nc{\bo}{{\mathbf{o}}}
\nc{\bp}{{\mathbf{p}}}
\nc{\bq}{{\mathbf{q}}}
\nc{\bt}{{\mathbf{t}}}
\nc{\bu}{{\mathbf{u}}}
\nc{\bv}{{\mathbf{v}}}
\nc{\bx}{{\mathbf{x}}}
\nc{\bs}{{\mathbf{s}}}
\nc{\by}{{\mathbf{y}}}
\nc{\bw}{{\mathbf{w}}}
\nc{\bA}{{\mathbf{A}}}
\nc{\bK}{{\mathbf{K}}}
\nc{\bB}{{\mathbf{B}}}
\nc{\bC}{{\mathbf{C}}}
\nc{\bG}{{\mathbf{G}}}
\nc{\bD}{{\mathbf{D}}}
\nc{\bH}{{\mathbf{H}}}
\nc{\bM}{{\mathbf{M}}}
\nc{\bN}{{\mathbf{N}}}
\nc{\bO}{{\mathbf{O}}}
\nc{\bT}{{\mathbf{T}}}
\nc{\bV}{{\mathbf{V}}}
\nc{\bW}{{\mathbf{W}}}
\nc{\bX}{{\mathbf{X}}}
\nc{\bZ}{{\mathbf{Z}}}
\nc{\bS}{{\mathbf{S}}}

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\nc{\sB}{{\mathsf{B}}}
\nc{\sC}{{\mathsf{C}}}
\nc{\sD}{{\mathsf{D}}}
\nc{\sF}{{\mathsf{F}}}
\nc{\sG}{{\mathsf{G}}}
\nc{\sK}{{\mathsf{K}}}
\nc{\sM}{{\mathsf{M}}}
\nc{\sO}{{\mathsf{O}}}
\nc{\sW}{{\mathsf{W}}}
\nc{\sQ}{{\mathsf{Q}}}
\nc{\sP}{{\mathsf{P}}}
\nc{\sV}{{\mathsf{V}}}
\nc{\sS}{{\mathsf{S}}}
\nc{\sT}{{\mathsf{T}}}
\nc{\sZ}{{\mathsf{Z}}}
\nc{\sfp}{{\mathsf{p}}}
\nc{\sll}{{\mathsf{l}}}
\nc{\sr}{{\mathsf{r}}}
\nc{\bk}{{\mathsf{k}}}
\nc{\sg}{{\mathsf{g}}}

\nc{\sff}{{\mathsf{f}}}
\nc{\sfb}{{\mathsf{b}}}
\nc{\sfc}{{\mathsf{c}}}
\nc{\sd}{{\mathsf{d}}}
\nc{\se}{{\mathsf{e}}}

\nc{\BK}{{\bar{K}}}

\nc{\tA}{{\widetilde{\mathbf{A}}}}
\nc{\tB}{{\widetilde{\mathcal{B}}}}
\nc{\tg}{{\widetilde{\mathfrak{g}}}}
\nc{\tG}{{\widetilde{G}}}
\nc{\TM}{{\widetilde{\mathbb{M}}}{}}
\nc{\tO}{{\widetilde{\mathsf{O}}}{}}
\nc{\tU}{{\widetilde{\mathfrak{U}}}{}}
\nc{\TZ}{{\tilde{Z}}}
\nc{\tx}{{\tilde{x}}}
\nc{\tbv}{{\tilde{\bv}}}
\nc{\tfP}{{\widetilde{\mathfrak{P}}}{}}
\nc{\tz}{{\tilde{\zeta}}}
\nc{\tmu}{{\tilde{\mu}}}

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\nc{\uC}{{\underline{\mathbb{C}}}}
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\nc{\uj}{\underline{j}}
\nc{\ofP}{{\overline{\mathfrak{P}}}}
\nc{\oB}{{\overline{\mathcal{B}}}}
\nc{\og}{{\overline{\mathfrak{g}}}}
\nc{\oI}{{\overline{I}}}

\nc{\eps}{\varepsilon}
\nc{\hrho}{{\hat{\rho}}}


\nc{\one}{{\mathbf{1}}}
\nc{\two}{{\mathbf{t}}}

\nc{\Rep}{{\mathop{\operatorname{\rm Rep}}}}
%\nc{\Sym}{{\mathop{\operatorname{\rm Sym}}}}
\nc{\Tot}{{\mathop{\operatorname{\rm Tot}}}}
%\nc{\Spec}{{\mathop{\operatorname{\rm Spec}}}}
\nc{\Ker}{{\mathop{\operatorname{\rm Ker}}}}
\nc{\Hilb}{{\mathop{\operatorname{\rm Hilb}}}}
%\nc{\Mod}{{{\mathcal M}od}}
\nc{\End}{{\mathop{\operatorname{\rm End}}}}
\nc{\Ext}{{\mathop{\operatorname{\rm Ext}}}}
%\nc{\Hom}{{\mathop{\operatorname{\rm Hom}}}}
\nc{\CHom}{{\mathop{\operatorname{{\mathcal{H}}\it om}}}}
\nc{\GL}{{\mathop{\operatorname{\rm GL}}}}
\nc{\gr}{{\mathop{\operatorname{\rm gr}}}}
\nc{\Id}{{\mathop{\operatorname{\rm Id}}}}
%\nc{\rk}{{\mathop{\operatorname{\rm r}}}}
\nc{\de}{{\mathop{\operatorname{\rm def}}}}
\nc{\length}{{\mathop{\operatorname{\rm length}}}}
\nc{\supp}{{\mathop{\operatorname{\rm supp}}}}

%\nc{\Bun}{{\mathsf{Bun}}}
\nc{\Cliff}{{\mathsf{Cliff}}}
%\nc{\Gr_{X^{\fset}}}{{\mathsf{Gr}}}
\nc{\Fl}{\on{Fl}}
\nc{\Fib}{{\mathsf{Fib}}}
\nc{\Coh}{{\on{Coh}}}
\nc{\QCoh}{{\on{QCoh}}}
\nc{\IndCoh}{{\on{IndCoh}}}
\nc{\FCoh}{{\mathsf{FCoh}}}

\nc{\reg}{{\text{\rm reg}}}

\nc{\cplus}{{\mathbf{C}_+}}
\nc{\cminus}{{\mathbf{C}_-}}
\nc{\cthree}{{\mathbf{C}_*}}
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\nc\Eisr{\on{Eis}^{rat}{}}
\nc\wh{\widehat}
\nc{\Def}{\on{Def_{\check{\fb}}(E)}}
\nc{\barZ}{\overline{Z}{}}
\nc{\barbarZ}{\overline{\barZ}{}}
\nc{\barpi}{\overline\pi}
\nc{\barbarpi}{\overline\barpi}
\nc{\barpip}{\overline\pi{}^+}
\nc{\barpim}{\overline\pi{}^-}


\nc{\fq}{\mathfrak q}

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\nc{\hattimes}{\wh\otimes}



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\nc{\seq}[1]{\stackrel{#1}{\sim}}



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\nc{\cG}{{\check{G}}}
\nc{\cM}{{\check{M}}}
\nc{\cB}{{\check{B}}}
\nc{\cP}{{\check{P}}}

\nc{\ct}{{\check{\mathfrak t}}}
\nc{\cg}{{\check{\fg}}}
\nc{\cb}{{\check{\fb}}}
\nc{\cn}{{\check{\fn}}}
\nc{\cp}{{\check{\fp}}}
\nc{\cm}{{\check{\fm}}}

%\nc{\cLambda}{{\check\Lambda}}

%\nc{\cla}{{\check\lambda}}
\nc{\cmu}{{\check\mu}}
\nc{\cnu}{{\check\nu}}
\nc{\ceta}{{\check\lambda}}



\nc{\imathb}{{\ol{\imath}}}
\nc{\rlr}{\overset{\longrightarrow}{\underset{\longrightarrow}\longleftarrow}}


\nc{\LocSys}{\on{LocSys}}

\nc{\Vect}{\ms{Vect}}
\nc{\Whit}{\ms{Whit}}
%\nc{\bT}{\on{CT}}
\nc{\bTb}{\ol{\on{CT}}}
\nc{\Ran}{\on{Ran}}
\nc{\bTr}{\on{CT}^{rat}{}}
\nc\jmathr{\jmath^{rat}{}}
\nc{\ux}{\underline{x}}
%\nc{\clambda}{{\check\lambda}}
\nc{\calpha}{{\check\alpha}}
\nc{\ind}{{\mathsf{ind}}}
\nc{\oblv}{{\mathsf{oblv}}}
\nc{\StinftyCat}{\on{DGCat}}
\nc{\inftygroup}{\infty\on{-Grpd}}

\nc{\fset}{\on{fSet}}
\nc{\LocSysG}{\LocSys_{\cG}}
\nc{\Sing}{\on{Sing}}
\nc{\dr}{{\on{dR}}}
\nc{\Dmod}{\on{D-mod}}
\nc{\Ind}{\on{Ind}}
\nc{\Sat}{\on{Sat}}
\nc{\Ho}{\on{Ho}}
\nc{\Res}{\on{Res}}
\nc{\sotimes}{\overset{!}\otimes}
\nc{\mmod}{{\on{-}}{\mathbf{mod}}}
\nc{\Maps}{\on{Maps}}
\nc{\CMaps}{{\mathcal Maps}}
\nc{\bMaps}{{\mathbf{Maps}}}



\nc{\dgSch}{\on{DGSch}}
\nc{\dgindSch}{\on{DGindSch}}
\nc{\indSch}{\ms{indSch}}
\nc{\IndSch}{\ms{IndSch}}
\nc{\Sch}{\mathsf{Sch}}
\nc{\affdgSch}{\on{DGSch}^{\on{aff}}}
\nc{\affSch}{\ms{Sch}^{\ms{aff}}}

%\nc{\Groupoids}{\on{Grpd}}
\nc{\inftypic}{\infty\on{-PicGrpd}}
\nc{\inftyCat}{\infty\ms{-Cat}}
%\nc{\StinftyCat}{\on{DGCat}}
\nc{\MoninftyCat}{\infty\on{-Cat}^{Mon}}
\nc{\SymMoninftyCat}{\infty\on{-Cat}^{\on{SymMon}}}
\nc{\SymMonStinftyCat}{\on{DGCat}^{\on{SymMon}}}
\nc{\MonStinftyCat}{\on{DGCat}^{Mon}}




\nc{\csupp}{\supp}
\nc{\Arth}{\on{Arth}}
\nc{\ArthG}{{\on{Arth}_\cG}}
\nc{\ul}{\underline}


\nc{\m}{\mu}

\nc{\xto}{\xrightarrow}

\renewcommand{\dim}{\mathsf{dim}}
\newcommand{\codim}{\mathsf{codim}}
\nc{\corr}{\mathsf{corr}}
\nc{\ft}{\mathsf{ft}}
\nc{\fty}{\mathsf{ft}}

\nc{\cores}{\mathsf{cores}}
\nc{\coind}{\mathsf{coind}}

\nc{\actotimes}{\overset{*!}\otimes}

\nc{\hto}{\hookrightarrow}
\nc{\bP}{\mathbf{P}}

%%%% change notations^^^ above???


\nc{\VV}{\mathbb V}

\nc{\rrep}{\on{-} \mathbf{rep}}


\nc{\ccomod}{\on - \mathbf{comod}}

\nc{\citep}{\cite}

\nc{\Tate}{\mathsf{Tate}}
\nc{\rec}{\mathsf{rec}}
\newcommand{\HC}{\mathsf{HC}}

\renewcommand{\exp}{\mathit{exp}}
\newcommand{\pro}{\mathsf{pro}}

\nc{\flip}{\text{<}}
%
\nc{\IFT}{\mathsf{IFT}}

\nc{\biota}{\boldsymbol{\iota}}

\nc{\LC}{\mathsf{LC}} %% iso from limit to colim

\nc{\ins}{\mathsf{ins}}
\nc{\ev}{\mathsf{ev}}

\nc{\inv}{\mathsf{inv}}

\nc{\rel}{\mathit{rel}}
\newcommand{\R}{\mathcal{R}}

\nc{\ren}{\mathit{ren}}
\nc{\diam}{\diamond}

\nc{\Heis}{\mathsf{Heis}}
\nc{\Corr}{\mathsf{Corr}}
\nc{\Coalg}{\ms{Coalg}}
\nc{\Alg}{\ms{Alg}}
%\nc{\ind}{\mathbf{ind}}
\nc{\footcite}{\footnote}
\nc{\fch}{\check{f}}
\nc{\bL}{\mathbf{L}}
\renewcommand{\'}{``}

\nc{\pr}{\mathsf{pr}}
\nc{\Fun}{\on{Fun}}
\nc{\Cu}{\C_{univ}}

\nc{\coact}{\mathsf{coact}}

\nc{\ppart}{(\!(t)\!)}
\nc{\bphi}{\boldsymbol{\phi}}
\nc{\botimes}{\boldsymbol{\otimes}}

\newcommand{\Dgr}{\Dmod(\Gr_{X^I})}

\nc{\chiral}[1]{\langle #1 \rangle}

\nc{\chiralx}{\chiral{X^I}}
\nc{\Gch}{\check{G}}
\nc{\FT}{\sF\sT}
\nc{\qcoh}{\QCoh}
\nc{\RepGran}{\Rep(G)_{\Ran}}
\nc{\RepGchran}{\Rep(G^\vee)_{\Ran}}

\nc{\rr}{\xymatrix{ \ar@<-0.1ex>[r] \ar@<.8ex>[r]  & } }
\nc{\rrr}{\xymatrix{ \ar@<.2ex>[r] \ar@<.9ex>[r] \ar@<-0.5ex>[r] & } }

\nc{\Z}{\CZ}
\nc{\CCC}{\wt\CC}
\renewcommand{\Q}{\QCoh}
%\newcommand{\R}{\Ran}
\nc{\calN}{\N}
\nc{\calW}{\mathcal{W}}
\nc{\calF}{\mathcal{F}}
\nc{\calH}{\mathcal{H}}
\nc{\calO}{\mathcal{O}}
\nc{\Comod}{\on{Comod}}
\nc{\bGamma}{\mathbf{\Gamma}}
\nc{\coGamma}{\on{co-}\bGamma}

\nc{\Jets}{\on{Jets}}
\nc{\act}{\mathsf{act}}
\nc{\Av}{\mathsf{Av}}
\nc{\Ad}{\on{Ad}}
\nc{\BGRan}{BG_{\Ran}}

\nc{\uscolim}[1]{\underset{#1}{\on{colim}} \, }
\nc{\usotimes}[1]{\underset{#1}{\otimes}}
\nc{\ustimes}[1]{\underset{#1}{\times}}
\nc{\colim}{\on{colim}}

\nc{\cpt}{{\on{cpt}}}
\nc{\dR}{{\on{dR}}}
\nc{\Div}{{\on{Div}}}
\nc{\DGCat}{\ms{DGCat}}
\nc{\DGCatcont}{\on{DGCat}_{cont}}
\nc{\glob}{{\on{glob}}}
\nc{\GS}{{\on{GS}}}
\nc{\heart}{{\heartsuit}}
\nc{\loc}{{\on{loc}}}

\nc{\Nout}{\bN_{out}}

\nc{\ran}{\on{Ran}}
\renewcommand{\op}{{\on{op}}}
\nc{\pt}{{\ms{pt}}}
\nc{\PreStk}{{\ms{PreStk}}}
\nc{\Cat}{{\on{Cat}}}
\nc{\DGSchaff}{{\on{DGSch}^{\text{aff}}}}
\nc{\ShvCat}{{\mathsf{ShvCat}}}
\nc{\Sect}{{\on{Sect}}}
\nc{\Stab}{{\on{Stab}}}
\nc{\tstr}{{\on{t-str}}}

\renewcommand{\Dmod}{\mathfrak{D}}

\nc{\Whitglob}{\on{Whit}_{\on{glob}}}
%\nc{\Dmod}{\D\mod}


\nc{\restr}[2]{\left. #1 \right |_{#2}}
\nc{\uprestr}[2]{\left. #1 \right |^{#2}}
\nc{\bLoc}{{\mathbf{Loc}}}
%\nc{\bGamma}{{\mathbf{\Gamma}}}

%% loop groups

\newcommand{\GO}{G({\mathcal O})}
\newcommand{\GK}{G({\mathcal K})}

\newcommand{\NO}{N({\mathcal O})}
\newcommand{\NK}{N({\mathcal K})}

\newcommand{\TO}{T({\mathcal O})}
\newcommand{\TK}{T({\mathcal K})}


\begin{document}

\title{Loop group actions on categories and Whittaker invariants}

\author{Dario Beraldo}

\begin{abstract}

We develop some aspects of the theory of $\fD$-modules on ind-schemes of pro-finite type. These notions are used to define $\fD$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. 

%We also extend the Fourier-Deligne transform to Tate vector spaces. \\
Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $\chi: N\ppart \to \GG_a$ and a category $\C$ acted upon by $N \ppart$, we define the category $\C^{N \ppart, \chi}$ of $(N \ppart, \chi)$-invariant objects, along with the coinvariant category $\C_{N \ppart, \chi}$. These are the Whittaker categories of $\C$, which are in general not equivalent. However, there is always a family of functors $\Theta_k: \C_{N \ppart, \chi} \to \C^{N \ppart, \chi}$, parametrized by $k \in \ZZ$.

We conjecture that each $\Theta_k$ is an equivalence, provided that the $N \ppart$-action on $\C$ extends to a $G \ppart$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained by taking invariants of $\C$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G \ppart$.
%
\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}

Categorical (or higher) representation theory is the study of symmetries of categories. In mathematical terms, such symmetries are encoded by the notion of \emph{group action} on a category. 

\smallskip

Our geometric context is the world of complex algebraic geometry, so that by ``group" we mean ``group prestack over $\CC$". As for the categorical context, we shall consider stable presentable $\CC$-linear $\infty$-categories containing all colimits. In other words, our categories are co-complete differential graded (DG) categories and functors among them are required to be continuous (i.e. colimit preserving). The foundational basis of these notions is contained in the books \cite{L0}, \cite{L1}. For succinct reviews, we recommend \cite{DG} and \cite{BFN}.

\ssec{Some higher algebra}

The above set-up, i.e. the $\infty$-category $\DGCat$ of co-complete DG categories and continuous functors, is extremely convenient for performing algebraic operations on categories, directly generalizing standard operations of classical algebra. For instance,

\begin{enumerate}
\item
one can form colimits and limits of categories.

\smallskip

\item
there is a tensor product that makes $\DGCat$ into a symmetric monoidal $\infty$-category. It commutes with colimits separately in each variable and the monoidal unit is $\Vect$, the category of (complexes of) $\mathbb C$-vector spaces.

\smallskip

\item
the monoidal structure is closed, i.e. there exists an ``internal hom functor", $\Hom(\C, \D)$: the category of continuous functors from $\C$ to $\D$.

\smallskip

\item
there is a notion of dualizable category; if $\C$ is dualizable,  its dual $\C^\vee$ is equivalent to $\Hom(\C, \Vect)$.


\smallskip

\item
monoidal categories are precisely algebra objects in the symmetric monoidal $\infty$-category $\DGCat$. They admit $\infty$-categories of modules: if $\A$ is a monoidal category, we denote by $\A \mmod$ the $\infty$-category of its (left) modules. Roughly speaking, this consists of categories $\M$ equipped with a functor $\A \otimes \M \to \M$ satisfying the natural compatibility conditions. Right $\A$-modules are defined similarly.

\smallskip

\item
combining (3) and (5), given two $\A$-modules $\M$ and $\N$, one can form the category of $\A$-linear functors $\Hom_\A(\M , \N)$.

\smallskip

\item
there is a well-defined notion of \emph{relative} tensor product: one can tensor a right and a left $\A$-module \emph{over} $\A$; the output is a plain DG category.

\end{enumerate}


\sssec{}

Coming back to higher representation theory, we would like to add one more item to the above list: given an affine group $G$ of finite type, we wish to make sense of the notion of ``category equipped with a $G$-action".
To do so, let us mimic the classical setting: the structure of a $G$-representation on a vector space $V$ consists of a coaction on $V$ of the coalgebra $\Gamma(G, \O_G)$ of functions on $G$ with convolution coproduct (i.e., pull-back along the multiplication).

\medskip

In our context, one has to replace functions of the group with sheaves: there are thus at least two possible notions of group action, accounting for the two standard sheaf-theoretic contexts of \emph{quasi-coherent sheaves} and \emph{$\fD$-modules} on $G$, both equipped with convolution.

\medskip

By definition, categories with a \emph{weak} action are comodules for $(\QCoh(G), m^*)$, whereas categories with a \emph{strong} action (or infinitesimally trivialized action, or \emph{de Rham} action) are comodules for $(\Dmod(G), m^!)$. 
We occasionally denote them by $G \on{-}\mathbf{rep}^{weak}$ and $G \on{-}\mathbf{rep}$, respectively.
Here $m^*$ (resp., $m^!$) is the quasi-coherent (resp., $\fD$-module) pull-back along the multiplication $m: G \times G \to G$.

\medskip

For $G$ of finite type, both $\qcoh(G)$ and $\Dmod(G)$ are canonically self-dual; under this duality, the coproducts specified above get sent to the convolution products in the context of quasi-coherent sheaves or $\fD$-modules, respectively. As a consequence,
$$
G \on{-}\mathbf{rep}^{weak} \simeq (\qcoh(G), \star) \mmod \hspace{.8 cm} G \on{-}\mathbf{rep} \simeq (\Dmod(G), \star) \mmod.
$$
%
This follows from a general result of category theory: 

\begin{lem} \label{lem:mod-comod}
Let $\S^\otimes$ be a symmetric monoidal $\infty$-category, and $A \in \S$ an algebra object, with dual $A^\vee$ (automatically a coalgebra object). Then $A \mod (\S) \simeq A^\vee \on{-comod}(\S)$.
\end{lem}
%

%
%
%
The main source of examples of categorical $G$-representations comes from geometry: if $X$ is a scheme of finite type with a $G$-action, then $\Dmod(X)$ is a module category for $(\Dmod(G), \star)$, via push-forward along the action map $G \times X \to X$. Likewise, $\QCoh(X)$ carries a weak $G$-action.

\sssec{}

Several standard operations with ordinary $G$-representations generalize immediately to this categorical context. 

\medskip

First, $\Rep(G)$ is a symmetric monoidal category: this follows from the fact that $\Gamma(G, \O_G)$ is a commutative Hopf algebra. The same holds true for $G \on{-}\mathbf{rep}^{weak}$ and $G \on{-}\mathbf{rep}$, since $(\QCoh(G), \Delta^*, m^*)$ and $(\Dmod(G), \Delta^!, m^!)$ are \emph{Hopf monoidal} categories with commutative multiplication.

\smallskip

Secondly, given an ordinary $G$-representation $V$, we can take its invariants $V^G := \Hom_G(\mathbb C, V)$ and coinvariants $V_G = \CC \otimes_G V$. 
%
In our categorical framework, we have the notions of weak invariants and coinvariants: for $\C \in G \on{-}\mathbf{rep}^{weak}$,
$$
\C^{G,w} := \Hom_{\QCoh(G)}(\Vect, \C), \, \, \, \, \C_{G,w} := \Vect \usotimes{\QCoh(G)}  \C;
$$
as well as strong invariant and coinvariants: for $\C \in G\on{-}\mathbf{rep}$,
$$
\C^{G} := \Hom_{\Dmod(G)}(\Vect, \C), \, \, \, \, \C_{G} := \Vect \usotimes{\Dmod(G)}  \C.
$$

\begin{rem}

The difference between weak and strong invariants is controlled by the monoidal category of \emph{Harish-Chandra} bimodules, the category of endomorphisms of $\fg \mod$ as a strong $G$-representation (the action of $\Dmod(G)$ on $\fg \mod$ is induced by the adjoint action of $G$ on $\fg$):
$$
\HC:= \Hom_{\Dmod(G)}(\fg \mod, \fg \mod).
$$
Precisely, if $\C \in G \on- \mathbf{rep}$, then $\C^{G,w}$ carries an action of $\HC$ and the invariant category $\C^G$ is the category of $\HC$-linear functors from $\Rep(G)$ to $\C^{G,w}$.

\end{rem}

If $X$ is a $G$-scheme of finite type, then $\Dmod(X)^G \simeq \Dmod(X/G)$, the category of $\fD$-modules on the quotient stack $X/G$. This follows immediately from fppf descent for $\fD$-modules. It turns out that $\Dmod(X)_G \simeq \Dmod(X/G)$ as well. (The very same statements hold for $\QCoh$ in place of $\fD$.)


\ssec{Sheaves of categories}

It is worthwhile to introduce one more generalization of a classical algebro-geometric concept to the categorical world, that of quasi-coherent sheaf. 

\smallskip

First of all, recall that quasi-coherent sheaves are defined on any prestack. A prestack is the most general ``space" of algebraic geometry, that is, an arbitrary contravariant functor from affine DG schemes to $\infty$-groupoids. Given a prestack $\Y$, we have
$$
\QCoh(\Y) := \lim_{S \in (\affdgSch_{/\Y})^\op} \QCoh(S).
$$
Informally, a quasi-coherent sheaf on $\Y$ is a collection of a $\Gamma(S, \O_S)$-module $M_S$, for any affine DG scheme $S \to \Y$, together with compatibilities along restrictions $S \to S' \to \Y$.

\medskip

In striking analogy with the above, we introduce the notion of \emph{quasi-coherent sheaf of categories} over a prestack $\Y$:\footcite{We will omit the adjective ``quasi-coherent" for brevity.}
$$
\ShvCat(\Y) := \lim_{S \in (\affdgSch_{/\Y})^\op} \QCoh(S) \mmod.
$$
Informally, a sheaf of categories on $\Y$ is a collection, for any affine DG scheme $S \to \Y$, of a $\qcoh(S)$-module category $\M_S$, together with compatibilities along restrictions $S \to S' \to \Y$.

\medskip

We say that a prestack $\Y$ is \emph{$1$-affine} if the functor of global sections 
$$
\bGamma: \ShvCat(\Y) \to \QCoh(\Y) \mmod
$$
is an equivalence: sheaves of categories over a $1$-affine $\Y$ can be reconstructed from their global sections.


\sssec{}

Ordinary $G$-representations are quasi-coherent sheaves over $BG =\pt/G$, the stack classifying $G$-bundles.
Likewise, categorical $G$-representations ought to be sheaves of categories over $BG$: in other words, a weak action of $G$ on $\C$ is the datum of a sheaf of categories $\wt\C$ over $BG$, whose restriction along $\pt \to BG$ is the original $\C$.

If $G$ is an affine group, the two definitions coincide, namely $\ShvCat(BG) \simeq (\QCoh(G), \star) \mmod$.
Furthermore, D. Gaitsgory shows that $BG$ is $1$-affine (\cite{shvcat}), so that
$$
\ShvCat(BG)\simeq \Rep(G) \mmod.
$$
%
In the same optic, a strong $G$-action on $\C$ ought to be given by a sheaf of categories over $BG_\dR$ that restrict to $\C$ along the inclusion $\pt \to BG$. As above, for groups of finite type, this matches the definition given above in terms of the group algebra or coalgebra (\textit{loc. cit.}):
$$
\ShvCat(BG_\dR )\simeq (\Dmod(G), \star) \mmod \simeq (\Dmod(G), m^!) \ccomod.
$$
However, one easily shows that $BG_\dR$ is not $1$-affine.

\medskip

More generally, the notion of crystal of categories over a quotient stack $X/G$ can also be expressed in terms of the group coalgebra (Proposition \ref{prop:ShvCat(X/G)}):
$$
\ShvCat( (X/G)_\dR) \simeq \Dmod(X) \mmod (\Dmod(G) \ccomod). 
$$


\ssec{$\fD$-modules on loop groups}

In this paper, we adopt the group-algebra definition, as we are interested in strong actions by the loop group $G \ppart$ of a reductive group. This is an ind-scheme of infinite type; for such prestacks, it is not clear at the time of writing what the appropriate notion of de Rham functor is. The definition of the DG category $\Dmod( G \ppart)$ has not been fully discussed either. The two issues are related, as the interpretation of $\fD$-modules as crystals prescribes that $\fD(\Y) := \qcoh(\Y_\dR)$ for any prestack $\Y$.

\medskip

More precisely, the de Rham functor is defined on any prestack (\cite{Crys}), but it may need to be adapted to capture the \textit{correct} notion of $\fD$-modules on loop groups.
In fact, $\fD$-modules on $G \ppart$ can be defined in an ad hoc way \emph{directly} from the finite-type case.

\medskip

A detailed construction will be given in the main body of the text, where we discuss the theory of $\fD$-modules on ind-schemes of pro-finite type. Nevertheless, let us briefly hint at it here: $\bG:=G \ppart$ comes with the canonical ``decreasing" sequence of congruence subgroups $\bG^r$ shrinking to the identity element. For each $r$, let $\pi: \bG /\bG^{r+1} \to \bG/\bG^r$ be the (smooth) projection. Since $\bG/\bG^r$ are ind-schemes on ind-finite type, $\fD$-modules on them make sense according to the theory developed in the published literature. Then, one puts
$$
\Dmod^*(\bG) :=\lim_{r,\pi_*}\, \Dmod(\bG /\bG^r) \simeq \uscolim{r,\pi^*} \,\Dmod(\bG /\bG^r).
$$
%
For the latter expression, one sees that $\Dmod^*(\bG)$ is dualizable (even compactly generated) with dual $\Dmod^!(\bG) \simeq \colim_{r,\pi^!}\, \Dmod(\bG /\bG^r)$. Any trivialization of the dimension torsor of $\bG$ leads to an equivalence between $\Dmod^!(\bG)$ and $\Dmod^*(\bG)$.

\begin{remark}
One of the reasons why we privilege strong actions comes from our interest in the local Langlands correspondence (reviewed later), which explicitly involves strong actions of loop groups on categories. Hereafter, unless specified otherwise, by the term ``action" we shall mean ``strong action".
\end{remark}

\ssec{Fourier transform}

In our effort to generalize classical notions of algebra and representation theory to the categorical world, let us revisit the classical Fourier transform.

An ordinary representation of a vector group $A$ can be decomposed according to characters of $A$, which are by definition linear maps $\chi: A \to \CC$, that is, elements of $A^\vee$. More fundamentally, once the appropriate function-theoretic context is assumed, Fourier transform gives an equivalence
$$
\FT: (\Fun(A), \star) \xrightarrow{\, \, \, \simeq \, \, \, } (\Fun(A^\vee), \cdot)
$$
between the algebra of functions on $A$ with convolution and the algebra of functions of $A^\vee$ with pointwise multiplication. The spectral decomposition of an $A$-representation cited above follows by looking at categories of modules for both algebras:
$$
A\on{-rep} := (\Fun(A), \star)\mod \xrightarrow{\, \, \, \simeq \, \, \, } (\Fun(A^\vee), \cdot)\mod.
$$
%
Moreover, if $V$ is an $A$-representation, the vector space of its invariant $V^A$ is equivalent to the fiber of $V$ at $0 \in A^\vee$, where $V$ is now, via Fourier transform, a sheaf over $A^\vee$. 

The fiber of $V$ over a nonzero character $\chi \in A^\vee$ corresponds to $V^{A,\chi}$: the vector space of $A$-invariants \emph{against} the character $\chi$. Explicitly, if plain $A$-invariants are those elements $v \in V$ for which $a \cdot v = v$, then $(A, \chi)$-invariants are those $v \in V$ for which 
$$
a \cdot v = e^{\chi(a)} v.
$$

\sssec{}

A categorification of the above situation, in case $A = \AA^n$ is the $n$-dimensional affine space, is provided by the Deligne-Fourier transform, \cite{Laumon}: an equivalence of monoidal categories
\begin{equation} \label{eqn:FD}
\FT: (\Dmod(A), \star) \xrightarrow{\, \, \, \, \simeq \, \, \, \, } (\Dmod(A^\vee), \stackrel ! \otimes),
\end{equation}
%
which yields
\begin{equation} \label{eqn:FD-mod}
A\on{-\mathbf{rep}}= (\Dmod(A), \star)\mmod \xrightarrow{\, \, \, \, \simeq \, \, \, \, } (\Dmod(A^\vee), \stackrel ! \otimes) \mmod.
\end{equation}
%
The kernel of this transform is no longer a function on $A \times A^\vee$, but a sheaf: precisely, the pull-back of $\exp \in \Dmod(\GG_a)$ via the duality pairing $Q: A \times A^\vee \to \GG_a$. The substitute of ``exponential function" $\exp$ is the $\fD$-module encoded by the defining differential equation for $\exp$. Its homomorphism property corresponds to the isomorphism $m^! (\exp) \simeq \exp \boxtimes \exp$.



\begin{rem}
In the main body of the paper, we extend (\ref{eqn:FD}) and thus (\ref{eqn:FD-mod}) to $A = \AA^n \ppart$ or, slightly more generally, to a Tate vector space.
\end{rem}

Invariants and coinvariants of $\C \in A \on{-\mathbf{rep}}$, correspond to \emph{fiber} and \emph{cofiber} of $\C$ over $0 \hookrightarrow A^\vee$, respectively.
The fiber at non-zero $\chi \in A^\vee$ is identified with the category of $(A, \chi)$-invariants of $\C$: objects $c \in \C$ for which the \emph{coaction} (dual to the action) is isomorphic to 
$$
\coact (c) \simeq \chi^! (\exp) \otimes c \, \in \Dmod(A) \otimes \C.
$$
%
By definition, the fiber (resp., cofiber) of $\C$ at $\chi$ is the category 
$$
\uprestr{\C}0 := \Hom_{\Dmod(A^\vee)}(\Dmod(\{\chi\}), \C)
\hspace{.6cm}
\big(\text{resp., }\, \restr{\C}0 := \Dmod(\{\chi\}) \usotimes{\Dmod(A^\vee)} \C \big).
$$
More generally, one can define \emph{restriction} and \emph{corestriction} of $\C$ along any map $B \to A^\vee$.



\ssec{Local geometric Langlands duality}


The importance of categorical representation theory in mathematics became first evident in the framework of the \emph{local geometric Langlands correspondence}, a conjecture put forward by E. Frenkel and D. Gaitsgory in the course of several papers (primarily, \cite{FG2}, \cite{FG3}, \cite{FG6}). See also the book \cite{F} for a review.

\medskip

The local Langlands conjecture predicts the existence of an equivalence
%
$$
\Dmod (G \ppart) \mmod \simeq \ShvCat( \LocSys_{\Gch} (\D^\times)),
$$
%
where $\LocSys_{\Gch} (\D^\times)$ is the stack of local systems on the punctured disk $\D^\times = \Spec(\CC \ppart)$, for the \emph{Langlands dual group} $\Gch$.

\medskip

This is, philosophically, a kind of Fourier transform: assuming that $\LocSys_{\Gch} (\D^\times)$ is $1$-affine, so that 
$$
\ShvCat( \LocSys_{\Gch} (\D^\times)) \simeq \QCoh(\LocSys_{\Gch} (\D^\times)) \mmod,
$$

\smallskip

\noindent
categorical representations of $G \ppart$ would admit a spectral decomposition over the stack $\LocSys_{\Gch} (\D^\times)$.

\medskip

According to \cite{FG2}, such a correspondence ought to be implemented by a universal kernel $\Cu$: a category lying over $\LocSys_{\Gch} (\D^\times)$ and acted on by $G \ppart$ fiberwise. In \textit{loc. cit.}, the authors did not identify $\Cu$, but described its ``base-change" along various maps $\Y \to \LocSys_{\Gch}(\D^\times)$.
For instance, one of the main results of \cite{FG2} suggests that $\wh\fg_{\on{crit}} \mod$ is the base-change of $\Cu$ along the forgetful map from \emph{opers} $\on{Op}_{\Gch}$ on the punctured disk to $\LocSys_{\Gch}(\D^\times)$:
$$
\Cu \underset{\QCoh(\LocSys_{\Gch}(\D^\times))}\otimes \QCoh(\on{Op}_{\Gch}) \simeq \wh\fg_{\on{crit}} \mod.
$$
In \cite{FG3}, the category of $G[[t]]$-invariant objects of $\Cu$ is argued to be equivalent to $\Rep(\Gch)$. This actually gives a clue as to what $\Cu$ is: we will identify a conjectural candidate below, after introducing the procedure $\C \rightsquigarrow \C^{N \ppart, \chi}$ of \emph{Whittaker invariants}.


\ssec{Whittaker actions}

Let $G= GL_2$ and $\C$ be a category with an action of $G \ppart$. Since $N = \GG_a$ is abelian, $N \ppart$-invariants of $\C$ can be studied via Fourier transform, as explained above. 
%
The dual of $\GG_a \ppart$ is itself, the duality being the residue pairing. In particular, the canonical element $1 \in \GG_a\ppart$ yields the residue character 
$$
\Res: \GG_a \ppart \to \GG_a, \, \, \, \, f(t) = \sum f_n t^n \mapsto f_{-1}.
$$
The \emph{Whittaker invariant} category of $\C$ is, by definition $\Whit(\C):=\uprestr \C 1$ the fiber of $\C$ at $1 \in \GG_a \ppart^\vee$. Alternatively,
$$
\Whit(\C) := \C^{\,\GG_a \ppart, \Res}.
$$ 

\sssec{}

It is the latter expression that admits a generalization to any reductive group $G$. Indeed, if $G$ has rank greater than one, the maximal unipotent subgroup $N \subset G$ is no longer abelian, hence Fourier transform is not available. 
%
However, $N \ppart$  still admits a non-degenerate character $\chi$, defined as follows. Denote by $\{\alpha_1, \ldots, \alpha_r \}$ be the simple roots of $G$, thought of as maps $N \to \GG_a$. We let
\begin{equation} \label{eqn:chi}
\chi: N \ppart \to \GG_a \, \, \, \,  n(t) \mapsto \on{Res} \Big( \sum_{j=1}^r \alpha_j(n(t)) \Big).
\end{equation}
%
Such character is non-degenerate (i.e., nonzero on any root space) and of conductor zero (i.e., null on $N [[t]]$).

\medskip

If $\C$ is acted upon by $G \ppart$, we set the Whittaker invariant category of $\C$ to be
$$
\Whit(\C) := \C ^{N \ppart, \chi}.
$$

\sssec{}

After Gaitsgory, we conjecture that $\Cu \simeq \Whit (\Dmod(G \ppart))$. It is immediately clear that the latter category admits an action of $G \ppart$. However, the very fact that the proposed candidate for $\Cu$ lives over $\LocSys_{\Gch} (\D^\times)$ has not been yet estabilished. We will partially address this question in another publication.

\medskip

There is a parallel theory of coinvariants and Whittaker coinvariants. One could propose that $\Cu \simeq \Dmod(G \ppart)_{N \ppart, \chi}$, the coinvariant Whittaker category of $\Dmod(G \ppart)$, leading to a different local geometric Langlands correspondence. After Gaitsgory, we conjecture (and prove for $G = GL_n$) that these categories are equivalent.

More precisely, for any category $\C$ acted on by $N \ppart$, we construct an explicit functor $\Theta: \C_{N \ppart, \chi} \to \C^{N \ppart, \chi}$. (Rather, we construct a sequence $\Theta_k$ of such, indexed by $k \in \ZZ$.)  We repropose the following conjecture:

\begin{conj}
If $\C$ is a category acted on by $G \ppart$, then each functor $\Theta_k: \C_{N \ppart, \chi} \to \C^{N \ppart, \chi}$ is an equivalence.
\end{conj}

The statement is known to be false if $\chi$ is degenerate, e.g. $\chi =0$; in other words, the conjecture is special to the Whittaker categories.

\ssec{Overview of the results}

In this paper we prove a refined version of the above conjecture for $G= GL_n$. Our main theorem reads:

\begin{thm} \label{thm:intro}
Let $P \subseteq GL_n$ be the mirabolic group. For any category $\C$ equipped with a $P \ppart$-action, each functor $\Theta_k: \C_{N \ppart, \chi} \to \C^{N \ppart, \chi}$ is an equivalence.
\end{thm}

To prove this for $\Theta_k$, we make use of an explicit group-scheme $\bH_k \subset P \ppart$. For $n=2$ and $n=3$, $\bH_k$ looks like
$$
\bH_k = \left( \begin{array}{cc}
1+t^k \O & t^{-k}\O  \\
0 & 1 \end{array} \right),
\, \, \,
\bH_k = \left( \begin{array}{ccc}
1+t^k\O & t^{-k}\O & t^{-2k}\O \\
t^{2k}\O & 1+t^k\O & t^{-k}\O \\
0 & 0 & 1 \end{array} \right),
$$
and the generalization to any $n$ is straightforward ($\O$ denotes the ring of formal Taylor series).
The sum of the residues of the entries in the over-diagonal yields a character on $\bH_k$ that we continue to denote $\chi$.

\sssec{}

We shall relate $(\bN, \chi)$-invariants and coinvariants on $\C$ to $(\bH_k, \chi)$-invariants on $\C$. Namely, we first produce natural functors
$$
\C_{\bN, \chi} \longleftarrow \C^{\bH_k, \chi} \longleftarrow \C^{\bN, \chi}.
$$
%
Secondly, we show that both functors are equivalences. The proof of that uses induction on $n$ and Fourier transform: indeed, $N$ is the semi-direct product $N' \ltimes \AA^{n-1}$, where $N'$ refers to the maximal unipotent subgroup of $GL_{n-1}$.
%
Thirdly, the composition of the inverse functors is shown to be our functor $\Theta$, for the chosen value of $k$.

\sssec{}

The strategy to prove the equivalence between $\C^{\bN, \chi}$ and $\C_{\bN, \chi}$ goes as follows. We first prove that $\C^{\bN, \chi} \simeq \C^{\bH_k, \chi}$ (outlined below). Using the same logic, we deduce that $\C_{\bN, \chi} \simeq \C_{\bH_k, \chi}$. Finally, there is a natural equivalence $\C^{H, \mu} \simeq \C_{H, \mu}$ for any pro-algebraic group $H$ (Theorem \ref{thm:theta_mu}, or Theorem \ref{thm:inv=coinv} for pro-unipotent groups). Our functor $\C_{\bN, \chi} \leftarrow \C^{\bH_k, \chi} $ incorporates these equivalences.

\medskip

The proof that $\C^{\bN, \chi} \simeq \C^{\bH_k, \chi}$ is very simple and, modulo some combinatorics, it could be explained here. The argument rests of the following facts, to be estabilished in the text:

\begin{itemize}

\item[(a)] 

If $\C$ is acted on by $K \ltimes H$, then $\C^{K \ltimes H} \simeq (\C^H)^K$, and similarly in the presence of characters.
See Lemma \ref{lem:semidirect-pro}.

\medskip

\item[(b)]
Let $W \subseteq V$ an inclusion of Tate vector spaces, and $\chi \in V^\vee$ a character. Under Fourier transform, $\C ^{W, \chi}$ is equivalent to the restriction $\uprestr \C{\chi + W^\perp}$, where $W^\perp$ is the annihilator of $W$ inside $V^\vee$.
This is Proposition \ref{prop:FT_char_subspace_indpro}.

%%%If $\C$ is acted on by $\GG_a \ppart$, then, under Fourier transform, $\C^{t^{-k} \O, \Res} \simeq \restr \C {1+ t^k \O}$.
\medskip

\item[(c)]

If $\C$ fibers over $X/K$ and $K$ acts on $X$ transitively, then $S=\Stab(x \in X)$ continues to act on $\uprestr \C x$ and $\C^K \simeq (\uprestr \C x)^S$. See Proposition \ref{prop:sigma_equiv}.
\end{itemize}

\sssec{}

Finally, let us sketch the proof of $\C^{\bN, \chi} \simeq \C^{\bH_k, \psi}$ for $G= GL_2$.
By (a),
$$
\C^{\bH_k, \chi} \simeq  \Big( \C^{t^{-k}\O, \Res} \Big)^{1 + t^k \O}.
$$
Using (b) in the case of the inclusion $t^{-k} \O \subset \GG_a \ppart$, we further obtain
$$
\Big(  \C^{t^{-k}\O, \Res} \Big)^{1+ t^k \O} \simeq \Big( \! \uprestr \C {1+t^k \O} \Big)^{1 + t^k \O}.
$$
Next, fact (c), applied to the regular action of $1+t^k \O$ on itself, implies
$$
\Big( \! \uprestr \C {1+t^k \O} \Big)^{1 + t^k \O} \simeq \Big( \! \uprestr \C {1+t^k \O} \Big) \Big|^{1} \simeq \uprestr \C 1.
$$
By Fourier transform (or using (b) again), we have $\uprestr \C 1 \simeq \C^{\bN, \Res}$.

\begin{rem}
The proof for $G=GL_n$ with $n > 2$ uses the full force of fact (c).
\end{rem}



\ssec{Notation and detailed contents}

%%%%In this paper, $G$ sometimes denotes an affine complex algebraic group of finite type, sometimes a complex reductive group. In the latter case, we let $B$ a chosen Borel subgroup and $N \subseteq B$ its unipotent radical. We indicate by $\bG := G \ppart$ the loop group of $G$ and, for $r \geq 0$, by $\bG^r \subseteq G[[t]]$ the congruence subgroups. Similar notations hold for $B$ and $N$. As noted above, $\bN$ is equipped with an additive character, $\chi$.
%%%%
%%%%\medskip

Let us explain how this paper is organized.


\sssec{} 

In Sect. \ref{SEC:action_fin_dim} we discuss the foundations of weak and strong group actions on categories, in the finite dimensional situation: we define Hopf monoidal categories, invariants and coinvariants, the Harish-Chandra monoidal category and analyze the relation between weak and strong (co)invariants.


\sssec{}

In Sect. \ref{SEC:Dmod-infinite}, we discuss some foundations of the theory of $\fD$-modules on schemes (and ind-schemes) of pro-finite type. The main examples of such are $\bG$, $\bN$ and variations thereof. There are two categories of $\fD$-modules on $\bG$, dual to each other. The first, $\Dmod^*(\bG)$, carries a convolution monoidal structure; its dual $\Dmod^!(\bG)$ is hence comonoidal and also carries a symmetric monoidal structure via the diagonal. It turns out that $\Dmod^*(\bG) \simeq \Dmod^!(\bG)$, but we will never use this fact.

\sssec{}

We proceed in Sect. \ref{SEC:grp-actions} to define loop group actions on categories and the concept of invariants and coinvariants. Since $\bN$ is exhausted by its compact open group sub-schemes, we analyze group actions by pro-unipotent group schemes in great detail. For instance, we define and study natural functors among the original category, the invariant category and the coinvariant category. We show that the latter two are equivalent. 
%A noteworthy feature of out set-up is that any category with a $\bG$-action is \emph{smoothly generated}, that is, generated by objects each of which is invariant for a high congruence subgroup $\bG^r \subset \bG$.

Next, we take up Whittaker actions of $\bN$ on categories: this is a special case of the above theory that accounts for the presence of the character $\chi: \bN \to \GG_a$. For any category acted on by $\bN$, we construct a functor (denoted by $\Theta$, as above) from Whittaker coinvariants to Whittaker invariants. 
%We conjecture that this functor is an equivalence, provided that the $\bN$-action is the restriction of a $\bG$-action.

\sssec{}

We discuss the abelian theory in Sect. \ref{SEC:loop_vectors}. We first review the theory of the Fourier-Deligne transform for finite dimensional vector spaces (in schemes) and then extend it to $\fD$-modules on $\AA^n \ppart$ (more generally, to $\fD$-modules on a Tate vector space). We prove that it still gives a monoidal equivalence. Finally, we re-interpret the concepts of the previous sections (invariants, coinvariants, averaging) in ``Fourier-transformed" terms.

\sssec{}

In Sect. \ref{SEC:trans_act}, we first discuss categories fibering over a $K$-space $X$ and acted on by the group $K$ in a compatible fashion. In this situation, we study how the operations of restriction of $\C$ to $Y \subseteq X$ and taking $K$-invariants interact.
Secondly, we discuss actions of semi-direct groups on categories.

\sssec{}

Finally, in Sect. \ref{SEC:GL_n}, we take up the proof of Theorem \ref{thm:intro}. We discuss some combinatorics of $GL_n$ and define some group schemes of $GL_n \ppart$ that will play a central role. Our proofs are on induction on $n$ and rely heavily on the theory of all previous sections.


\ssec*{Acknowledgements}

I am immensely indebted to Dennis Gaitsgory for proposing the problem and generously teaching me most of the techniques to solve it. Many of the ideas described in this paper come directly from his suggestions.

I wish to express my deepest gratitude to Constantin Teleman for several years of patient explanations at UC Berkeley: it was him who introduced me to higher representation theory. 

It is a pleasure to thank Edward Frenkel for his invaluable help and the impact he had on my thinking about the geometric Langlands program. 

I am obliged to Sam Raskin for several conversations that very strongly influenced the development of this paper.

I also benefited enormously from discussions with Jonathan Barlev, David Ben-Zvi and David Nadler. 


\section{Actions by groups of finite type} \label{SEC:action_fin_dim}

In this section we cover some background on categorical actions of affine groups of finite type. Let $G$ be such a group. We first show that $\Dmod(G)$, as well as $\QCoh(G)$, admits two dual Hopf structures. Next, we analyze the difference between strong and weak invariants: this is controlled by the Harish-Chandra monoidal category $\HC$. Lastly, whenever $G$ is equipped with an additive character $\mu: G \to \GG_a$, we discuss the notion of $\mu$-twisted $G$-actions.

\ssec{Hopf algebras and crossed products} \label{ssec:Hopf}

Given a symmetric monoidal $\infty$-category $\C^\otimes$, the $\infty$-category $\ms{Coalg}(\C^\otimes)$ of its coalgebra objects inherits a symmetric monoidal structure, compatible with the forgetful functor $\Coalg(\C^\otimes) \to \C$ (see \citep{L1}).
A \emph{Hopf algebra} in $\C$ is, by definition, an object in $\Alg(\Coalg(\C))$. The definition is known to be symmetric under the switch $\Alg \leftrightarrow \Coalg$, thus
\begin{equation} \label{eqn:def-Hopf}
\ms{HopfAlg}(\C) := \Alg(\Coalg(\C)) \simeq \Coalg(\Alg(\C)).
\end{equation}
%
Consequently, $\ms{HopfAlg}(\C)^\mathit{dualizable}$ (the full subcategory of $\ms{HopfAlg}(\C)$ spanned by those Hopf algebras that are dualizable as objects of $\C$) is closed under duality.

\begin{rem}
The term ``Hopf" here is abusive, for we are not requiring that an \emph{antipode} be specified. More properly, what has been defined is the notion of \emph{bi-algebra}. However, an obvious antipode will be evident in all the examples considered below.
\end{rem}

For $\C = (\DGCat, \otimes)$, we obtain the concept of \emph{Hopf monoidal} category (or just Hopf category, for short). Roughly speaking, $\H \in \DGCat$ is Hopf if it is both comonoidal and monoidal in a compatible manner.


\sssec{}

Any group object $(G, m)$ in $\C = \on{Set}$ (or $\Sch$, $\IndSch$, $\IndSch^{pro}$ etc.) is a Hopf algebra in $\C$, with multiplication being $m$ and comultiplication being $\Delta: G \to G \times G$. The compatibility between the two sctructures follows at once from commutativity of the diagram
%
%
\begin{gather}
\xy
(40,0)*+{ G \times G, }="00";
(0,0)*+{ G }="10";
%
(40,15)*+{ G \times G \times G \times G  }="01";
(00,15)*+{ G \times G  }="11";
%
{\ar@{<-}_{m_{13} \times m_{24} } "00";"01"};
{\ar@{<-}_{ \Delta  } "00";"10"};
%
{\ar@{<-}_{\Delta \times \Delta \,\, \, } "01";"11"};
{\ar@{<-}_{ m} "10";"11"};
%
%
\endxy
\end{gather} 
%
which shows that $m$ is a morphism of coalgebras (and that $\Delta$ is a morphism of algebras).

\begin{example} \label{example: Dmod(G)-Hopf}

For a group scheme $G$ of finite type, we claim that $\H = \Dmod(G)$ can be naturally endowed with the structure of a Hopf category. Indeed, the contravariant functor $\Dmod: (\Sch^{\fty})^\op \to \DGCat$ is symmetric monoidal. As any such functor maps algebras to coalgebras and viceversa, $\Dmod(G)$ is indeed Hopf, with comultiplication induced by $m^!$ and multiplication by $\Delta^!$. For clarity, we denote this Hopf category as $(\Dmod(G), \Delta^!, m^!)$. 
(The mentioned antipode $\H \to \H$ in this case is pull-back along the inversion automorphism $\sigma : G \to G$.)
\end{example}

\begin{example}
The self-duality $\Dmod(G)^\vee \simeq \Dmod(G)$ transforms $!$-pull-backs into $*$-push-forwards. Hence, $(\Dmod(G), m_*, \Delta_*)$ is also a Hopf category.
\end{example}


\sssec{}

The coalgebra structure on a Hopf category $\H$ allows to form the $\infty$-category $\H\ccomod := \on{Comod}_{\H}(\DGCat)$ of comodules categories for $\H$. The rest of the structure endows $\H \ccomod$ with a monoidal structure compatible with the tensor product of the underlying DG categories: informally, given $\C, \, \E \in \H \ccomod$, their product $\C \otimes \E$ has the following $\H$-comodule structure
$$
\C \otimes \E \xrightarrow{\coact_\C \otimes \coact_\E \, } \C \otimes \E \otimes \H \otimes \H \xrightarrow{ \on{mult_\H} \,} (\C \otimes \E) \otimes \H.
$$
%
Hence, we can consider algebra objects in $\H \ccomod$, that is, monoidal categories with a compatible coaction of $\H$. 

\sssec{}

Let $\H$ be a Hopf category, which is dualizable as a plain category. Then, as pointed out before, $\H^\vee$ (the dual of $\H$ as a plain category) is naturally a Hopf category. 
Given an object $\B \in \on{Alg}(\H^\vee \ccomod)$, we shall form the monoidal category $\H \ltimes \B$, called the \emph{crossed product algebra} of $\H$ and $\B$. Informally, $\H \ltimes \B$ is described as follows: its underlying category is simply $\B \otimes \H$ and the multiplication is given by
$$
\B \otimes \H \otimes (\B \otimes \H)
\xrightarrow{\act_{\H \curvearrowright \B \otimes \H}}
\B \otimes \B \otimes \H
\xrightarrow{ m_\B}
\B \otimes \H.
$$
%
More rigorously, consider the obvious adjuction
$$
\mathsf{free}: \DGCat \rightleftarrows \B \mmod (\H^\vee \ccomod): \mathsf{forget},
$$
which satisfies the hypotheses of the monadic Barr-Beck theorem. Thus, $\B \mmod (\H^\vee \ccomod)$ concides with the category of modules for a monad whose underlying functor is $\C \mapsto \B \otimes (\H^\vee)^\vee \otimes \C$.
The monad structure endows $\B \otimes \H$ with an algebra structure, which is tautologically the one displayed above.


\begin{example}

Given a group $G$ as above, consider the Hopf category $\H = (\Dmod(G), m_*, \Delta_*)$.  Let $X$ an scheme (of finite type) acted upon by $G$. We claim that $\B= \Dmod(X)$, equipped with the point-wise tensor product, belongs to $\on{Alg}(\H^\vee \ccomod)$. In fact, the datum of the action $G \times X \xrightarrow{\, \, \act \, \,} X$ yields the coaction of $\H^\vee$ on $\Dmod(X)$, and the required compatibility arises from the commutative diagram
%
%
%
\begin{gather} \label{diag:GactsX}
\xy
(40,15)*+{ X }="00";
(0,15)*+{ G \times X }="10";
%
(40,0)*+{ X \times X.  }="01";
(00,0)*+{ G \times G \times X \times X  }="11";
%
{\ar@{->}_{\Delta_X } "00";"01"};
{\ar@{<-}_{ \act  } "00";"10"};
%
{\ar@{<-}_{\, \, \,\, \act \times \act } "01";"11"};
{\ar@{->}_{ \Delta_{G} \times \Delta_X} "10";"11"};
%
%
\endxy
\end{gather} 
%
%
Thus, we have a well-defined category $\Dmod(G) \ltimes \Dmod(X)$. This example, or rather its generalization to the ind-pro-setting, will be of importance later.

\end{example}


\sssec{}

A \emph{category over a quotient stack} $X/G$ (for us, always with connection) is by definition an object of $\ShvCat( (X/G)_\dR)$. 
Alternatively, this notion can be expressed via the crossed product $\Dmod(G) \ltimes \Dmod(X)$. Indeed:

\begin{prop} \label{prop:ShvCat(X/G)}
With the notation above, recall that $(\Dmod(X), \otimes)$ is an algebra object in $\Dmod(G) \ccomod$. The functor $\bGamma(X_\dR, -): \ShvCat( (X/G)_\dR) \to \DGCat$ upgrades to an equivalence
$$
\ShvCat( (X/G)_\dR) \xrightarrow{\, \, \simeq \, \, } \Dmod(G) \ltimes \Dmod(X) \mmod.
$$
\end{prop}

\sssec{}

To explain the proof, we need to recollect some facts about sheaves of categories.
For any map of prestacks $f: \Y \to \Z$, consider the adjunction (see \cite{shvcat})
$$
\cores_f: \ShvCat(\Z) \rightleftarrows \ShvCat (\Y): \coind_f.
$$
Both functors preserve colimits and the monad structure on the composition 
$$
\sM := \coind_f \circ \cores_f: \wt\C \mapsto \bGamma(- \times_{\Z} \Y, \wt\C)
$$
is obtained by applying the contravariant functor $\bGamma(-, \wt\E)$ to the comonad
$$
-  \times_\Z \Y : \PreStk_{/\Z} \to \PreStk_{/\Z}, \, \, \, \, \Delta_f : -  \times_\Z \Y \to -  \times_\Z \Y \times_\Z \Y.
$$
In \textit{loc. cit.} it is shown that $\coind_f$ is conservative if $S \times_\Z \Y$ is $1$-affine for any $S \in \affSch_{/\Z}$. We complement this statement with the following lemma.

\begin{lem} \label{lem:1-affine-etale}
$1$-affineness of $S \times_\Z \Y$ is étale-local on $S$. 
\end{lem}

\begin{proof}
Let $U \to S$ be an étale cover such that $U \times_\Z \Y$ is $1$-affine and $U^\centerdot$ the induced Cech simplicial scheme. Each prestack $U^{n} \times_\Z \Y$ is clearly $1$-affine. Consider the commutative diagram
%
\begin{gather}
\xy
(70,0)*+{ \Tot\Big( \QCoh(U^{\centerdot} \times_\Z \Y) \mmod \Big). }="00";
(0,0)*+{ \Tot \, \, \ShvCat(U^{\centerdot} \times_\Z \Y)   }="10";
%
(70,20)*+{  \QCoh(S \times_\Z \Y) \mmod  }="01";
(00,20)*+{ \ShvCat(S \times_\Z \Y)  }="11";
%
{\ar@{<-}_{ } "00";"01"};
{\ar@{<-}^{\simeq}_{ \Tot \,\bGamma(U ^\centerdot \times_\Z \Y, -) \,\,\,\,\,\,\, } "00";"10"};
%
{\ar@{<-}_{\bGamma  } "01";"11"};
{\ar@{<-}_{ \simeq } "10";"11"};
%
%
\endxy
\end{gather} 
%
It suffices to prove that the rightmost vertical arrow $\E \mapsto \Tot \big( \QCoh(U^\centerdot \times_\Z \Y) \otimes_{\QCoh(S \times_\Z \Y)} \E \big)$ is an equivalence. This follows from the canonical equivalence
\begin{eqnarray}
 \QCoh(U^\centerdot \times_\Z \Y) \usotimes{\QCoh(S \times_\Z \Y)} \E
 & \simeq &
 \QCoh(U^\centerdot) \usotimes{\QCoh(S)} \QCoh(S \times_\Z \Y) \usotimes{\QCoh(S \times_\Z \Y)} \E  \nonumber \\
& \simeq &
 \QCoh(U^\centerdot) \usotimes{\QCoh(S)} \E,  \nonumber
\end{eqnarray}
%
combined with étale descent for $\QCoh(S) \mmod$ (estabilished in \citep{L2}).
\end{proof}


\begin{proof}[Proof of Proposition \ref{prop:ShvCat(X/G)}]
Let now $f: (X/G)_\dR \to (BG)_\dR$ be the natural map. From the lemma above, we see that $\coind_f$ is conservative: indeed, any $G_\dR$-torsor on $S$ is locally trivial in the étale topology and $X_\dR$ is $1$-affine.

Hence, the Barr-Beck-Lurie theorem can be applied to the adjunction $(\cores_f, \coind_f)$.
Note first that the functor $\bGamma(\pt, -): \ShvCat(BG_\dR) \to \DGCat$ enhances to an equivalence $\ShvCat(BG_\dR) \simeq \Dmod(G) \ccomod$. Thus, the monad $\sM$, viewed as a functor on $\Dmod(G) \ccomod$, sends 
$$
\bGamma(\pt, \wt \E) \mapsto \bGamma (\pt \times_{BG_\dR} (X/G)_\dR, \wt\E) \simeq \bGamma(X_\dR, \wt\E) \simeq \Dmod(X) \otimes \bGamma(\pt, \wt\E),
$$
whence it corresponds to the algebra object $(\Dmod(X), \Delta^!) \in \Alg( \Dmod(G) \ccomod)$, as wanted.
\end{proof}


\ssec{Groups actions on categories}
%
We say that a (co-complete DG) category $\C$ admits a \emph{strong}\footcite{To obtain the concept of \emph{weak} action, one changes $\fD$ with $\QCoh$ and consider the push-forwards and pull-back functors of quasi-coherent sheaves. Details are left to the reader.} $G$-action if $\C$ is equipped with the structure of a comodule category for $(\Dmod(G), m^!)$. The same datum can be encoded by an action of the monoidal category $(\Dmod(G), m_*)$ on $\C$. Thus,
$$
G \on{-} \mathbf{rep} := (\Dmod(G), \star) \mmod \simeq (\Dmod(G), m^!) \ccomod.
$$
%
The theory of \citep{L1} guarantees that the $\infty$-category $G \on{-} \mathbf{rep}$ admits limits (computed object-wise) and colimits (also computed object-wise).
%
By the previous section, $G \on{-} \mathbf{rep} := (\Dmod(G), \star)$ is monoidal: in terms of the coaction, for $\C, \E \in G \on{-} \mathbf{rep}$, the coaction of $\Dmod(G)$ on $\C \otimes \E$ given by 
\begin{equation} \label{eqn:coact on a tensor}
c \otimes e \mapsto \Delta_{G}^! \big(  \coact(c) \boxtimes \coact(e) \big).
\end{equation}


\sssec{}

An example of a category with a $G$-action is the \emph{regular representation}: $\Dmod(G)$, considered as a module over itself. In analogy with this, we sometimes denote the action map $\Dmod(G) \otimes \C \to \C$ by the convolution symbol, $\star$.

\medskip

The \emph{trivial representation} is $\Vect$, the category of complexes of $\mathbb C$-vector spaces, endowed with the (left, as well as right) $G$-action specified by the monoidal functor $\Gamma_\dR : \Dmod(G) \to \Vect$. This is called the \emph{trivial action} of $G$ on $\Vect$.\footcite{
More generally, we say that a $\Dmod(G)$-module $\C$ is acted on \emph{trivially} by $G$ if the action $\Dmod(G) \otimes \C \to \C$ is prescribed by $M \otimes c \mapsto \Gamma_\dR(G, M) \otimes c$.}


For $\C \in G \rrep$, we define its (strong) \emph{coinvariant} and \emph{invariant} categories as
$$
\C_{G} := \Vect \usotimes{\Dmod(G)} \C,
\hspace{.7cm}
\C^{G} := \Hom_{\Dmod(G)} ( \Vect,  \C ).
$$
%

\sssec{} \label{paradigm}
We shall discuss the relations between $\C$, $\C^G$ and $\C_G$ using the following general framework.
Let $(\A,\star) \in \Alg(\DGCat)$ and $\C \in \A \mmod$. We shall tacitly use the identifications $\A \otimes_\A \C \simeq \C$ and $ \Hom_\A(\A, \C) \simeq \C$ induced by the unit object of $\A$.
Consider the functors
$$
- \usotimes\A \C : \A^\op \mmod \to \DGCat, 
\hspace{.6cm}
\Hom_\A (-, \C): (\A \mmod)^\op \to \DGCat.
$$
At the level of morphisms, we denote them by $\phi \rightsquigarrow \phi_\C$ and $\phi \rightsquigarrow \phi^\C$, respectively.
The two facts below will be repeatedly used throughout the text.

\begin{itemize}

\item
Let $\phi: \D \rightleftarrows \E: \psi$ be mutually adjoint functors in $\A^\op \mmod$. Then, the functors
$$
\phi_\C: \D \usotimes\A \C \rightleftarrows \E \usotimes\A \C: \psi_\C
$$ 
are also mutually adjoint in $\DGCat$. Similarly, if $\phi: \D \rightleftarrows \E: \psi$ is an adjunction in $\A\mmod$, so is
$$
\psi^\C: \Hom_\A( \E,\C) \rightleftarrows \Hom_\A( \D,\C)  : \phi^\C.
$$ 


\item
Assume that $\Vect$ has been given the structure of a $(\A, \A)$-bimodule and $\phi: \Vect \to \A$ a left $\A$-linear functor. Consider the diagram of solid arrows depicted below, where the vertical map is the tautological functor $\Vect \otimes \C \to \Vect \otimes_\A \C$.
It is easy to see that there exists a dashed arrow as indicated
%
\begin{gather}
\xy
(0,0)*+{ \Vect \usotimes{\A} \C  }="00";
%
%
(00,23)*+{ \C  }="01";
(13.5,23)*+{\simeq \Hom_\A (\A, \C)  }="01'";
(53,23)*+{   \Hom_\A (\Vect, \C)  }="11";
%
{\ar@{<-}_{   } "00";"01"};
%
{\ar@{->}^{ \phi^\C } "01'";"11"};
{\ar@{-->}^{  } "00";"11"};
%
%
\endxy
\end{gather} 
%%%%%
%%%%
%%%%
%%%%$$
%%%%\Vect \usotimes \A \C \to \Hom_\A (\Vect, \C)
%%%%$$
if and only if $\phi$ is $(\A, \A)$-bilinear.
\end{itemize}


\sssec{}

We now apply the above paradigms to the adjunction $p^*: \Vect \rightleftarrows \Dmod(G): p_*$.
Define the \emph{constant sheaf} of $G$ to be $k_G := p^*(\CC) \in \Dmod(G)$: the $\fD$-module corepresenting de Rham cohomology $\Gamma_\dR := p_*$.

\begin{lem} \label{lem:p^*-equivariant}
The two adjoint functors $p_*$ and $p^*$ are (both left and right) $G$-equivariant. (In particular, there is a functorial identification $k_G \star - \simeq k_G \otimes \Gamma_\dR (-)$.)
\end{lem}

\begin{proof}
The fact that $p_*$ is $G$-equivariant is obvious (and has been used to define the notion of trivial $G$-action). Equivariance of $p^*$ follows from the fact that the automorphism $\xi : (g, h) \mapsto (g, g \cdot h)$ of $G \times G$ converts the diagram
%
\begin{gather}
\xy
(0,0)*+{ \pt \times G  }="00";
(32,0)*+{  \pt }="10";
%
%
(00,15)*+{ G \times G }="01";
(32,15)*+{  G }="11";
%
{\ar@{<-}_{ p \times \id    } "00";"01"};
{\ar@{->}^{ p } "00";"10"};
%
{\ar@{<-}_{ p  } "10";"11"};
{\ar@{->}^{ m } "01";"11"};
%
%
\endxy
\end{gather} 
%
into the standard product one.
\end{proof}

Hence, we obtain the adjunctions
$$
(\pr_G)^L = (p^*)_\C : \C_G = \Vect \usotimes{\Dmod(G)} \C \rightleftarrows \C : (p_*)_\C =: \pr_G
$$
$$
\oblv^G := (p_*)^\C : \C^G \rightleftarrows \C: (p^*)^\C =: \Av_*^G.
$$
Tautologically, $\oblv^G$ is conservative and $\pr_G$ is essentially surjective.

\begin{cor} \label{cor:k_G-induces}
$k_G \star -: \C \to \C$ induces a functor $\theta_G : \C_G \to \C^G$.
\end{cor}

We shall prove that $\theta_G$ is an equivalence of categories (Theorem \ref{thm:equiv-Ginv-coinv}).

\begin{conv}
What we have treated so far is the concept of \emph{left} $G$-action. Right actions are defined in the obvious way. 
Whenever the $G$-action on $\C$ is clear from the context, we write $\C^G$ for the invariant category (regardless of whether the $G$-action is left or right). 
On the contrary, if it is important to distinguish between right and left $G$-action (in the case $\C$ is equipped with both), we will denote by $^G \C$ and $\C^G$ the left and right invariant categories, respectively. The same conventions hold for the coinvariant categories.
\end{conv}

\sssec{} \label{sssec:paradigm_dual actions}

Let $\S$ be a symmetric monoidal $\infty$-category admitting internal mapping objects, $\A \in \Alg(\S)$ an algebra and $\M \in \S$ a left module for $\A$. This structure amounts to a monoidal functor $\act: \A \to \Hom(\M, \M)$. If $\M$ is dualizable as an object of $\S$, there is an induced monoidal functor $\act': \A^\op \to \Hom(\M^\vee, \M^\vee)$; in other words, $\M^\vee$ acquires a right $\A$-module structure.
Tautologically, $\act'(a) := \act'_a$ is specified by the formula
$$
\langle \act'_a (\phi), m \rangle = \langle \phi, \act_a(m) \rangle,
$$
where $\langle -,- \rangle: \M^\vee \otimes \M \to \Vect$ is the evaluation.

\medskip

Applying this paradigm to $\S = \DGCat$ and $\A = (\Dmod(G), \star)$, we obtain the following. If $\C \in G \rrep$ is dualizable as a plain category, its dual $\C^\vee := \Hom(\C, \Vect)$ inherits a \emph{right} action of $G$, described informally by
$$
\C^\vee \otimes \Dmod(G) \to \C^\vee, \, \, \, \, \phi(-) \otimes F \mapsto \phi(F \star - ).
$$
With this structure,
%
\begin{lem} \label{lem:duality-inv-coinv}
If $\C$ and $\C_G$ are dualizable as DG categories, then $(\C_G)^\vee \simeq (\C^\vee)^{G}$. Under this equivalence, the tautological functors $\oblv^G: (\C ^\vee)^G \to \C^\vee$ and $\pr_G: \C \to \C_G$ are dual to each other.
\end{lem}

\begin{proof}
This is immediate from the Bar realizations of $\C_G$ and $(\C^\vee)^G$. See formulas (\ref{eqn:simplicial_cat}) and (\ref{eqn:cosimplicial_cat}).
\end{proof}



\sssec{}

In the present finite-type context, we freely use the notion of ``de Rham" functor, as developed in \cite{Crys}. Specifically, there is a natural map $\q: G \to G_\dR$, where $G_\dR$ is a group pre-stack such that $\Dmod(G) = \QCoh(G_\dR)$. This is the realization of $\fD$-modules as \emph{left crystals}.

This point of view shows that $\QCoh(G)$ acts on $\Dmod(G)$ via the monoidal functor $\ind_L: (\QCoh(G), \star) \to (\Dmod(G), \star)$, left adjoint (as well as dual) to the forgetful functor $\oblv_L = \q^*: \Dmod(G)  \to \QCoh(G)$. To prove $\ind_L$ is in fact monoidal, recall that $\oblv_L$ intertwines quasi-coherent with de Rham pull-backs, so that by duality $\ind_L$ intertwines the corresponding push-forward functors.


\sssec{}

The action of $G$ on a scheme $X$ of finite type induces a weak (resp., strong) action of $G$ on $\qcoh(X)$ (resp., $\Dmod(X)$).
More generally, the action of $G$ (resp., $G_\dR$) on an arbitrary prestack $\Y$ gives rise to a weak (resp., strong) $G$-action on $\QCoh(\Y)$. Tautologically, the invariant category is 
$$
\QCoh(\Y)^{G,w} \simeq \QCoh(\Y/G), \text{        (resp., } \QCoh(\Y)^{G,s} \simeq \QCoh(\Y/G_\dR) \text{)}, 
$$
where the quotient is simply the geometric realization of the Cech cosimplicial prestack:\footcite{In other words, we are considering the \emph{prestack quotient}. Recall, however, that $\QCoh(-)$ and $\Dmod(-)$ are insensitive to the operation of sheafification in the flat topology.}

We shall discuss a fundamental example of this situation, recovering the adjoint action of $G$ on its Lie algebra $\fg$.

\sssec{}

Let $\wh G$ denote the formal group of $G$ at $1 \in G$. By definition,
$$
\wh G := \{1\} \times_{G_\dR} G.
$$
%
It is well-known that $G_\dR \simeq G/ \wh G$ (see, e.g., \cite{Crys}), so that $\pt/  \wh G \simeq G_\dR / G$. Thus, $\pt/  \wh G$ carries an action of $G_\dR$, induced by the multiplication of $G_\dR$:
%
\begin{equation} \label{eqn:act-strong-on-BGhat}
G_\dR \times \pt/\wh G \simeq G_\dR \times G_\dR/G \xrightarrow{\, \, m \, \,} G_\dR/G \simeq  \pt/ \wh G.
\end{equation}
%
This yields a strong $G$-action on $\QCoh(\pt/ \wh G)$.


\begin{rem}

We claim that this action is familiar. Let $\fg \mod$ is the category of representations of the Lie algebra of $G$. There is an equivalence $\QCoh(\pt/ \wh G) \simeq \fg \mod$, under which the strong $G$-action just described corresponds to the adjoint action of $G$ on $\fg$.
This follows from the equivalence between Lie algebras and formal groups, which holds over a field of characteristic zero. The correspondence associates to a Lie algebra $\fg$ the formal group $\on{Spf}(U(\fg)^\vee)$ and it is well-known that $U(\fg)$ and $\O(\wh G)$ are dual Hopf algebras.

\end{rem}

We will not use any Lie algebra theory in the present paper, and by the symbol ``$\fg \mod$" we understand the category $\QCoh(\pt/ \wh G)$. Clearly,
$$
(\fg \mod)^{G,s} \simeq \QCoh(G \backslash G_\dR)^{G_\dR} \simeq \QCoh( G \backslash G_\dR / G_\dR)   \simeq \Rep(G),
$$
while 
$$
(\fg \mod)^{G,w} \simeq \QCoh( G \backslash G_\dR / G) 
$$ 
is the \emph{Harish-Chandra} category to be studied and used in the next paragraphs.


\ssec{Some results on weak invariants} \label{ssec:weak-vs-s}

By construction, the datum of a weak $G$-action on $\C$ consists of the datum of a monoidal functor $(\QCoh(G), \star) \to (\End(\C), \circ)$. 

In case $\C = (\A, \ast)$ admits the structure of a monoidal category, there is also a natural monoidal functor 
$$
(\A, \ast) \to (\End(\A), \circ), \hspace{.5cm} a \mapsto \act_a := a \ast -.
$$ 
Hence, a monoidal functor $\phi: (\QCoh(G), \star) \to (\A, \ast)$ induces a weak left $G$-action on $\A$. Likewise, $\phi$ also induces a weak right $G$-action on $\A$.
We shall discuss this situation in general, after recalling two prime examples:

\begin{itemize}
\item[(1)]

Take $\A = (\Dmod(G),\star)$. We mentioned above that the induction functor $\ind_L: \QCoh(G) \to \Dmod(G)$ is monoidal.

\medskip

\item[(2)]

Take $\A = (\QCoh(H), \star)$, where $H$ is a group scheme together with a morphism of groups $f: G \to H$. Then $f_*: \qcoh(G) \to \qcoh(H)$ is monoidal.\footnote{Thus, in latter case the monoidal functor is \emph{right} adjoint to $f^*$; in the former one, $\ind$ is \emph{left} adjoint to $\q^*: \QCoh(G_\dR) \to \QCoh(G)$.}
\end{itemize}
%


\sssec{}


Clearly, the weak (co)invariant category of any weak $G$-representation admits an action of $\Hom_{\QCoh(G)} (\Vect, \Vect) \simeq \Rep(G)$. Thus, there is an adjunction
\begin{equation} \label{adj:RepG-QCoh(G)}
{\rec} : (\Rep(G), \otimes) \mmod \rightleftarrows (\QCoh(G), \star) \mmod: {\inv}^{G,w},
\end{equation}
where $\inv^{G,w}$ is the functor of weak $G$-invariants and ${\rec}$, the ``reconstruction" functor, sends $\E$ to $\Vect \otimes_{\Rep(G)} \E$.
%
\begin{thm}[Gaitsgory-Lurie, \cite{shvcat}] \label{thm:Gaitsgory-Lurie}
These adjoint functors are mutually inverse equivalences of categories. 
\end{thm}


\sssec{}

Let us combine this with the standard adjunction 
\begin{equation} \label{adj:A-Qcoh}
 \A \otimes_{\QCoh(G)} - : \QCoh(G) \mmod \rightleftarrows \A \mmod : \oblv^{\A \to \QCoh(G)}.
\end{equation}
%
Taking into account that $\Vect$ is a $(\qcoh(G), \Rep(G))$-bimodule category, we obtain the adjunction
\begin{equation} \label{adj:RepG-Dmod}
\A_{G,w} \otimes_{\Rep(G)} - : \Rep(G) \mmod \rightleftarrows \A \mmod : \inv^{G,w} \circ \oblv^{\A \to \QCoh(G)}.
\end{equation}
The conditions of the Barr-Beck-Lurie theorem are satisfied: indeed, it suffices to observe that the absolute forgetful functor $\oblv^{\A}: \A \mmod \to \DGCat$ is continuous and conservative. Thus, $\A \mmod$ is equivalent to the category of modules for ${}^{G,w}\A_{G,w} \in \Alg(\Rep(G) \mmod)$. 

\sssec{}

Let us describe the algebra structure explicitly.
Note the following facts:
\begin{itemize}
\item[(i)]
the adjunction (\ref{adj:RepG-QCoh(G)}) yields the functor (actually an equivalence)
$$
\Vect \usotimes{\Rep(G)} {}^{G,w}\A \to \A;
$$
\item[(ii)]
the multiplication of $\A$ descends to a functor
$$
\A \usotimes{\QCoh(G)} \A \to \A,
$$
which is $G \times G$-equivariant.
\end{itemize}
Combining (i) and (ii), we obtain a $G \times G$-equivariant functor
$$
\wh\ast: \A_{G,w} \usotimes{\Rep(G)} {}^{G,w}\A \simeq
\A \usotimes{\qcoh(G)} \Vect \usotimes{\Rep(G)} {}^{G,w}\A
\xrightarrow{\, \, (i) \, \,}
\A \usotimes{\qcoh(G)} \A
\xrightarrow{\, \, (ii) \, \,}
\A,
$$
which descends to the multiplication of ${}^{G,w}\A_{G,w}$.

\medskip

Another standard application of Barr-Beck shows that ${}^{G,w}\A_{G,w}$ is also an algebra object in $\DGCat$, with multiplication given by
$$
{}^{G,w}\A_{G,w} \otimes {}^{G,w}\A_{G,w} 
\xrightarrow{\, \mathsf{tautological} \, }
{}^{G,w}\A_{G,w} \usotimes{\Rep(G)} {}^{G,w}\A_{G,w} 
\xrightarrow{\, \mathsf{mult}\, }
{}^{G,w}\A_{G,w}.
$$

\sssec{}

Hence, we have proven that Theorem \ref{thm:Gaitsgory-Lurie} admits the following generalization:

\begin{thm} \label{thm:A-weak-inv}
For any $\A$ as above, the functor of weak invariants $\C \rightsquigarrow \C^{G,w}$ yields an equivalence
$$
\inv^{G,w}: \A \mmod \xrightarrow{\, \, \simeq \,\,} {}^{G,w}\A_{G,w}  \mmod.
$$
\end{thm}

\sssec{}

We now collect a few consequences.

\begin{cor}
Under the above set-up, the invariant category $^{G,w} \A$ is dualizable as a right $\A$-module, with dual the left $\A$-module $\A_{G,w}$.
\end{cor}


\begin{proof}
The duality datum consists of a functor
$$
\eps: \A_{G,w} \otimes \,  {}^{G,w} \A \to \A
$$
of $\A$-bimodules and of a plain functor
$$
\mu : \Vect \to \, {}^{G,w} \A \usotimes{\A} \A_{G,w} \simeq \, {}^{G,w} \A_{G,w},
$$
satisfying the usual compatibilities. We let $\eps = \wh \star$ and $\mu$ simply be the unit of the algebra structure on ${}^{G,w} \A_{G,w}$. It is immediate to check that the duality axioms are satisfied.
\end{proof}

\begin{rem}
A tautology: under the equivalence ${}^{G,w} \A_{G,w} \simeq \Hom_{\A} (\A_{G,w}, \A_{G,w})$, the monoidal structure of ${}^{G,w} \A_{G,w}$ comes from composition.
\end{rem}

For any $\C$ acted on weakly by $G$, there is an equivalence $\C_{G,w} \simeq \C^{G,w}$ compatible with the $\Rep(G)$-action (see \cite{shvcat}). Thus, $^{G,w} \A ^{G,w}$ also acquires a monoidal structure, both in $\DGCat$ and $\Rep(G) \mmod$.

\begin{cor} \label{cor:apply_invariants}
For any left $\A$-modules $\M, \N$ and any right $\A$-module $\P$, there are equivalences
$$
\Hom_{\A} (\M, \N) \simeq \Hom_{ {}^{G,w} \A^{G,w}} ({}^{G,w} \M, {}^{G,w} \N),
\hspace{.7cm}
 \P \usotimes{\A} \N \simeq \P^{G,w}  \usotimes{{}^{G,w} \A^{G,w}} {}^{G,w}\N.
$$
\end{cor}


\sssec{Example}

Let us take up example (2) at the beginning of Section \ref{ssec:weak-vs-s}. In that case, $\A^{G,w}$ and $\A_{G,w}$ are both equivalent to $\QCoh(H/G)$, the action of $\Rep(G)$ being by pull-back along $H/G \to BG$.
The monoidal structure on ${}^{G,w} \A ^{G,w} \simeq \QCoh(G \backslash H /G)$ is induced by the correspondence
$$
G \backslash H /G \times G \backslash H /G \leftarrow G \backslash (H  \stackrel G \times H)  /G \to G \backslash H /G.
$$
In particular, setting $G=H$, we obtain the monoidal equivalence $\QCoh(G \backslash G /G) \simeq \Rep (G)$.


\sssec{} 

The next result provides a sufficient condition for ${}^{G,w} \A ^{G,w} $ to be a rigid monoidal category (\cite{DG}). This entails that ${}^{G,w} \A ^{G,w} $ is self-dual and that dualizability of an object of ${}^{G,w} \A ^{G,w}  \mmod$ is equivalent to its dualizability in $\DGCat$.

\medskip

By the above example, the functor $\phi: \QCoh(G) \to \A$ induces $\phi' : \Rep(G) \to {}^{G,w} \A ^{G,w}$, which is still monoidal.

\begin{cor} \label{cor:rigidity}
Assume that the right adjoint to $\phi$ is continuous and conservative. If the diagram 
%
%
\begin{gather}
\xy
(0,0)*+{ \QCoh(G) }="00";
(40,0)*+{  \A }="10";
%
%
(00,15)*+{  {}^{G,w} \QCoh(G) ^{G,w} }="01";
(40,15)*+{  {}^{G,w} \A ^{G,w} }="11";
%
{\ar@{<-}_{ \oblv^{G \times G }    } "00";"01"};
{\ar@{->}^{ \phi} "00";"10"};
%
{\ar@{<-}_{ \oblv^{G \times G }  } "10";"11"};
{\ar@{->}^{ \, \phi' } "01";"11"};
%
%
\endxy
\end{gather} 
%
is right adjointable (i.e. $\phi^R$ and $(\phi')^R$ intertwine the $\oblv$ functors), then ${}^{G,w} \A ^{G,w}$ is compactly generated by objects that are (both left and right) dualizable. Hence, ${}^{G,w} \A ^{G,w}$ is rigid.
\end{cor}

\begin{proof}
As any compact object in $\Rep(G)$ is (both left and right) dualizable, it suffices to prove that the right adjoint to $\phi'$ is continuous and conservative, which is immediate from the fact that $\oblv^{G \times G}$ and $\phi^R$ are such.
\end{proof}


\ssec{Weak vs strong invariants}

Let us take up example (1) at the beginning of Section \ref{ssec:weak-vs-s}: $\A = (\Dmod(G), \star)$ and $\phi = \ind_L : \QCoh(G) \to \Dmod(G)$. This will show how to recover strong invariants from weak invariants. 

\medskip

We define the Harish-Chandra monoidal category to be
$$
\HC := {}^{G,w} \Dmod(G) ^{G,w} \simeq \QCoh( G \backslash G_\dR /G),
$$
equipped with the monoidal structure discussed above. Alternatively, $\HC \simeq \Hom_{\Dmod(G)} (\fg \mod, \fg \mod)$, monoidal under composition. If $\C$ admits a strong action of $G$, its weak invariant and coinvariant categories $\C^{G,w}$ and $\C_{G,w}$ both retain an action of $\HC$.

\begin{cor}
$\HC$ is compactly generated by objects that are left and right dualizable, hence it is rigid.
\end{cor}

\begin{proof}
Let $\q: G \to G_\dR$ be the quotient map. The right adjoint to $\ind_L$ is the pull-back functor $\q^* : \QCoh(G_\dR) \to \QCoh(G)$, which is conservative by \cite{Crys} and continuous by general non-sense. The second condition required by Corollary \ref{cor:rigidity} follows from the commutative diagram
%
%
\begin{gather} 
\xy
(0,0)*+{ G \backslash G/G \, \,  }="00";
(35,0)*+{  \,\, G \backslash G_\dR/G }="10";
%
%
(00,18)*+{  G }="01";
(35,18)*+{  G_\dR }="11";
%
{\ar@{<-}_{    } "00";"01"};
{\ar@{->}^{ \q } "00";"10"};
%
{\ar@{<-}_{   } "10";"11"};
{\ar@{->}^{ \q' } "01";"11"};
%
%
\endxy
\end{gather} 
%
by taking pull-backs.
\end{proof}

As a consequence of Corollary \ref{cor:apply_invariants}, we obtain:

\begin{cor} \label{cor:strong-from-weak}
If $\C$ is equipped with a strong action of $G$, then
$$
\C^{G} \simeq \Hom_{\HC} ( \Rep(G), \C^{G,w})
\, \, \, 
\text{ and } \, \, \, 
\C_{G} \simeq  \Rep(G) \usotimes{\HC} \C_{G,w}.  
$$
\end{cor}

In particular, the rigidity of $\HC$ and the self-duality of $\Rep(G) \in \DGCat$ imply the existence of an equivalence $\C^{G} \simeq \C_{G}$.  Our preferred equivalence is the following:
%
\begin{thm} \label{thm:equiv-Ginv-coinv}
The functor $\theta_G: \C_G \to \C^G$ of Corollary \ref{cor:k_G-induces} is an equivalence.
\end{thm}

\begin{proof}
Consider $\theta$ as a continuous functor $G \rrep \to \DGCat^{\Delta^1}$. Since the operation $\C \rightsquigarrow \C^G$ commutes with colimits and tensor products with categories, we may assume without loss of generality that $\C = \Dmod(G)$.
In this case, $\C_G$ and $\C^G$ are both equivalent to $\Vect$. 
Let us identify $\Dmod(G)_G$ and $\Dmod(G)^G$ with $\Vect$ using the objects $\pr_G(\delta_1)$ and $p^*$ (see Lemma \ref{lem:p^*-equivariant}), respectively. Then, $\theta$ goes over to the identity functor of $\Vect$.
%
\end{proof}


\sssec{} \label{sssec:normal}

We digress briefly to discuss actions by normal subgroups. Let $i: K \hookrightarrow G$ be a normal subgroup, so that $\Dmod(K)$ acts on $\Dmod(G)$ via $i_*: \Dmod(K) \to \Dmod(G)$. We prove that $G$-invariants and $G$-coinvariants can be taken in two steps.

\begin{lem} \label{lem:normal-inv}
Let $\C$ be a category with a strong right action of $G$. The quotient group $Q:=G/K$ acts on $\C^K$ and $\C_K$ from the right in such a way that
$$
\C^G \simeq  \big(\C^K \big)^Q,
\hspace*{.6cm}
 \C_G \simeq  \left( \C_K \right)_{G/K}.
$$
With the obvious modifications, the statement holds for weak actions and weak (co)invariants.
\end{lem}

\begin{proof}
For coinvariants, we have
$$
\C_K := \C \usotimes{\Dmod(K)} \Vect \simeq \C \usotimes{\Dmod(G)} \Big( \Dmod(G) \usotimes{\Dmod(K)} \Vect \Big) 
 \simeq \C \usotimes{\Dmod(G)} \Dmod(Q),
$$
where the last step uses Corollary \ref{cor:strong-from-weak}. The sought-after $Q$-action on $\C_K$ is the one induced by the regular action of $Q$ on itself and, tautologically,
$$
\left( \C_K \right)_{Q}
\simeq \C_K \usotimes{\Dmod(Q)} \Vect
\simeq \C \usotimes{\Dmod(Q)} \Dmod(Q) \usotimes{\Dmod(G)} \Vect 
\simeq \C \usotimes{\Dmod(G)} \Vect   = \C_G.
$$
The version of argument for invariants and for weak actions is entirely analogous.
\end{proof}

\ssec{Twisted group actions} \label{ssec:twisted_act_FD}

Given a category $\C \in G \rrep$, we shall explain how to twist the action by an additive character $\m : G \to \GG_a$.

\sssec{Strong actions on $\Vect$}

Tautologically, a strong action of $G$ on $\Vect$ is given by a monoidal functor $(\Dmod(G), \star) \to \Vect$. Equivalently, by a comonoidal functor $\Vect \to (\Dmod(G), m^!)$. Such functors correspond precisely to \emph{character $\fD$-modules}, i.e. $\fD$-modules $\F$ equipped with an isomorphism
$$
m^! (\F) \simeq \F \boxtimes \F,
$$ 
satisfying the natural compatibility conditions.
For such $\F$, the action map $\Dmod(G) \otimes \Vect \to \Vect$ is given by 
$$
M \otimes V \mapsto M \star_\F V := \eps_{G} (M \otimes \F) \otimes V,
$$
where, $\eps_G = \Gamma_\dR \circ \Delta^!$ is the evaluation pairing between $\Dmod(G)$ and $\Dmod(G)^\vee \simeq \Dmod(G)$.


\sssec{}

%
Consider the exponential (right) $\fD$-module $exp$ on $\GG_a = \AA^1 = \Spec (k[z])$:
\begin{equation} \label{def:exp}
exp = \frac{\fD_{\AA^1}}{(\partial_z-1) \fD_{\AA^1}}.
\end{equation}
%
%
(This is a substitute of the Artin-Schreier sheaf in characteristic zero.) It is a character $\fD$-module,
\begin{equation}\label{eqn:exp_homo}
m^! (exp) \simeq exp \boxtimes exp,
\end{equation}
%
and the prototype of all the character $\fD$-modules we shall consider. In fact:

\begin{lem} \label{lem:char_pull-back}
Let $\m: G \to \GG_a$ be an additive character of $G$. Then $\m^! (exp) \in \Dmod(G)$ is a character $\fD$-module.
\end{lem}

\nc{\Tw}{\ms{Tw}}

We write $\Vect_{\m}$ to emphasize that $\Vect$ is being considered as a category with the $\Dmod(G)$-coaction corresponding to $\m^! (exp)$.
Recall that $G \on{-}\mathbf{rep}$ is a monoidal $\infty$-category. Hence, we can define the $\m$-twist functor
\begin{equation} \label{eqn:twist}
\Tw_\m : G \rrep \to G \rrep, 
\hspace*{.4cm}
\C \to \C \otimes \Vect_\m.
\end{equation}
%
Clearly, $\Tw_\m$ is an automorphism with inverse $\Tw_{-\m}$.

\medskip

In terms of the $G$-coaction on $\C$, the $G$-coaction on $\C \otimes \Vect_\m$ is given by
$$
\C   \to  \Dmod(G) \otimes \C, 
\hspace*{.4cm} c \mapsto \m^!(exp) \otimes \coact(c),
$$
where we have identified $\C \simeq \C \otimes \Vect_{\m}$ (as DG categories). 


\sssec{}

Define the \emph{$(G,\m)$-invariant} and \emph{$(G,\m)$-coinvariant} categories of $\C$ respectively as
$$
\C^{G,\m} := \Hom_{\Dmod(G)}(\Vect_\m, \C)
\hspace{.6cm}
\C_{G,\m} :=  \Vect_\m \usotimes{\Dmod(G)} \C.
$$
%
\begin{lem} \label{lem:Whit_with_dualiz_char}
$\C^{G,\m} \simeq\big( \C \otimes \Vect_{-\m} \big)^G$ and  $\,\,\C_{G,\psi} \simeq \big( \C \otimes \Vect_{-\m} \big)_G$.

\end{lem}

We have natural functors
$$
\begin{array}{ll}
& \pr_{G,\m}: \C \xto{\Tw_{-\m}} \C \otimes \Vect_{-\m}
\xto{\pr_G} 
\big( \C \otimes \Vect_{-\m} \big)_G
\simeq 
\C_{G,\m} \vspace*{.3cm} \\

%%%
& \oblv^{G,\m}: \C^{G,\m} \simeq \big( \C \otimes \Vect_{-\m} \big)^G
\xto{\oblv^G} 
\C \otimes \Vect_{-\m} 
\xto{\Tw_\m}
\C  \vspace*{.3cm} \\

& \Av_*^{G,\m}: \C \xto{\Tw_{-\m}} \C \otimes \Vect_{-\m}
\xto{\Av^G_*} 
\big( \C \otimes \Vect_{-\m} \big)^G
\simeq 
\C^{G,\m}. \\
%%%
\end{array}
$$
As in the untwisted case, $(\oblv^{G,\m}, \Av_*^{G,\m})$ forms an adjoint pair. From Theorem \ref{thm:equiv-Ginv-coinv}, we obtain:

\begin{cor} \label{cor:theta-mu-fin-type}
There is an equivalence $\theta_{G,\m} : \C_{G,\m} \to \C^{G,\m}$ such that $\theta_{G,\m} \circ \pr_{G,\m} \simeq \Av_*^{G,\m}$.
\end{cor}




\section{$\fD$-modules on ind-schemes of pro-finite type} \label{SEC:Dmod-infinite}

Leaving aside the notion of sheaves of categories, we wish to say that a group prestack $(\G,m)$ acts strongly on a category $\C$ if the latter is endowed with an action of the monoidal category $(\Dmod(\G), \star)$. Similarly, we say that $\G$ acts weakly on $\C$ is the latter is a module for $(\QCoh(\G), \star)$. 
We mostly focus on strong actions. For such, the above definition makes sense whenever we can provide a construction of $\Dmod(\G)$ as a \emph{dualizable} category endowed with the convolution monoidal functor $m_*$.

\medskip

The ultimate goal of this section is to supply this definition in our cases of interest: $\G = G \ppart$ and $\G = N \ppart$, where $G$ is a reductive group and $N$ its maximal unipotent subgroup. 
%
To address this, we proceed in two steps (see \citep{KV} for a very similar discussion). 

First, we identify the kind of algebraic structure that $G \ppart$ and $N \ppart$ possess: the answer is that they are \emph{ind-pro schemes}. Roughly speaking, these are (0-)prestacks constructed from schemes of finite type out of affine smooth projections and closed embeddings. Secondly, we develop the theory of $\fD$-modules on ind-pro schemes. In the treatment below we present these two steps in opposite order, namely we start from the abstract theory and later we apply it to loop groups.




\ssec{$\fD$-modules on pro-schemes}

Let $\Sch^{qc, qs}$ be the $1$-category of quasi-compact quasi-separated schemes and $\Sch^{\fty}$ its full subcategory of schemes of finite type.
%
Between the two lies the $1$-category $\Sch^\pro := \on{Pro}^{\on{aff,sm,s}} \Sch^{\fty}$ of schemes of \emph{pro-finite type}: schemes that can be written as filtered limits of schemes of finite type along affine smooth surjective maps. 
%
The natural functor $\Sch^\pro \to \Sch^{qc, qs}$ is fully faithful, as shown in Appendix C of \cite{TT}.\footcite{This reference was pointed out by S. Raskin, who has independently developed the theory of $\fD$-modules on schemes of infinite type.} 

\medskip

To shorten the terminology, we refer to objects of $\Sch^\pro$ just as \emph{pro-schemes}. By \emph{pro-group}, we shall mean ``group object in the category of pro-schemes".
%


\sssec{}

We define $\fD^*$-modules on pro-schemes as follows: we let
$$
\Dmod^* : \Sch^\pro \to \DGCat
$$
be the right Kan extension of $\Dmod: \Sch^{\fty} \to \DGCat$ along the inclusion $\Sch^{\fty} \hookrightarrow \Sch^\pro$. 
Here, $\Dmod: \Sch^{\fty} \to \DGCat$ is the usual functor that assigns $S \rightsquigarrow \Dmod(S)$ and $f \rightsquigarrow f_*$. So, $\Dmod^*$ is covariant by construction: for any morphism $f: X \to Y$ in $\Sch^\pro$, we denote by $f_*$ the corresponding functor $\Dmod^*(X) \to \Dmod^*(Y)$.

\medskip

Explicitly, suppose $Z \in \Sch^\pro$ be presented as 
\begin{equation} \label{eqn:pro-schemeZ}
Z \simeq \lim_{r \in \R} Z^r,
\end{equation}
%
where $\R^{op}$ is a filtered category. 
(In most cases of interest, $\R = (\NN, <)^{op}$.) 
For any arrow $s \to r$ in $\R^{op}$, let $\pi_{s \to r}: Z^s \to Z^r$ be the corresponding projection. Then, 
%
\begin{equation} \label{eqn:Dmod*-as-limit}
\Dmod^*(Z) \simeq \lim_{r \in \R, \pi_*}\, \Dmod(Z^r),
\end{equation}
%
the limit being taken with respect to the pushforwards $(\pi_{s \to r})_*$. 

\sssec{} \label{sssec:Lurie_G0}

On several occasions, we will make use of the following paradigm, due to J. Lurie (cf. \cite{DG}). Let $\C_\bullet: \I \to \DGCat$ be a diagram of categorie: for each $\gamma: i \to j$, denote the corresponding functor by $F_\gamma : \C_i \to \C_j$. Let $G_\gamma$ be the possibly discontinuous right adjoint of $F_\gamma$. There is then an equivalence of categories
$$
\Omega: \lim_{i \in \I^\op, G} \C_i \xrightarrow{ \, \, \simeq \, \, } \uscolim{i \in \I, F} \C_i,
$$
under which the tautological functors of ``insertion" and ``evaluation"
$$
\ins_i : \C_i \to \uscolim{i \in \I, F} \C_i \hspace{.6cm} \ev_i : \lim_{i \in \I^\op, G} \C_i  \to \C_i
$$
form an adjoint pair.

\sssec{}

In the case at hand, smoothness of each $\pi_{s \to r}$ implies the existence of the left adjoints $(\pi_{s \to r})^*$, so that
$$
\Dmod^*(Z) \simeq \lim_{\R, \pi_*} \Dmod(Z^r) \simeq \uscolim{\R^{op}, \pi^*}\, \Dmod(Z^r).
$$
The latter expression explains the notation $\Dmod^*$, which is meant to indicate a colimit along $*$-pullback functors. It also shows that $\Dmod^*(Z)$ is compactly generated, hence dualizable. We denote the insertion
$$
\ins_s: \Dmod(Z^s) \to \uscolim{\R^{op}, \pi^*}\, \Dmod(Z^r)
$$
\emph{formally} by $(\pi_{\infty \to s})^*$. Analogously, the evaluation functor will be formally denoted by $(\pi_{\infty \to s})_*$.
%
Compact objects of $\Dmod^*(Z)$ are those of the form $(\pi_{\infty \to s})^*(M)$, for $M$ compact in $\Dmod(Z^s)$. 




\sssec{} \label{sssec:fromLimToColim}

Thanks to the continuity of the push-forward functors $\pi_*$ and the filteredness of $\R^\op$, the isomorphism $ \underset{\R, \pi_*}  \lim\Dmod(Z^r) \to \uscolim{\R^{op}, \pi^*} \Dmod(Z^r)$ can be made explicit (\cite{DG}). 
Namely,
\begin{equation} \label{eqn:fromLimToColim}
\Omega: M = \{M^r \}_r  \mapsto \uscolim{r \in \R^{op}}  \Big( (\pi_{\infty \to r})^*(M^r) \Big),
\end{equation} 
%
while the inverse equivalence is completely determined by the assignment:
%
\begin{equation} \label{eqn:fromColimToLim}
\Omega^{-1}: (\pi_{\infty \to r})^* (M^r)  \mapsto  \left\{ \uscolim{k \, : \, k \to r, \, k \to s} (\pi_{k \to s})_* (\pi_{k \to r})^* (M^r)  \right\}_s.
\end{equation}
%
%
Clearly, the latter expression greatly simplifies if each pull-back $(\pi_{r_1 \to r_2})^*$ is fully faithful: in that case, the colimit in (\ref{eqn:fromColimToLim}) is independent of $k$.
%
Thus, we propose the following definition: we say that a pro-scheme $Z$ is \emph{pseudo-contractible} if it admits a presentation as in (\ref{eqn:pro-schemeZ}) with transition maps giving rise to fully faithful $*$-pullback functors. By smoothness, the $!$-pullbacks are also fully faithful. 


\sssec{}

The dual of $\Dmod^*(Z)$ is easily computed and it is by the definition the category of $\fD^!$-modules on $Z$:
$$
\Dmod^!(Z):= \big( \Dmod^*(Z) \big)^\vee \simeq \uscolim{\R^{op}, \pi^!} \Dmod(Z^r).
$$
This is a consequence of the fact that $\Dmod(S)^\vee \simeq \Dmod(S)$ via the classical Verdier duality, and that $(\pi_*)^\vee \simeq \pi^!$ under this self-duality.

\medskip

The assigment $Z \rightsquigarrow \Dmod^!(Z)$ upgrades to a contravariant functor mapping $f: X \to Y$ to 
$$
f^! := (f_*)^\vee : \Dmod^!(Y) \to \Dmod^!(X).
$$
%
As before, let
$$
(\pi_{\infty \to s})^!: \Dmod(Z^s) \to \uscolim{\R^{op}, \pi^!} \Dmod(Z^r)
$$
symbolize the tautological insertion map. Compact objects of $\Dmod^!(Z)$ are those objects of the form $(\pi_{\infty \to s})^!(M)$, for all $s \in \R$ and all $M$ compact in $\Dmod(Z^s)$.

\sssec{}

Since each $\pi_{s \to r}$ is smooth, $(\pi_{s \to r})^! \simeq (\pi_{s \to r})^* [2 d_{rs}]$, where $d_{rs}$ is the dimension of $\pi_{s \to r}$. Thus, the right adjoint of $(\pi_{s \to r})^!$ is isomorphic to $(\pi_{s \to r})_*[-2 d_{rs}]$ (hence, it is continuous).
We can realize $\Dmod^!(Z)$ as a limit:
\begin{equation}\label{eqn:Dmod^!-as-limit}
\Dmod^!(Z) \simeq \lim_{r \in \R, \pi_*[-2d_\pi ]} \Dmod(Z^r).
\end{equation}
%

Assume a trivialization of the dimension torsor of $Z$ has been specified. This consists of a function $\dim Z^ {\centerdot}: \R \to \ZZ$ such that $\dim (Z^s)- \dim(Z^r) = \dim (\pi_{s \to r})$.\footcite{We also say that $Z$ has been given a \emph{dimension theory}.} Then, comparing the above formula with (\ref{eqn:Dmod*-as-limit}), we construct the equivalence
%
\begin{equation} \label{eqn:eta-pro-scheme}
\lambda_Z : \Dmod^!(Z) \xrightarrow{ \, \, \simeq \, \, } \Dmod^*(Z),
\end{equation}
%
induced by the inverse family of shift functors $ \id [-2 \dim(Z^r)]: \Dmod(Z^r) \to \Dmod(Z^r)$. Equivalently, 
$$
(\pi_{\infty \to r})_* \circ \lambda_Z = \big( (\pi_{\infty \to r})^! \big)^R \circ  [- 2 \dim(Z^r)],
$$
or, upon passage to left adjoints,
\begin{equation} \label{eqn:eta-compat-with-ins}
\lambda_Z \circ (\pi_{\infty \to r})^!  = (\pi_{\infty \to r})^* \circ [-2 \dim(Z^r)].
\end{equation}
\sssec{}

By (\cite{DG}), the duality pairing $\eps_Z$ between $\Dmod^*(Z)$ and $\Dmod^!(Z)$ consists of the assignment
%
\begin{equation} \label{eqn:dual-pairing-limColim}
\eps_Z ( \{M^s\}_s, (\pi_{\infty \to r})^! N^r ) \simeq \eps_{Z^r} ( M^r, N^r). 
\end{equation}
%
In other words, $(\pi_{\infty \to r})^!$ is dual to the functor $ (\pi_{\infty \to r})_* :  \Dmod^*(Z) \to \Dmod(Z^r)$.
%
Alternatively, by (\ref{eqn:fromColimToLim}), we obtain
%
\begin{eqnarray} \label{eqn:duality_pairing}
\eps_Z ( (\pi_{\infty \to r})^* M^r, (\pi_{\infty \to r})^! N^r ) 
 &=& \uscolim{s \in \R_{/r}} \, \eps_{Z^s} \big( (\pi_{s \to r})^* M^r, (\pi_{s \to r})^! N^r \big)     
  \\
   &=& \uscolim{s \in \R_{/r}} \, \Gamma_\dR \big( Z^s, (\pi_{s \to r})^* (M^r \otimes N^r) \big). \nonumber
\end{eqnarray}

%%%%%\begin{rem} \label{rem:eta-selfdual}
%%%%%Whenever $\lambda_Z$ is defined (equivalently, for any dimension theory of $Z$), it is self-dual: $(\lambda_Z)^\vee \simeq \lambda_Z$.
%%%%%\end{rem}

\begin{rem}
If $Z$ is pseudo-contractible, the formula (\ref{eqn:duality_pairing}) simplifies as
$$
\eps_Z ( (\pi_{\infty \to r})^* M^r, (\pi_{\infty \to r})^! N^r )  \simeq \Gamma_\dR (Z^r, M^r \otimes N^r ). 
$$
\end{rem}

\begin{lem}
For any $M^r \in \Dmod(Z^r)_\cpt$, Verdier duality $\DD_Z : \Dmod^!(Z) \xrightarrow{\, \, \simeq \, \, } \Dmod^*(Z)$ is computed ``component-wise", i.e., it sends
$$
\DD_Z: (\pi_{\infty \to r})^! (M^r) \mapsto (\pi_{\infty \to r})^* (\DD_{Z^r} (M^r)).
$$
\end{lem}

\begin{proof}
By definition, it suffices to exhibit a canonical equivalence
%
\begin{equation} \label{eqn:Verdier_tech}
\Hom_{\Dmod^*(Z)} \Big(  (\pi_{\infty \to r})^* (\DD_{Z^r} (M^r)), P \Big)
\simeq 
\eps_{Z} \big(  ( \pi_{\infty \to r})^! (M^r), P \big),
\end{equation}
%
for any $P \in \Dmod^*(Z)$ pulled back from a finite-type quotient.
%
This follows immediately from the adjunction $\big ( (\pi_{\infty \to r})^* , (\pi_{\infty \to r})_* \big )$ for $\fD^*$-modules, combined with the duality between $(\pi_{\infty \to r})^!$ and $(\pi_{\infty \to r})_*$.
%
\end{proof}


\ssec{Basic functoriality}

In this subsection we work out part of the theory of $\fD^*$ and $\fD^!$-modules on pro-schemes: we discuss various push-forward and pull-back functors, the tensor products, the projection and base-change formulas.

\sssec{}

By definition of right Kan extension, if $f: X \to Y$ is a morphism of pro-schemes, the functor $f_*: \Dmod^*(X) \to \Dmod^*(Y)$ is determined by the formula
%
\begin{equation} \label{eqn:ev----f_*}
(\pi^Y_{\infty \to r})_* \circ f_* \simeq (f^r)_* \circ (\pi^X_{\infty \to r})_*
\end{equation}
for any morphism $f^r: X^r \to Y^r$ of schemes of finite type covered by $f$.
Below are two main examples.

\medskip

The functor of \emph{de Rham cohomology} $\Gamma_\dR$ of $X \simeq \lim_r X^r$ is defined as
$$
\Gamma_\dR:= (p_X)_* \simeq (p_{X^r})_* \circ (\pi^X_{\infty \to r})_* : \Dmod^*(X) \to \Vect,
$$
Explicitly, if $M \in \Dmod^*(X)$ is represented by the collection $ \{M^r\}_r \in \lim_{r, \pi_*} \Dmod(X^r)$, then $\Gamma_\dr(M) := \Gamma_\dr(X^r, M^r)$, the RHS being independent of $r$.

\medskip


If $i_x: \pt \hookrightarrow X$ is a closed point, the \emph{delta $\fD$-module} at $x$ is given by the usual formula $\delta_{x,X}:=(i_x)_*(\CC) \in \Dmod^*(X)$. In the realization of $\Dmod^*(X)$ as a limit, $\delta_x$ is represented by the collection of $\delta_{x^r,X^r} \in \Dmod(X^r)$, where $x^r$ is the image of $x$ under the projection $X \to X^r$.


\begin{rem}

Contrarily to the finite-type case,  $(i_x)_*$ is not \emph{proper}\footnote{A functor between cocomplete categories is proper if it sends compact objects to compact ones.} as pointed out before, compact $\fD^*$-modules on $X$ are (in particular) $*$-pulled back along some projection $X \to X^r$ and $\delta_{x, X}$ is not such. As a consequence, $(i_x)_*$ does not admit a continuous right adjoint. 
However, $(i_x)_*$ is fully faithful, being the limit of the functors $(i_{x^r})_*$, which are fully faithful by Kashiwara's lemma. 

\end{rem}

%%%More generally, if $f: Y \hookrightarrow X$ is \emph{q-closed} (which means that $f$ can be written as a limit of a filtered family of closed embeddings between schemes of finite type), then $f_*$ is fully faithful.



\sssec{}

Consider again an arbitrary map $f: X \to Y$ of pro-schemes. The left adjoint to $f_*$, denoted $f^*$, is only partially defined. We say that $f^*$ is defined on $M \in \Dmod^*(Y)$ if the functor $\Hom(M, f_*(-))$ is corepresentable (by an object that we denote as $f^*(M)$). 

A sufficient condition for $f^*$ to be defined on all of $\Dmod^*(Y)$ is that each $f^r$ be smooth. Indeed, by (\ref{eqn:ev----f_*}) and the $(\ins, \ev)$ adjunction, $f^*$ is specified by
\begin{equation} \label{eqn:f^*-ins}
(\pi^Y_{\infty \to r})^* \circ (f^r)^* \simeq f^* \circ (\pi^X_{\infty \to r})^*
\end{equation}

In particular, whenever $X^r$ admits a constant sheaf $k_{X^r}$, the functor of de Rham cohomology $p_* : \Dmod^*(X) \to \Vect $ is corepresented by the \emph{constant $\fD^*$-module} 
\begin{equation} \label{eqn:k_X}
k_X := (\pi_{\infty \to r})^*(k_{X^r}),
\end{equation}
%
where the RHS is independent of $r$. 
%If $X = \lim X^r$ is pseudo-contractible, $k_X$ is easily written as a limit: $k_X \simeq \{k_{X^r}\}_r$.

\sssec{} 

Let us now discuss the functor $f^!: \Dmod^!(Y) \to \Dmod^!(X)$, dual to $f_*$. By (\ref{eqn:dual-pairing-limColim}), one readily gets
%
\begin{equation}\label{eqn:f^!}
f^! \circ  (\pi^Y_{\infty \to r})^! = (\pi^X_{\infty \to r})^! \circ (f^r)^!.
\end{equation}
For instance, consider $p: X \to \pt$. Then, $p^!: \Vect \to \Dmod^!(X)$ gives the \emph{dualizing sheaf}, $\omega_X := p^! (\CC)$.
Explicitly,
$$
\omega_X \simeq (\pi_{\infty \to r})^! (\omega_{X^r}),
$$
being clear that the RHS does not depend on $r$.
%If $X = \lim X^r$ is pseudo-contractible, $\omega_X$ is computed naively in the realization of $\Dmod^!(X)$ as a limit: $\omega_X \simeq \{ \omega_{X^r}\}_r$.

\sssec{}

The functor $f_!$, left adjoint to $f^!$, is only partially defined. For instance, here is a typically infinite dimensional phenomenon.

\medskip

If $X$ is infinite dimensional, $(i_x)^!$ does not have a left adjoint. Indeed, the value of the hypothetical left adjoint on $\CC$ would have to be compact, hence of the form $(\pi_{\infty \to r})^! (F)$ for some $r$ and some $F \in \Dmod(X^r)_{\cpt}$. It is easy to see that this causes a contradiction. For simplicity, assume that $\R = (\NN, <)^{op}$ and that all $\pi^!$ are fully faithful. For any $s \to r$, adjunction forces
$$ 
\Hom_{\Dmod^!(X)} \big( (\pi_{\infty \to r})^! (F), (\pi_{\infty \to s})^! (-) \big) \simeq (i_{x^s})^!
$$
as functors from $\Dmod(X^s)$ to $\Vect$. It follows that $(\pi_{s\to r})^! (F) \simeq \delta_{x^s}$, which is absurd if $\pi_{s \to r}$ is of positive dimension.

\begin{rem} \label{rem:p^*-for-indschemes}
A similar logic shows that $(p_\X)^*: \Vect \to \Dmod(\X)$ is not defined whenever $\X$ is a (genuine) ind-scheme of ind-finite type. For instance, $\AA^{\infty, \ind} := \colim_n \AA^n$ does not admit a constant sheaf. This fact is ``Fourier dual" to the non-existence of the $!$-pushforward along $i_0 : \pt \to \AA^{\infty, \pro} := \lim_n \AA^n$.
\end{rem}

\sssec{}

The above example shows that $f_!$ may not be defined even if all $(f^r)_!$ are. Matters simplify for $f= p: X 	\to \pt$, with $X$ pseudo-contractible. Then
$$
(p^r)^! \simeq \big( (\pi_{\infty\to r})^! \big)^R \circ p^!,
$$
by contruction and fully faithfulness of $(\pi_{\infty \to r})^!$. Passing to the left adjoints, we obtain
$$
(p^r)_! \simeq  p_! \circ (\pi_{\infty \to r})^!:
$$
in other words,
$$
p_! (M) \simeq \uscolim{r \in \R^\op} (p^r)_! (M^r)
$$
for $M=\{M^r\}_r \in \Dmod^!(X) = \lim_{(\pi^!)^R} \Dmod(X^r)$,  provided that each $(p^r)_!$ is defined on $M^r$.

\sssec{}

Let $f: X \to Y$ be a map of pro-schemes for which the equivalences $\lambda_X$ and $\lambda_Y$ have been specified (for instance, if $X$ and $Y$ are limit of smooth schemes). We shall occasionally use the \emph{renormalized} push-forward
$$
f_*^\ren: \Dmod^!(X) \to \Dmod^!(Y), \, \, \, f_*^\ren := \lambda_Y^{-1} \circ f_* \circ \lambda_X.
$$


\sssec{}

Suppose that $f: X \to Y$ can be presented as the limit of maps $f^r : X^r \to Y^r$ such that all the squares
%
\begin{gather} \label{diag:square-for-pro-schemes}
\xy
(0,0)*+{ Y^s }="00";
(25,0)*+{  Y^r }="10";
%
%
(00,15)*+{  X^s }="01";
(25,15)*+{  X^r }="11";
%
{\ar@{<-}_{ f^s} "00";"01"};
{\ar@{->}^{\pi} "00";"10"};
%
{\ar@{<-}_{f^r} "10";"11"};
{\ar@{->}^{\pi} "01";"11"};
%
%
\endxy
\end{gather} 
%
are Cartesian (equivalently, $f$ is finitely presented, \cite{Sam:Dmod}). In this case, base-change yields commutative squares
\begin{gather}
\xy
(0,0)*+{ \Dmod(Y^s) }="00";
(25,0)*+{  \Dmod(Y^r) }="10";
%
%
(00,15)*+{  \Dmod(X^s) }="01";
(25,15)*+{  \Dmod(X^r) }="11";
%
{\ar@{->}_{ (f^s)^!} "00";"01"};
{\ar@{->}^{\pi_* } "00";"10"};
%
{\ar@{->}_{(f^r)^! } "10";"11"};
{\ar@{->}^{\pi _*} "01";"11"};
%
%
(50,0)*+{ \Dmod(Y^s) }="20";
(75,0)*+{  \Dmod(Y^r), }="30";
%
%
(50,15)*+{  \Dmod(X^s) }="21";
(75,15)*+{  \Dmod(X^r) }="31";
%
{\ar@{->}_{ (f^s)_*} "21";"20"};
{\ar@{->}^{\pi^! } "30";"20"};
%
{\ar@{<-}_{(f^r)_* } "30";"31"};
{\ar@{<-}^{\pi ^!} "21";"31"};
%
%
\endxy
\end{gather} 
%
%
which allow to define the functors $f^\flip : \Dmod^*(Y) \to \Dmod^*(X)$ and $f_+ : \Dmod^!(X) \to \Dmod^!(Y)$
by the formulas
$$
 (\pi^X_{\infty \to r})_* \circ  f^\flip :=  (f^r)^! \circ  (\pi^Y_{\infty \to r})_*
$$
and
$$
f_+ \circ (\pi^X_{\infty \to r})^! := (\pi^Y_{\infty \to r})^! \circ (f^r)_*.
$$
These functors are easily seen to be independent of the presentations.
Also it is immediate from the finite-type case that the pairs of functors $(f_*, g^\flip)$ and $(f_+, g^!)$ satisfy the base-change formula (see \cite{Sam:Dmod} for a thorough treatment). 

\sssec{} \label{sssec:closed-embedd-pro}

A map $f: X \to Y$ between pro-schemes is said to be a \emph{closed embedding} (resp., \emph{proper}) if it is finitely presented and $f^r: X^r \to Y^r$ is a closed embedding (resp., proper) for some $r \in \R$ (equivalently, assuming that $r$ is final: for all $r \in \R$). For instance, the inclusion of a point into $Y \in \Sch^\pro$ of infinite type is \emph{not} a closed embedding.

\begin{lem} \label{lem:f_bullet}
Let $f: X \to Y$ be finitely presented. The following statements hold true:
\begin{itemize}
\item
if $f$ is proper, then $f_+$ is left adjoint to $f^!$. If moreover $X$ and $Y$ have dimension theories, then 
$$
f_+ \simeq f_*^\ren [2 \dim_f],
$$
where $\dim_f  := \dim (X^r)- \dim(Y^r)$ is clearly well-defined;

\smallskip
\item

if $f$ is a closed embedding, then $f_+$ is fully faithful.
\end{itemize}
\end{lem}

\begin{proof}
Base change along the diagrams (\ref{diag:square-for-pro-schemes}) guarantees that $f_+$ and $f^!$ are compatible with the evaluations for the $!$-categories of $\fD$-modules. All statements follow immediately.
\end{proof}

%%%%%%\begin{cor} \label{cor:weird-left-adjoint}
%%%%%%Let $\iota: Z \hookrightarrow X$ be a closed embedding of pro-schemes, pulled back from an embedding $Z^r \hto X^r$ of \emph{smooth} schemes. In this case, $X$ and $Z$ have canonical dimension theories. Then, 
%%%%%%$$
%%%%%%(\iota_+)^L \simeq \lambda^{-1}_Z \circ \iota^* \circ \lambda_X [2 \on{codim}(X:Z)],
%%%%%%$$
%%%%%%whenever both functors are defined. E.g., they are defined on $\omega_X \in \Dmod^!(X)$.
%%%%%%\end{cor}

\sssec{}

The $1$-category $\Sch^\pro$ admits products. Moreover, for two pro-schemes $X$ and $Y$, there are canonical equivalences
\begin{eqnarray} 
\Dmod^*(X) \otimes \Dmod^*(Y) \xrightarrow{\,\,\, \boxtimes \,\,\,} \Dmod^*(X \times Y) \nonumber \\
\Dmod^!(X) \otimes \Dmod^!(Y) \xrightarrow{\,\,\, \boxtimes \,\,\,} \Dmod^!(X \times Y), \label{eqn:boxtimes_equiv}
\end{eqnarray}
%
which follow at once from dualizability of each $\Dmod(X^r)$ and the fact that $\Dmod^*$ and $\Dmod^!$ can be represented as colimits.

\begin{rem} \label{rem:pro-schemes-fiber-products}

It is easy to see that $\Sch^\pro$ admits fiber products. Indeed, let $X \to Z \leftarrow Y$ be a diagram of pro-schemes.
Let $Z = \lim Z^r$ be a presentation of $Z$. For each $r \in \R$, the composition $X \to Z \twoheadrightarrow Z^r$ factors through a projection $X \twoheadrightarrow X'$, with $X'$ of finite type. We let $X^r := X'$. In this way we contruct compatible pro-scheme presentations of $X$ and $Y$ and $\lim_r (X^r \times_{Z^r} Y^r)$ is a presentation of $X \times_Z Y$,

\end{rem} 

Note that $\Dmod^!(X)$ is symmetric monoidal: indeed, it is a colimit formed along the monoidal functors $(\pi_{s \to r})^!$. By construction, each insertion functor $(\pi_{\infty \to r})^! : \Dmod(X^r) \to \Dmod^!(X)$ is monoidal and, consequently, the tensor product on $\Dmod^!(X)$ is defined as usual: 
$$
\Dmod^!(X) \otimes \Dmod^!(X) \xrightarrow{\, \, \boxtimes \, \, } \Dmod^!(X \times X) \xrightarrow{\, \, \Delta^! \, \, } \Dmod^!(X).
$$


\sssec{}

For any $f: X \to Z$, the functor $f^!: \Dmod^!(Z) \to \Dmod^!(X)$ is symmetric monoidal, hence $(\Dmod^!(Z), \otimes)$ acts on $\Dmod^!(X)$.
%
The relative analogue of  (\ref{eqn:boxtimes_equiv}) holds true as well:
%
\begin{lem} \label{lem:BN}
Let $X \to Z \leftarrow Y$ be a diagram of pro-schemes.
There is a canonical equivalence
\begin{equation}
\Dmod^!(X) \usotimes{\Dmod^!(Z)} \Dmod^!(Y) \simeq \Dmod^!(X \times_Z Y).
\end{equation}
\end{lem}

\begin{proof}
The canonical functor is induced by pullback along $X \times_Z Y \to X \times Y$. To prove it is an equivalence, we reduce it to the finite dimensional case, where it is true by a result of \cite{BN}. By fixing compatible presentations of $X, Y, Z$ and $X \times_Z Y$ as in Remark \ref{rem:pro-schemes-fiber-products}, we obtain
%
$$
\Dmod^!( X \times_Z Y) 
\simeq
 \uscolim{r \in \R^\op} \Dmod(X^r \times_{Z^r} Y^r) 
\simeq
 \uscolim{r \in \R^\op} \Big( \Dmod^!(X^r) \usotimes{\Dmod^!(Z^r)} \Dmod^!(Y^r) \Big) 
 \simeq
\Dmod^!(X) \usotimes{\Dmod^!(Z)} \Dmod^!(Y).
$$
\end{proof}


\sssec{}

By Sect. \ref{sssec:paradigm_dual actions}, there is an induced right action of $\Dmod^!(Z)$ on $\Dmod^*(X)$, which we indicate by $\overset{*!}\otimes$. Since $\Dmod^!(Z)$ is symmetric monoidal, we sometimes consider the latter as a left action: in that case, we use the symbol $\overset{!*} \otimes$.
%
It is easy to check that
$$
(\pi_{\infty \to r})^! (M) \overset{!*} \otimes  (\pi_{\infty \to r})^* (P) = (\pi_{\infty \to r})^* (M \otimes_{X^r} P)
$$
and that
$$
M \overset {!*} \otimes N \simeq \lambda_X (M \otimes \lambda_X^{-1}(N)),
$$
where the RHS is independent of the choice of $\lambda_X$.

\medskip

Both statements follow formally from the functorial equivalence
$$
\pi^!(M) \otimes \pi^*(P) \simeq \pi^*(M \otimes P),
$$
valid for any smooth map $\pi$ between schemes of finite type.
%
%
\begin{lem} \label{lem:proj_formula}
For $f: X \to Y$, the functor $f_*$ is a functor of $\Dmod^!(Y)$-module categories. 
\end{lem}

\begin{proof}
This is an instance of the following general phenomenon. In the setting of Sect. \ref{sssec:paradigm_dual actions}, if a morphism $\alpha: \M \to \N$ is $\A$-linear and $\M, \N$ are dualizable in $\S$, then $\alpha^\vee: \N^\vee \to \M^\vee$ is also $\A$-linear with respect to the dual $\A$-actions.
\end{proof}

\begin{cor} \label{cor:proj_formula_ren}
For $f: X \to Y$ and any choice of dimension theories for $X$ and $Y$, the functor $f_*^\ren : \Dmod^!(X) \to \Dmod^!(Y)$ is $\Dmod^!(Y)$-linear.
\end{cor}

\begin{proof}
The three functors $\lambda_X$, $f_*$ and $(\lambda_Y)^{-1}$ are all $\Dmod^!(Y)$-linear.
\end{proof}

The content of these two results is to give the projection formulas, that is, the equivalences
\begin{equation}\label{eqn:proj_formulas}
f_*(M) \overset{*!} \otimes N \simeq f_*(M \overset{*!} \otimes f^!(N))
\hspace*{.6cm}
f_*^\ren (P) \overset{!} \otimes N \simeq f^\ren_*  \big( P \overset{!} \otimes f^!(N) \big)
\end{equation}
\emph{up to coherent homotopy}.

\begin{rem} \label{rem:eval:compact}
The evaluation pairing (\ref{eqn:duality_pairing}) between $\Dmod^*(X)$ and $\Dmod^!(X)$ can be tautologically rewritten as $\eps_X (M , N) \simeq \Gamma_\dR \big(X, M \overset{*!}\otimes N \big)$.
%%%\begin{equation}\label{eval:compact}
%%%\eps_X (M , N) \simeq \Gamma_\dR(X, M \overset{*!}\otimes N).
%%%\end{equation}
\end{rem}



\ssec{$\fD$-modules on ind-pro-schemes}

Let us now extend the above theory to the set-up of ind-pro-schemes.

\sssec{}

We adopt the convention of \cite{indschemes}: a (classical) \emph{ind-scheme} is a prestack that can be presented as a filtered colimit of quasi-compact and quasi-separated schemes along closed embeddings. A frequently used subcategory of such prestacks is that of ind-schemes of \emph{ind-finite type}: it consists of those ind-schemes formed out of schemes of finite type.

\sssec{}

Before explaining the two theories of $\fD$-modules on ind-pro-schemes, let us recall the definition of the category of $\fD$-modules on ind-schemes of ind-finite type. Let $\IndSch^{\fty}:= \ms{Ind}^{cl}\Sch^{\fty}$ denote the $1$-category of such. The functor
$$
\Dmod: \IndSch^{\fty} \longrightarrow \DGCat
$$
is defined to be the left Kan extension of $\Dmod: \Sch^{\fty} \to \DGCat$ along the inclusion $\Sch^{\fty} \hookrightarrow \IndSch^{\fty}$.
%
Explicitly, if $\Y \in \IndSch^{\fty}$ is written as a filtered colimit $\Y = \colim_{n \in \I} Y_n$, with $Y_n \in \Sch^{\on{ft}}$ and closed embeddings $\iota_{m \to n}: Y_m \hookrightarrow Y_n$, then $\Dmod(\Y) \simeq \colim_{n \in \I} \Dmod(Y_n)$, with respect to the pushforward morphisms $(\iota_{m \to n})_*$.


\sssec{}

We repeat the same process for ind-pro-schemes. Namely, we define the ordinary category $\IndSch^\pro:= \ms{Ind}^{cl}\Sch^\pro$ of \emph{ind-pro-schemes} to be the one comprising ind-schemes that can be formed as colimits of pro-schemes under closed embeddings.\footnote{The definition of closed embedding of pro-schemes is given in Sect. \ref{sssec:closed-embedd-pro}.}
By \emph{ind-pro-group}, we mean a group object in the category $\IndSch^\pro$.

\medskip

The functor
$$
\Dmod^* : \IndSch^{\pro} \to \DGCat
$$
is defined as the left Kan extension of $\Dmod^*: \Sch^\pro \to \DGCat$ along the inclusion $\Sch^\pro \hookrightarrow \IndSch^\pro$.
Analogously, the functor
$$
\Dmod^! : (\IndSch^\pro)^\op \to \DGCat
$$
is defined to be the right Kan extension of $\Dmod^!: (\Sch^\pro)^\op \to \DGCat$ along the inclusion $(\Sch^\pro)^\op \hookrightarrow (\IndSch^\pro)^\op$. 

\medskip

For $f: X \to Y$ a map in $\IndSch^\pro$, we continue to denote by $f_*: \Dmod^*(X) \to \Dmod^*(Y)$ and $f^!: \Dmod^!(Y) \to \Dmod^!(X)$ the induced functors.

\begin{lem}
The duality $\Dmod^! \simeq (\Dmod^*)^\vee$ and the isomorphism $(f_*)^\vee \simeq f^!$ continue to hold for ind-pro-schemes.
\end{lem}

\begin{proof}
Let $\Y = \colim_{n \in \I} Y_n$ be a presentation of $\Y$ as an ind-pro-scheme. The two categories $\Dmod^!(\Y)$ and $\Dmod^*(\Y)$ are, by construction,
$$
\Dmod^!(\Y) \simeq \lim_{n \in \I^\op, \iota^!} \Dmod^!(Y_n), \hspace{1 cm}
\Dmod^*(\Y) \simeq \uscolim{n \in \I, \iota_*} \Dmod^*(Y_n).
$$
They are evidently dual to each other, thanks to the validity of the present lemma for pro-schemes. The duality $(f_*)^\vee \simeq f^!$ is a formal consequence of this.
\end{proof}

\begin{prop} \label{prop_D^!-symm-monoidal}
The ordinary category $\IndSch^\pro$ is symmetric monoidal via Cartesian product and the functor $\Dmod^!: (\IndSch^\pro)^\op \to \DGCat$ is symmetric monoidal.
\end{prop}

\begin{proof}
The first assertion is obvious. 
As for the second, recall that $\Dmod^!$ is symmetric monoidal as a functor $(\Sch^\pro)^\op \to \DGCat$. Furthermore, for any map of pro-schemes $f: X \to Y$, the pull-back $f^! : \Dmod^!(Y) \to \Dmod^!(X)$ is symmetric monoidal. The combination of these two facts yields the assertion.
\end{proof}

The following result is a generalization of Lemma \ref{lem:BN}. The proof is completely analogous.

\begin{lem} \label{lem:BN-ind}
Let $\X \to \Z \leftarrow \Y$ be a diagram of pro-schemes.
There is a canonical equivalence
\begin{equation}
\Dmod^!(\X) \usotimes{\Dmod^!(\Z)} \Dmod^!(\Y) \simeq \Dmod^!(\X \times_\Z \Y).
\end{equation}
\end{lem}

\sssec{}

Let $\Y = \colim_{n \in \I} Y_n$ an ind-pro-scheme.
By Lemma \ref{lem:f_bullet} and Sect. \ref{sssec:fromLimToColim}, there is an equivalence
$$
\Omega : \Dmod^!(Y) \simeq \uscolim{n \in \I, \iota_+} \Dmod^!(Y_n).
$$
Assume that each $Y_n$ is equipped with a dimension theory and the corresponding self-duality $\lambda_n : \Dmod^!(Y_n) \to \Dmod^*(Y_n)$. For any arrow $m \to n$ in $\I$, the cited lemma yields the commutative diagram
%
\begin{gather}
\xy
(0,0)*+{ \Dmod^!(Y_m)}="00";
(40,0)*+{  \Dmod^!(Y_n). }="10";
%
%
(00,15)*+{   \Dmod^*(Y_m) }="01";
(40,15)*+{  \Dmod^*(Y_n)  }="11";
%
{\ar@{<-}_{\lambda_m} "01";"00"};
{\ar@{->}^{\iota_+} "00";"10"};
%
{\ar@{<-}_{\lambda_n[2 \dim_{\iota}]} "11";"10"};
{\ar@{->}^{\iota_*} "01";"11"};
%
%
\endxy
\end{gather}
%
A dimension theory for $\Y$ (or, trivialization of the dimension torsor of $\Y$) is an assignment $Y_n \rightsquigarrow \dim(Y_n) \in \ZZ$ such that $\dim(\iota_{m \to n}) = \dim(Y_m) - \dim(Y_n)$ for any arrow $m \to n$.

For instance, the dimension theory specified by $\dim(Y_m) = 0$ gives rise to a self-duality $ \Lambda_{Y_m} = \Lambda_m : \Dmod^!(\Y) \to \Dmod^*(\Y)$ given by the formula
$$
\Lambda_n \circ 
(i_{n\to \infty})_+
 = (i_{n\to \infty})_* \circ \lambda_n [2 \dim(Y_n)], 
\hspace*{0.4cm} 
\text{ for }
n \in \I_{m/}.
$$
In the formula above, $(i_{n\to \infty})_+$ and $(i_{n\to \infty})_*$ indicate the insertion functors $\Dmod(Y^n) \to \Dmod^!(\Y)$ and $\Dmod(Y^n) \to \Dmod^*(\Y)$.

\begin{rem}
Any dimension theory $\Lambda$ produces a canonical object $\Lambda(\omega_\Y)$ of $\Dmod^*(\Y)$. Assuming each $Y_n$ is a limit of smooth schemes (so that $Y_n$ is equipped with the canonical dimenstion theory), we obtain
$$
\Lambda_m(\omega_\Y) \simeq \uscolim{n \in \I_{m/ }} (i_{n\to \infty})_* k_{Y^n} [2 \dim(Y_n)] \in \Dmod^*(\Y) .
$$
\end{rem}


\sssec{}

For any map $f: \X \to \Y$ of ind-pro-schemes, $\Dmod^!(\Y)$ acts on $\Dmod^!(\X)$ by pull-back. We denote the dual action by the usual symbol:
$$
\overset{!*} \otimes : \Dmod^!(\Y) \otimes \Dmod^*(\X) \to \Dmod^*(\X).
$$
As in the pro-scheme case, this action is the one induced by the self-duality $\Lambda: \Dmod^!(\X) \to \Dmod^*(\X)$ (for any choice of $\Lambda$). Consequently, we have the ``projection formulas":

\begin{prop} \label{prop:proj-formula-indschemes}
For $f: \X \to \Y$ as above, the functors $f_*$ and $f_*^\ren := \Lambda^{\Y} \circ f_* \circ (\Lambda^{\X})^{-1}$ are $\Dmod^!(\Y)$-linear.
\end{prop}

\begin{rem} %%%ciao
The evaluation pairing between $\Dmod^*(\X)$ and $\Dmod^!(\X)$ is
\begin{equation} \label{eqn:eval-compact}
\eps_\X (M,N) \simeq p_* \big(M \overset{*!}\otimes N \big) \simeq  p_*^\ren \big(\Lambda^{-1}(M) \otimes N \big).
\end{equation}
\end{rem}


\ssec{$\fD$-modules on $G \ppart$ and $N \ppart$}


Finally, let us take up the case of loop groups. The following construction is well-known:

\begin{lem}
For any affine group $G$ of finite type, $\bG := G \ppart$ is an ind-pro-scheme.
\end{lem}

\begin{proof}
Consider the (schematic) quotient map to the affine Grassmannian $\q: \bG \to \Gr$ and choose a presentation of $\Gr$ as an ind-scheme of ind-finite type: $\Gr \simeq \colim_{n, \iota} Z_n$. 
Pulling-back each $Z_n$ along $\q$, we obtain an ind-scheme presentation of $\bG$:
$$
\bG \simeq \uscolim{n \in \NN} \q^{-1}(Z_n).
$$
Of course, each $\q^{-1}(Z_n)=Z_n \underset{\Gr}\times \bG$ is of infinite type. However, as $\q$ factors through $\bG \to \bG^r$ for any $r \in \NN$, we can write:
$$
\q^{-1}(Z_n)  \simeq \lim_{r \in \NN} \big(  Z_n \underset{\Gr}\times \bG/\bG^r \big),
$$
where the limit is taken along the maps induced by the projections $\pi_{s \to r}: \bG/ \bG^s \to \bG/\bG^r$. This is a presentation of $\q^{-1}(Z_n)$ as a pro-scheme, and the proof in concluded.
\end{proof}

We now justify the definitions given in the introduction.

\begin{lem} \label{lem:DmodG}
$$
\Dmod^*(\bG) \simeq \uscolim{r, \pi^*} \Dmod(\bG/\bG^r), \, \, \, \, \, \, \, \Dmod^!(\bG) \simeq \uscolim{r, \pi^!} \Dmod(\bG/\bG^r).
$$
\end{lem}

\begin{proof}
We only prove the first formula, the proof of the second being completely analogous. 
Let $Z_n^r :=   Z_n \underset{\Gr}\times \bG/\bG^r$, so that 
$$
\bG = \uscolim{n,\iota} \lim_{r, \pi} Z_n^r \, \, \hspace{.5cm} \, \, \bG/\bG^r = \colim_{n,\iota} Z_n^r
$$
are presentations of $\bG$ and $\bG/\bG^r$ as an ind-pro-scheme and an ind-scheme, respectively. 
For each $n$ and $r$, consider the evidently Cartesian square:
%
\begin{gather}
\xy
(0,0)*+{ Z_{n+1} \times_{\Gr} \bG/ \bG^{r+1}}="00";
(40,0)*+{  Z_{n+1} \times_{\Gr} \bG/ \bG^{r}. }="10";
%
%
(00,15)*+{   Z_{n} \times_{\Gr} \bG/ \bG^{r+1} }="01";
(40,15)*+{  Z_{n} \times_{\Gr} \bG/ \bG^{r}  }="11";
%
{\ar@{<-}^{\iota} "00";"01"};
{\ar@{->}^{\pi} "00";"10"};
%
{\ar@{<-}^{\iota} "10";"11"};
{\ar@{->}^{\pi} "01";"11"};
%
%
\endxy
\end{gather}
%
Consequently, the category of $\fD^*$-modules on $\bG$ is expressed as follows:
$$
\Dmod^*(\bG) 
\simeq  \uscolim{n, \iota_*}  \lim_{r, \pi_*} \Dmod(Z_n^r) 
\simeq  \uscolim{n, \iota_*}  \uscolim{r, \pi^*} \Dmod(Z_n^r)
\simeq \uscolim{r, \pi^*} \uscolim{n, \iota_*} \Dmod(Z_n^r)
\simeq \uscolim{r, \pi^*} \Dmod(\bG/\bG^r), 
$$
where the switch of colimits in the third equivalence is a consequence of base-change along the above square.
%
\end{proof}

%%%%%%OBSOLETE
%%%%%%%%%%%%%%\begin{lem}
%%%%%%%%%%%%%%\begin{equation} \label{eqn:DmodG_lim}
%%%%%%%%%%%%%%\Dmod^*(\bG) \simeq \lim_{r, \pi_*} \Dmod(\bG/\bG^r).
%%%%%%%%%%%%%%\end{equation}
%%%%%%%%%%%%%%Moreover, $\Dmod^*(\bG)$ and $\Dmod^!(\bG)$ are compactly generated and dual to each other.
%%%%%%%%%%%%%%\end{lem}
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%\begin{proof}
%%%%%%%%%%%%%%The formula follows at once from the first formula of Lemma \ref{lem:DmodG}. The second assertion was proven before.
%%%%%%%%%%%%%%\end{proof}


\sssec{}

We now prove that $\Dmod^*(\bG)$ has a convolution monoidal structure. In the next section, we will use this result to define categorical $\bG$-actions. 

\begin{lem} \label{lem:D^*(G)-monoidal}
Let $\G$ be a group ind-scheme of pro-finite type.
The functor $m_*: \Dmod^*(\G \times \G) \to \Dmod^*(\G)$, together with the equivalence
\begin{equation}
\Dmod^*(\G) \otimes \Dmod^*(\G) 
 \xrightarrow{\, \, \boxtimes \, \, \,} \Dmod^* (\G \times \G),
\end{equation}
endows $\Dmod^*(\G)$ with a monoidal structure.
\end{lem}

\begin{proof}
The functor $\Dmod^! : (\IndSch^\pro)^\op \to \DGCat$ is contravariant and, by Proposition \ref{prop_D^!-symm-monoidal}, symmetric monoidal. Hence, it sends algebras in $\IndSch^\pro$ to comonoidal categories: in particular, $\big( \Dmod^!(\G), m^! \big)$ is comonoidal. By duality, we obtain the required statement.
\end{proof}

\sssec{}

Unlike $\bG$, the ind-pro scheme $\bN := N \ppart$ is exhausted by its compact open subgroups, hence it is an ind-object in the category of pro-unipotent group schemes. We can choose a presentation $\bN \simeq \colim_k \bN_k$ indexed by the natural numbers. For future use, note that each $\bN_k$ has a canonical dimension theory.


\section{Strong actions by ind-pro-groups}\label{SEC:grp-actions}

With the theory of $\fD$-modules on ind-pro-schemes in place, we may define strong actions of ind-pro-groups on categories. As in the finite dimensional case, we discuss the (twisted) invariant and coinvariants categories and the natural functors relating them. For a pro-group $H$ with a character $\m: H \to \GG_a$, the $(H,\m)$-invariant and $(H,\m)$-coinvariant categories are naturally equivalent via a natural functor that we call $\theta_{H,\m}$ (see Theorem \ref{thm:theta_mu}).

This will be of fundamental importance for the study of Whittaker categories and for the proof of our main theorem. Indeed, approximating $\bN$ by its pro-unipotent subgroups and using the corresponding $\theta$'s, we construct a functor $\Theta_{\bN,\chi}$ (depending on a trivialization of the dimension torsor of $\bN$), which ought to realize an equivalence between $\C_{\bN, \chi}$ and $\C^{\bN, \chi}$.

\ssec{The main definitions}

Let $\G$ be an ind-pro-group. For instance, $\G = G \ppart$ or $\G = N \ppart$ where $G$ is a reductive group and $N$ its maximal unipotent subgroup.
By definition, $\C \in \DGCat$ is acted on by $\G$ if it is endowed with an action of the monoidal category $(\Dmod^*(\G), m_*)$ (see Lemma \ref{lem:D^*(G)-monoidal}). The totality of categories with $\G$ action forms an $\infty$-category, denoted by $\G \on{-} \mathbf{rep}$. 
As in the finite-type case, $\G \on{-} \mathbf{rep}$ is monoidal: indeed, $\Dmod^*(\G)$ is Hopf with multiplication $\star := m_*$ and comultiplication $\Delta_*$. By duality, $\Dmod^!(\G)$ is also a Hopf algebra, with multiplication $\Delta^!$ and comultiplication $m^!$.


\sssec{} \label{sssec:coin-inv-simplicial-constructions}

Since the map $p: \G \to \pt$ is $\G$-equivariant, the pushforward $p_*$ gives a $\G$-action on $\Vect$, the so called trivial action. This allows us to define invariant and coinvariants, as in the finite dimensional case.

\medskip

The coinvariant category
$$
\C_{\G} := \Vect \usotimes{\Dmod^*(\G)} \C.
$$
is computed by  the Bar resolution of the relative tensor product, i.e. the simplicial category
%
\begin{equation} \label{eqn:simplicial_cat}
\cdots \,\, \Dmod^*(\G) \otimes \Dmod^*(\G) \otimes \C \rrr \Dmod^*(\G) \otimes \C \rr \C,
\end{equation}
%
where the maps are given by action, multiplication and trivial action, according to the usual pattern. To show this, one uses the ``free" resolution of $\Vect$ as a right $\Dmod^*(\G)$-module and invokes the fact that relative tensor product commutes with colimits in each variable. As in the finite-type context, there is a tautological map
$$
\pr_\G := (p_*)_\C : \C \to \C_\G.
$$


\sssec{}

Analogously, we define the invariant category
$$
\C^{\G} := \Hom_{\Dmod^*(\G)} ( \Vect,  \C ).
$$
It can be computed by the totalization of 
$$
\C \rr \Hom(\Dmod^*(\G), \C) \rrr \Hom (\Dmod^*(\G) \otimes \Dmod^*(\G), \C) \,\, \cdots.
$$
%
Moreover, as $\Dmod^*(\G)$ and $\Dmod^!(\G)$ are in duality, the latter becomes
\begin{equation} \label{eqn:cosimplicial_cat}
\C \rr \Dmod^!(\G) \otimes \C \rrr  \Dmod^!(\G) \otimes \Dmod^!(\G) \otimes \C \, \,\cdots .
\end{equation}
We have the conservative tautological map
$$
\oblv^\G := (p_*)^\C : \C^\G \to \C.
$$
However, its right adjoint $\Av_*^\G$ (defined for abstract reasons) may not be continuous whenever $\G$ is an ind-scheme. Indeed, $p_*: \Dmod^*(\G) \to \Vect$ may not admit a left adjoint in that case (see Remark \ref{rem:p^*-for-indschemes}).
%
We also denote by $\Av_!^\G: \C \to \C^H$ the \emph{partially defined} left adjoint to $\oblv^\G$. 


\sssec{Example: $\bN$-actions} \label{sssec:N-actions}
 
Let  $\G = \bN$. In view of the extra property of $\bN$ of being a colimit of groups, an $\bN$-action on $\C$ corresponds to a family of compatible actions of $(\Dmod^*(\bN_k),\star)$ on $\C$. 
It follows that
$$
\C_{\bN} \simeq \uscolim{k \in \NN} \C_{\bN_k},
$$
where the functors in the directed system 
$$
\Vect \usotimes{\Dmod^*(\bN_k)} \C \to \Vect \usotimes{\Dmod^*(\bN_{k+1})} \C
$$
come from the push-forwards $i_*: \Dmod^*(\bN_k) \to \Dmod^*(\bN_{k+1})$.
%%
Likewise, 
$$
\C^{\bN} \simeq \lim_{k \in \NN} \C^{\bN_k},
$$
the limit being along the forgetful functors
$$
\Hom_{\Dmod^*(\bN_{k+1})} (\Vect, \C) \to \Hom_{\Dmod^*(\bN_{k})} (\Vect, \C).
$$
See Proposition \ref{prop:N-inv_vs_Nk-inv} below for a more explicit description of the transition maps in both cases.

\sssec{Actions on $\Vect$}

As in the finite-type case, a strong action of $\G$ on $\Vect$ consists of a comonoidal functor $\Vect \to \Dmod^!(\G)$. The latter is equivalent to specifying a character $\fD^!$-module on $\G$. For such $\F \in \Dmod^!(\G)$, the action map $\Dmod^*(\G) \otimes \Vect \to \Vect$ is 
$$
M \otimes V \mapsto M \star_\F V := \eps_{\G} (M \otimes \F) \otimes V,
$$
where, $\eps_\G = \Gamma_\dR \big( - \overset {*!} \otimes - \big)$ is the duality pairing between $\Dmod^*(\G)$ and $\Dmod^!(\G) := \Dmod^*(\G)^\vee$.

\medskip

Let $\mu: \G \to \GG_a$ be any additive character. By the (obvious) ind-pro version of Lemma \ref{lem:char_pull-back}, $\mu^! (exp) \in \Dmod^!(\G)$ is a character $\fD^!$-module.
We write $\Vect_{\mu}$ to emphasize that $\Vect$ is being considered as a category with the $\Dmod^!(\G)$-coaction corresponding to $\mu^!(\exp)$. 

\sssec{}

The automorphism $\Tw_\m$ of $\G \rrep$ is defined as in (\ref{eqn:twist}). Tautologically, the action of $\Dmod^*(\G)$ on $\C \otimes \Vect_\mu$ consists of the composition of the ``old" action of $\Dmod^*(\G)$ on $\C$ with the monoidal automorphism
$$
\Dmod^*(\G) \to \Dmod^*(\G), 
\hspace*{.4cm} 
M \mapsto \mu^!(exp) \overset {!*} \otimes M.
$$
%
For $\C \in \G \rrep$, we define the $(\G, \mu)$-invariant and $(\G, \mu)$-coinvariant categories as
$$
\C^{\G, \mu} := \Hom_{\Dmod^*(\G)} (\Vect_\m, \C),
\hspace{.6cm}
\C_{\G,\mu} := \Vect_{\m} \usotimes{\Dmod^*(\G)} \C.
$$
Lemma \ref{lem:Whit_with_dualiz_char} and the definitions of $\oblv$, $\pr$, $\Av$ generalize verbatim from the finite dimensional case.

\ssec{Actions by group pro-schemes}

The formulas of Example \ref{sssec:N-actions} show that, in order to understand $\bN$-actions on categories, one should first discuss $\bN_k$-actions. In this section we study categorical actions by an arbitrary group pro-scheme $H$.

\sssec{}

Since $H$ can be realized as a limit of smooth schemes, the usual adjunction $p^*: \Vect \rightleftarrows \Dmod^*(H): p_*$ is available.

\begin{prop} \label{prop:p^*-equiv-pro}
The functor $p^*: \Vect \to \Dmod^*(H)$ is $H$-equivariant.
\end{prop}

\begin{proof}
It suffices to prove that the diagram 
%
\begin{gather}
\xy
(40,0)*+{  \Vect }="00";
(0,0)*+{ \Dmod^*(H) \otimes \Vect }="10";
%
(40,15)*+{ \Dmod^*(H)  }="01";
(00,15)*+{ \Dmod^*(H) \otimes \Dmod^*(H) }="11";
%
{\ar@{->}_{ p^* } "00";"01"};
{\ar@{<-}_{  \Gamma_\dR  } "00";"10"};
%
{\ar@{<-}_{\,\,\,m_* } "01";"11"};
{\ar@{->}_{ \id \otimes p^*} "10";"11"};
%
%
\endxy
\end{gather} 
commutes. By means of the automorphism $\xi^*$ of $\Dmod^*(H \times H)$ (see Lemma \ref{lem:p^*-equivariant}), the top arrow can be converted into $p_* \otimes \id$ and commutativity is obvious.
\end{proof}

Using the same logic as in the finite-type case, we obtain:

\begin{itemize}
\item
the functor $\Av_*^H \simeq (p^*)^\C : \C \to \C^H$, right adjoint to $\oblv^H$, is continuous;
\medskip 
\item
the projection $\pr_H : \C \to \C_H$ admits a left adjoint, $(\pr_H)^L$;
\medskip
\item
the composition $\oblv^H \circ \Av_*^H \simeq k_H \star -$ factors through a functor $\theta_H : \C_H \to \C^H$. We shall prove later (Theorem {\ref{thm:theta_mu}}) that this functor is an equivalence.
\end{itemize}

\sssec{}

Let $\mu: H \to \GG_a$ be a character. The composition
$$
\oblv^{H,\m} \circ \Av_*^{H,\m}: \C 
\xto{\Tw_{-\m}}
\C \otimes \Vect_{-\m} 
\xto{k_H \star - }
\C \otimes \Vect_{-\m} 
\xto{\Tw_{\m}}
\C
$$
is immediately seen to be isomorphic to the functor
\begin{equation} \label{eqn:mumu}
\oblv^{H,\mu}\circ \Av_*^{H,\mu} \simeq
\big( (-\mu)^!(exp) \overset{!*}\otimes k_H \big) \star -: \C \to \C
\end{equation}
%
and it descends to $\theta_{H,\m}: \C_{H,\m} \to \C^{H,\m}$.
To shorten the notation, we set 
\begin{equation} \label{not:short}
(-\mu)_H :=  (-\mu)^!(exp) \overset{!*}\otimes k_H.
\end{equation}
%
%%%%%%%%Consider the obvious adjunction:
%%%%%%%\begin{equation}\label{eqn:linear-adjoints-char}
%%%%%%%\mu^! (exp) \overset {!*} \otimes p^*: \Vect_{-\mu} \rightleftarrows \Dmod^*(H) :  
%%%%%%%p_* ((-\mu)^! (exp) \overset {!*} \otimes - ).
%%%%%%%\end{equation}
%%%%%%%As an easy consequence of Proposition \ref{prop:p^*-equiv-pro} and of the multiplicative nature of $\mu^!(exp)$, both functors are $\Dmod^*(H)$-linear. Hence, they yield adjunction $\C^{H, \mu} \rightleftarrows \C$. 
%%%%%
%%%%%\begin{lem}
%%%%%The functor $\C^{H, \mu} \to \C$ induced by $p_* ((-\mu)^! (exp) \overset {!*} \otimes - ): \Dmod^*(H) \to \Vect_{-\mu}$ coincides with $\oblv^{H, \mu}$. Analogously, the functor $\C \to \C_{H, \mu}$ coincides with $\pr_{H,\mu}$. The latter admits a left adjoint.
%%%%%\end{lem}
%%%%%
%%%%%\begin{proof}
%%%%%The last claim is manifest.
%%%%%As for the rest of the lemma, unraveling the definitions, it boils down to proving $\mu^!(exp) \overset{!*}\otimes \delta_{1, H} \simeq \delta_{1,H}$, which is a routine computation.
%%%%%\end{proof}
%%%
%%%%%%We will denote the (continuous) right adjoint to $\oblv^{H,\mu}$ by $\Av_*^{H,\mu}$. Notice that
%%%%%%\begin{equation} \label{eqn:mumu}
%%%%%%\oblv^{H,\mu}\circ \Av_*^{H,\mu} \simeq
%%%%%%(\mu^!(exp) \overset{!*}\otimes k_H) \star -: \C \to \C.
%%%%%%\end{equation}
%%%%%%%
%%%%%%Indeed, this is the value of the comonad of the adjunction (\ref{eqn:linear-adjoints-char}) applied to $\delta_{1,H}$. 
%%%%%%As in the untwisted case, the composition (\ref{eqn:mumu}) descends to $\theta_{H, \mu}: \C_{H, \mu} \to \C^{H, \mu}$.
%
\begin{thm} \label{thm:theta_mu}
The above functor $\theta_{H, \mu}: \C_{H, \mu} \to \C^{H, \mu}$ is an equivalence.
\end{thm}

\begin{proof}
Recall the Levi decomposition $H \simeq H^r \ltimes H^u$, where $H^r$ is the maximal reductive quotient of $H$ and $H^u$ is pro-unipotent. The assertion follows from Corollary \ref{cor:theta-mu-fin-type} and Theorem \ref{thm:inv=coinv} below, together with the discussion of Sect. \ref{ssec:semi-direct} regarding semi-direct products.
\end{proof}


\ssec{Actions by pro-unipotent group schemes}

Let us discuss the pro-unipotent case in greater detail. Let $\mu: H \to \GG_a$ be a character.

\begin{cor}
If $H$ is pro-unipotent, $\oblv^{H,\mu}$ and $(\pr_{H,\mu})^L$ are fully faithful.
\end{cor}

\begin{proof}
By changing $\C$ with $\C \otimes \Vect_{-\m}$, it suffices to prove the claim for $\mu =0$. 
Both assertions follow from the cohomological contractibility of $H$: indeed, $p_* \circ p^* \simeq \Gamma_\dR(H, k_H) \simeq \CC$.
\end{proof}
%
%
Thus, for pro-unipotent $H$, we often regard $\C^{H,\mu}$ as a subcategory of $\C$ and $\Av_*^{H,\mu}$ as an endofunctor of $\C$. In the pro-unipotent case, the fact that $\theta_{H,\m}$ is an equivalence is very easy:

\begin{thm} \label{thm:inv=coinv}
Let $H$ be a pro-unipotent group. 
The functors 
$$
\theta_{H,\m} : \C_{H,\m} \rightleftarrows \C^{H,\m}: \pr_{H,\m} \circ \oblv^{H,\m}
$$
are mutually inverse equivalences. In particular, the operation $\C \rightsquigarrow \C^{H,\m}$ commutes with colimits and tensor products by categories.
\end{thm}

\begin{proof}
Without loss of generality, $\m =0$.
We prove that both compositions are naturally isomorphic to the identity functors.
On one hand:
$$
\pr_H \circ \oblv^H \circ \theta_H  (\pr_H (c)) \simeq 
\pr_H ( \Av_*^H (c)) 
\xrightarrow{\, \, \simeq \, \,}
\pr_H (\Gamma_\dR (H, k_H) \otimes c) \simeq \pr_H(c),
$$
for $\pr_H$ coequalizes the given $H$-action and the trivial $H$-action.
On the other hand:
$$
\theta_H \circ \pr_H \circ \oblv^H (c) \simeq
\Av_*^H \circ \oblv^H (c) 
\xrightarrow{\, \, \simeq \, \,}
c,
$$
for $\oblv^H$ is fully faithful.
\end{proof}

\begin{rem} \label{rem:Av_*-twisted-in-prounip-case}
Let $H$ be pro-unipotent and $\lambda_H$ the self-duality associated to the canonical dimension theory of $H$. Then, $\lambda_H^{-1}(k_H) \simeq \omega_H$ and
\begin{equation} \label{eqn:oblvAv-twisted-unip}
\oblv^{H,\m} \circ \Av_*^{H,\m} \simeq  \lambda_H ((-\m^!) exp)  \star -: \C \to \C.
\end{equation}
%
The notation introduced in (\ref{not:short}) simplifies to $(-\mu)_H := \lambda_H ((-\m)^!exp)$ in the pro-unipotent case.
\end{rem}

\sssec{}

Let $S \to H$ be an inclusion of pro-unipotent pro-groups. We wish to define functors $\C^H \rightleftarrows \C^S$, for any $\C \in H \rrep$. To this end, note that $p_* :\Dmod^*(H) \to \Vect$ factors through $\Dmod^*(H)_S$ and gives rise to an adjunction
$$
\pr_S \circ p^* : \Vect \rightleftarrows \Dmod^*(H)_S : p_*,
$$
which is $\Dmod^*(H)$-linear. This induces a pair of adjoint functors
$$
\oblv^{H \to S} :\C^H \rightleftarrows  \Hom_{\Dmod^*(H)} \big( \Dmod^*(H)_S, \C \big) \simeq \C^S : \Av_*^{S \to H}.
$$
When no confusion is likely, we indicate these by $\oblv^{\rel}$ and $\Av_*^{\rel}$, respectively. By contractibility, the composition $\oblv^{\rel} \circ \Av_*^{\rel}$ is given by convolution with $k_{H}$. By changing $\C$ with $\C \otimes \Vect_{-\m}$, these constructions generalize immediately to the $\mu$-twisted case.

\sssec{}

Let us prove two technical results to be used in later chapters.

\begin{lem} \label{lem:auxiliary}
Let $H$ be a pro-unipotent group and $S \hto H$ a normal subgroup. Then, $H/S$ acts on $\C^S$ and
$\C^H \simeq (\C^{S})^{H/S}$. 
\end{lem}

\begin{proof}
It suffices to prove that  $\Dmod^*(H)^{S} \simeq \Dmod(H/S)$ and then repeat the argument of Lemma \ref{lem:normal-inv}.
We may choose a presentation $H \simeq \lim_{r \in \R^\op} H/H^r$, where each $H^r$ is normal in $H$ and is contained in $S$.
Thus, $S$ acts on $H/H^r$ via the projection $S \twoheadrightarrow S/H^r$. We compute
$$
\Dmod^*(H)^{S} \simeq \lim_{r \in \R^\op} \, \Dmod(H/H^r)^{S} \simeq \lim_{r \in \R^\op} \, \Dmod  \Big( \frac{H/H^r}{S/H^r} \Big).
$$
The RHS is the limit of the constant family equal to $\Dmod(H/S)$.
\end{proof}


\sssec{}


Let $\I$ be an indexing category and $\{H_i\}_{i \in \I}$ an inverse family of pro-unipotent groups, where the transition maps are closed embeddings. Denote by $H_0$ the final object of this family and $H_{\infty} := \lim_i H_i$: this is the intersection of all the $H_i$'s. Let $\m$ be a character on $H_0$.

\begin{prop} \label{prop:smooth-generation}
If $\C$ is acted on by $H_0$, then
\begin{equation} \label{eqn:smooth-generation}
\C^{H_\infty,\m} \simeq \uscolim{i \in \I^\op, \oblv^{\rel}} \C^{H_i,\m}.
\end{equation}
\end{prop}

\begin{proof}

Without loss of generality, we may assume that $\mu =0$ and that $H_\infty = \{1\}$. We shall construct a pair of inverse functors $\C \rightleftarrows \colim_i \C^{H_i}$.
By Sect. \ref{sssec:fromLimToColim}, we have
$$
\uscolim{\I^\op, \oblv} \C^{H_i} \simeq \lim_{\I, \Av_*^{\rel}} \C^{H_i};
$$
we indicate by $\ins_i$ and $\ev_i$ the structure functors, as usual.
We define a functor $\alpha: \colim_i \C^{H_i} \to \C$ by imposing the equality $\alpha \circ \, \ins_i \simeq \oblv^{H_i}$. Then, its right adjoint $\beta:= \alpha^R$ satisfies the relation $\ev_i \circ \beta = \Av^{H_i}_*$. Equivalently, $\beta = \colim_i \ins_i \circ \Av_*^{H_i}$.

\medskip

Obviously, the composition $\beta \circ \alpha$ is the identity, by pro-unipotence of each $H_i$. It remains to show that $\alpha \circ \beta \simeq \id_\C$, which is equivalent to exhibiting a natural equivalence
$$
\delta_{1, H_0} \simeq \uscolim{i \in \I} \big(  (\iota_{H_i \to H_0})_* k_{H_i} \big).
$$
Note that $H_0 \simeq \lim_i H_0/H_i$ is a pro-scheme presentation of $H$, so that
$$
\delta_{1,H_0} \simeq \uscolim{i \in \I} (\pi_{\infty \to i})^* (\delta_{1, Q^i}),
$$
where $Q^i := H_0/H_i$.
We will show that $(\pi_{\infty \to i})^* (\delta_{1, Q^i}) \simeq (\iota_{H_i \to H_0})_* k_{H_i}$, for any $i \in \I$. By pseudo-contractibility,
$$
(\pi_{\infty \to i})^* (\delta_{1, Q^i}) \xrightarrow{ \Omega^{-1}} \big\{ (\pi_{j \to i})^* (\delta_{1, Q^i}) \big\}_{j \in \I_{/i}},
$$
%
and, by base-change,
\begin{equation} \label{eqn:aux}
(\pi_{\infty \to i})^* (\delta_{1, Q^i}) \simeq \{ k_{\on{fib}(Q^j \to Q^i)} \}_{j \in \I_{/i}}.
\end{equation}
%
Here, $\on{fib}(Q^j \to Q^i)$ is the fiber of the projection $Q^j \to Q^i$ over $1 \in Q^i$, which is of course $H^i /H^j$. Thus, (\ref{eqn:aux}) coincides with $k_{H_i}$, by the very construction of the latter.
\end{proof}




\ssec{Invariants and coinvariants with respect to $N \ppart$}

We focus now on categories with a (twisted) action of $\bN$ and study their (co)invariants in terms of the invariants for the sequence of $\bN_k$. 

\sssec{}

Recall that $\bN \simeq \colim_k \bN_k$ and, for each $k \in \NN$, let $i_k: \bN_k \to \bN_{k+1}$ the inclusion. 

\begin{prop} \label{prop:N-inv_vs_Nk-inv}
There are natural equivalences
$$
\C^{\bN} := \lim_{\oblv^{\rel}} \C^{\bN_k} 
\, \, \, \text{ and }\, \, \,
\C_{\bN} := \uscolim{\Av_*^{\rel}} \, \C^{\bN_k}.
$$
\end{prop}

\begin{proof}
Let us treat coinvariants first. As each functor $(i_k)_*: (\Dmod^*(\bN_k),\star) \to (\Dmod^*(\bN_{k+1}), \star)$ is monoidal, the equivalence $\Dmod^*(\bN) =\colim_{i_*} \Dmod^*(\bN_k)$ is an equivalence of \emph{monoidal} categories. Hence, we can commute the colimit under the tensor product:
$$
\C_{\bN} := \Vect \usotimes{\Dmod(\bN)} \C \simeq \Vect \usotimes{\uscolim{k,i_*} \Dmod(\bN_k)} \C
\simeq \uscolim{k,i_*} \Big( \Vect \usotimes{\Dmod(\bN_k)} \C \Big)  \simeq \uscolim{k,i_*} \, \C_{\bN_k}.
$$
Next, identifying $\C_{\bN_k}$ with $\C^{\bN_k}$ via Theorem \ref{thm:inv=coinv}, the map induced by $i_*$ goes over to $\Av_*^{\rel}: \C^{\bN_k} \to \C^{\bN_{k+1}}$, the right adjoint to the inclusion $\oblv^{\rel}: \C^{\bN_{k+1}} \to \C^{\bN_{k}}$.
%
Indeed, this follows from the commutativity of the diagram:
%
\begin{gather} \label{diag:1}
\xy
(0,0)*+{ \Dmod^*(\bN_{k+1}) \otimes \C \, \, }="00";
(32,0)*+{\, \,  \C \, \,  }="10";
(64,0)*+{\, \,  \C^{\bN_{k+1}}.}="20";
%
(00,15)*+{  \Dmod^*(\bN_{k}) \otimes \C \, \,  }="01";
(32,15)*+{  \, \,  \C \, \,  }="11";
(64,15)*+{\, \,  \C^{\bN_k}   }="21";
%
{\ar@{->}^{i_*} "01";"00"};
{\ar@<+0.3ex>^{} "00";"10"};
{\ar@<-0.5ex>^{} "00";"10"};
%
{\ar@<+0.3ex>^{} "01";"11"};
{\ar@<-0.5ex>^{} "01";"11"};
%
{\ar@{->}^{\id} "11";"10"};
%
%
%{\ar@<+0.7ex>^{} "10";"20"};
{\ar@<-0.0ex>^{\Av_*^{\bN_{k+1}}} "10";"20"};
%{\ar@<-0.7ex>^{} "10";"20"};
%
{\ar@<-0.0ex>^{\Av_*^{\bN_k}} "11";"21"};
%
{\ar@{->}^{\Av_*^{\rel}} "21";"20"};
%
\endxy
\end{gather}

\smallskip

The computation of $\bN$-invariants is easier: $\C^\bN$ is the limit of $\C^{\bN_k}$, along the transition maps $\C^{\bN_{k+1}} \to \C^{\bN_k}$ induced by $i^! : \Dmod^!(\bN_{k+1}) \to \Dmod^!(\bN_k)$. The relevant diagram
%
\begin{gather} 
\xy
(0,0)*+{ \Dmod^!(\bN_{k+1}) \otimes \C \, \, }="00";
(32,0)*+{\, \,  \C \, \,  }="10";
(64,0)*+{\, \,  \C^{\bN_{k+1}}}="20";
%
(00,15)*+{  \Dmod^!(\bN_{k}) \otimes \C \, \,  }="01";
(32,15)*+{  \, \,  \C \, \,  }="11";
(64,15)*+{\, \,  \C^{\bN_k}   }="21";
%
{\ar@{<-}^{i^!} "01";"00"};
{\ar@<+0.3ex>^{} "10";"00"};
{\ar@<-0.5ex>^{} "10";"00"};
%
{\ar@<+0.3ex>^{} "11";"01"};
{\ar@<-0.5ex>^{} "11";"01"};
%
{\ar@{->}^{\id} "10";"11"};
%
%
%{\ar@<+0.7ex>^{} "10";"20"};
{\ar@{->}^{\oblv^{\bN_{k+1}}} "20";"10"};
%{\ar@<-0.7ex>^{} "10";"20"};
%
{\ar@{<-}^{\oblv^{\bN_k}} "11";"21"};
%
{\ar@{<-}^{\oblv^{\rel}} "21";"20"};
%
\endxy
\end{gather}
%
is commutative (the assertion for the left square follows by duality from commutativity of the left square of (\ref{diag:1})). This identifies $\C^{\bN_{k+1}} \to \C^{\bN_k}$ as $\oblv^{\rel}$.
%
\end{proof}

\sssec{}

We now introduce the main mathematical objects of this paper, the Whittaker invariant and coinvariant categories of an object of $\bG \on{-}\mathbf{rep}$. Let $\chi$ be the character defined in (\ref{eqn:chi}).

\medskip

We define the \emph{Whittaker invariant} and \emph{Whittaker coinvariant} categories of $\C$ respectively as $\C^{\bN,\chi}$ and $\C_{\bN,\chi}$.

\medskip

In view of Proposition \ref{prop:N-inv_vs_Nk-inv}, we have:
%
$$
\C^{\bN,\chi} \simeq \lim_{\oblv^\rel} {\C}^{\bN_k,\chi} 
\, \, \, \text{ and }\, \, \,
\C_{\bN,\chi} \simeq \uscolim{\Av_*^{\rel}} \, {\C}^{\bN_k, \chi}.
$$


\sssec{}

For any choice of dimension theory on $\bN$, we shall construct a functor $\Theta: \C_{\bN, \chi} \to \C^{\bN,\chi}$ between the Whittaker categories, called the \emph{renormalized averaging} functor.

\medskip

Without loss of generality, we may assume that the self-duality $\Lambda: \Dmod^!(\bN) \to \Dmod^*(\bN)$ is determined by $\dim(\bN_0) = 0$ and that $\bN$ is presented as $\bN \simeq \colim_{n \geq 0} \bN_n$.
Consider the object $(-\chi)^\ren_\bN := \Lambda ((-\chi)^!exp)$ and the corresponding functor $(-\chi)^\ren_\bN \star - : \C \to \C$. Explicitly, this is the functor
\begin{equation} \label{theta}
(-\chi)^\ren_\bN \star - 
\simeq
 \uscolim{n \geq 0} \Big(  (-\chi)_{\bN_n} \star  - [2 \dim(\bN_n)] \Big).
\end{equation}
%
where $(-\chi)_{\bN_n}$ was defined in Remark \ref{rem:Av_*-twisted-in-prounip-case}. 

\begin{prop} \label{prop:theta-descends}
The functor (\ref{theta}) descends to a functor $\Theta: \C_{\bN, \chi} \to \C^{\bN,\chi}$. (When we wish to emphasize the dependence on the dimension theory chosen, we write $\Theta_\Lambda$.)
\end{prop}

The proof fits the following general paradigm.

\sssec{} \label{paradigm-colim-lim}

Let $\I$ be a filtered indexing category, with an initial object $0 \in \I$.
Consider a diagram $\D_\bullet : \I \to \DGCat$, with transition functors $\ins_{i \to j}: \D_i \to \D_j$, and a diagram $\E^\bullet : \I^\op \to \DGCat$, with transition functors $\ev^{j \to i}: \E^j \to \E^i$. 
Let $\{\theta_i: \D_i \to \E^i \}_{i \in \I}$ be a collection of functors together with transitive systems of maps
$$
\theta_i \Longrightarrow \ev^{j\to i} \circ \theta_j \circ \ins_{i \to j},
$$
so that we can form the functor 
$$
\wt \Theta := \uscolim{i \in \I} \big( \ev^{i \to 0}  \circ \theta_i \circ \ins_{0 \to i} \big)
: \D_0 \to \E^0.
$$
Set $\D_\infty := \colim_{i \in \I} \D_i$ and $\E^\infty := \lim_{i \in \I^\op} \E^i$.

\begin{lem}
In the situation just described, assume that all $\ins_{i \to j}$ are \emph{essentially surjective} and that all $\ev^{j \to i}$ are \emph{fully faithful}. Then $\wt \Theta$ factors as $\ev^{\infty \to 0} \circ \Theta \circ \ins_{0 \to \infty}$ for some functor $\Theta: \D_\infty\to \E^{\infty}$.  
\end{lem}

\begin{proof}
By filteredness of $\I$, we can write
$$
\wt\Theta \simeq \ev^{\ell \to 0} \circ \Big( \uscolim{i \in \I_{\ell/}} (\ev^{i \to \ell} \circ \theta_i \circ \ins_{\ell \to i}) \Big) \circ \ins_{0 \to \ell}
$$  
for any $\ell \in \I$. The conclusion is manifest.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:theta-descends}]
It suffices to realize that the functor (\ref{theta}) is the functor $\wt\Theta$ obtained via the above paradigm in the example where $\I = \NN$, $\D_ n := \C_{\bN_n, \chi}$, $\E^n := \C^{\bN_n, \chi}$ and $\theta_n := \theta_{\bN_n} [2 \dim(\bN_n)] :  \C_{\bN_n} \to \C^{\bN_n}$.
\end{proof}



\sssec{}

The following conjecture has been proposed by Gaitsgory:

\begin{conj} \label{conj:Tequiv}
Let $\C$ a category equipped with a $\bG$-action and $\chi$ the character of (\ref{eqn:chi}). For any trivialization of the dimension torsor of $\bN$, the functor $\Theta_{\Lambda}: \C_{\bN,\chi} \to \C^{\bN,\chi}$ is an equivalence of categories.
\end{conj}

We prove a refinement of this conjecture for $G= GL_n$ in Section \ref{SEC:GL_n}, but first we need to study actions of loop vector spaces. This is the subject of the next section.


\section{Fourier transform and actions by loop vector groups} \label{SEC:loop_vectors}


If $G= GL_2$, then $N \simeq \AA^1$ is abelian, so that all the notions discussed above (group actions, invariants, coinvariants, averaging functors...) can be understood via Fourier transform. More generally, we consider the case of a vector group $\AA^n$ and its loop group $\bA := \AA^n \ppart$, which is of course the main example of a Tate vector space.

\ssec{Fourier transform for finite dimensional vector spaces}

We start by checking that the well-known theory of the Fourier-Deligne transform (see, e.g., \citep{Laumon}) upgrades to the DG setting.

\sssec{}

Let $V$ be a finite dimensional vector space, thought of as a scheme, with dual $V^\vee$. We indicate by $m$ the addition in $V$, $V^\vee$ or $\GG_a$ (depending on the context) and by $Q: V \times V^\vee \to \GG_a$ the duality pairing. Let  $p_1$ and $p_2$ be the projections from $V \times V^\vee$ to $V$ and $V^\vee$, respectively.
%
Recall the $\fD$-module $exp$ on $\GG_a$, as in formula (\ref{def:exp}). The Fourier transform kernel is 
$$
exp^Q:=Q^!(exp) \in \Dmod(V \times V^\vee).
$$
A well-known key property of this ``integral kernel" is the equivalence
\begin{equation} \label{eqn:projection_exp}
(p_1)_*(exp^Q) \simeq \delta_{0,V}[2 d_V].
\end{equation}

\sssec{}

The \emph{Fourier transform} $\FT_V$ is the functor
$$
\FT_V: \Dmod(V) \rightarrow\Dmod(V^\vee), \, \, \, M \mapsto (p_2)_* (p_1^!(M) \otimes exp^{Q}).
$$
Note that $(\FT_V)^\vee \simeq \FT_{V^\vee}$.

\medskip
%
We also define the \emph{inverse Fourier transform} $\IFT = \IFT_V$ as
$$
\IFT: \Dmod(V^\vee) \rightarrow\Dmod(V), \, \, \, M \mapsto (p_1)_* (p_2^!(M) \otimes exp^{-Q}) [-2 d_V].
$$
%
This name will be justified by Proposition \ref{prop:FT_equiv_fin-dim}. 
%
Emphasizing the dependence on $Q$, we record the formula
\begin{equation} \label{eqn:IFT-Q}
\FT_V^{Q} \simeq \IFT_{V^\vee}^{-Q}[2 d_V].
\end{equation}


\sssec{}

To upgrade the known results on $\FT_V$ to the setting of DG categories, we need to recall the formalism of correspondences and how it allows to handle base-change.
Let $\Sch^\fty$ the 1-category of schemes of finite type. We form $\Sch^{\fty}_{\corr}$, the 1-category whose objects are the same as $\Sch^{\fty}$ and whose morphisms are given by correspondences:
$$
\Hom_{\Sch^{\fty}_{\corr}}(S, T) = \Big\{ S \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} T \, : H \in \Sch^{\fty}  \Big\}.
$$
Such correspondences compose under fiber product. Moreover, $\Sch^{\fty}_{\corr}$ inherits a symmetric monoidal structure from $\Sch^{\fty}$.

\begin{thm}[\cite{GR_corresp}] \label{thm:GR-corresp}
The assignment $S \rightsquigarrow \Dmod(S)$ upgrades to a \emph{symmetric monoidal functor}
$$
\Dmod: \Sch^{\fty}_{\corr} \to \DGCat
$$
which sends $S \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} T$ to the functor $\Dmod(S) \xrightarrow{\beta_* \,\circ\, \alpha^!} \Dmod(T)$.
\end{thm}
%

In particular, by restricting the domain of the above functor to the 1-category of finite dimensional vector spaces (in schemes) and linear maps under correspondences, we obtain the theory of $\fD$-modules on vector spaces. To discuss further properties of the Fourier transform, we shall need a mild generalization of Theorem \ref{thm:GR-corresp}.

\sssec{}

\nc{\KL}{\ms{KL}}

Let $G$ be a monoid in $\Sch^{\fty}$ and $\KL_{G}$ be the following symmetric monoidal 1-category. Its objects are schemes of finite type and the tensor structure is the ordinary product; given two $V$ and $W$, the set of morphisms $V \dasharrow W$ consists of all diagrams of the form
%
\begin{gather} \label{diag:decorated_corresp}
\xy
(0,0)*+{ V }="s";
(26,0)*+{ W, }="t";
%
(13,10)*+{ H }="H";
(13,22)*+{ G }="Ga";
%
{\ar@{->}_{\alpha } "H";"s"};
{\ar@{->}^{\beta } "H";"t"};
{\ar@{->}^{f } "H";"Ga"};
%
%
\endxy
\end{gather}
%
where $H \in \Sch^{\fty}$.
%
For short, we write $(V \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} W; f)$ to indicate the morphism $V \dasharrow W$ in (\ref{diag:decorated_corresp}). The composition of $(V \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} W; f)$ with $(W \xleftarrow{ \, \gamma \,} K \xrightarrow{\, \delta \,} Z; g)$ is the correspondence
$$
\left( V \xleftarrow{ \, \, \,} H \underset{W}\times K \xrightarrow{\, \, \,} Z; h \right) ,
$$
where $h: H \times_W K \to H \times K \xrightarrow{\, f \cdot g \,} G$.
%It is straightforward to check that  $\Sch^{\ft, \GG_a}_{\corr}$ is indeed a symmetric monoidal 1-category with identity morphisms $ id_V : V \dasharrow V$ given by $( V \xleftarrow{ \, \id \,} V \xrightarrow{\, \id \,} V; 0)$.

\sssec{}

It has been observed by N. Rozenblyum that $\KL_{G}$ is the \emph{Kleisli category} associated to $\Sch^{\fty}_{\corr}$ and the monad $G \times - $, whence the choice of the symbol $\KL$.
%
In general, for a $1$-category $\C$ and a monad $\sM : \C \to \C$, the Kleisli $1$-category $\KL_\sM$ has the same objects as $\C$ and 
$$
\Hom_{\KL_\sM}(c, c') := \Hom_{\C}(c, \sM(c')).
$$
The composition is defined via the monad structure.

\medskip

There is a fully faithful functor $\Xi: \KL_{\sM} \to \sM \mod (\C)$ that sends
$$
c \rightsquigarrow \sM(c), \hspace*{.6cm}
(c \xto{\phi} c') \rightsquigarrow (\sM(c) \xto{\Xi(\phi)} \sM(c')),
$$
where $\Xi(\phi)$ is obtained by adjunction from the morphism $c \to \sM(c')$ in $\C$ defined by $\phi \in (\KL_\sM)^{\Delta^1}$.

\begin{rem} \label{rem:kleisli-symm-mon}
Assume that $\C$ is symmetric monoidal and that $\sM = A \otimes -$ for some commutative algebra $A \in \Alg(\C)$. Then, $\KL_A$ is symmetric monoidal in a natural way and the functor $\Xi: (\KL_A, \otimes) \to (A \mod(\C), \otimes_A)$ respects the monoidal structures. 
\end{rem}

\begin{thm}
The assignment $V \rightsquigarrow \Dmod(V)$ upgrades to a symmetric monoidal functor
$$
\Dmod^{enh}: \KL_{\GG_a} \to \DGCat
$$
which sends $( V \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} W; f ) $ to the functor 
$$
\Dmod(V) 
\xrightarrow{\beta_* ( f^!(exp) \otimes \alpha^!(-)) } 
\Dmod(W).
$$
\end{thm}


\begin{proof}
By general non-sense,\footcite{For any symmetric monoidal functor $\sF : \C \to \E$ and any commutative algebra object $A \in \C$, the resulting functor $(A \mod (\C), \otimes_A ) \to (\sF(A) \mod (\E), \otimes_{\sF(A)})$ is symmetric monoidal as well.} the functor $\Dmod$ of Theorem \ref{thm:GR-corresp} upgrades to a symmetric monoidal functor 
$$
\Dmod_{\GG_a}: \Big( \GG_a \mod (\Sch^{\fty}_\corr), \overset {\GG_a} \times \,\,\Big) \to \Big( \Dmod(\GG_a) \mmod, \usotimes{\Dmod(\GG_a)} \,\, \Big).
$$
%
By Remark \ref{rem:kleisli-symm-mon}, the functor 
$$
\Xi: (\KL_{\GG_a}, \times) \to \Big(\GG_a \mod (\Sch^{\fty}_\corr), \overset {\GG_a} \times \, \Big)
$$
is symmetric monoidal as well.
Let us describe $\Xi$ concretely. At the level of objects, it sends $V \mapsto V \times \GG_a$, where $\GG_a$ acts freely on the second factor. At the level of morphisms, it sends
%
$$( V \xleftarrow{ \, \alpha \,} H \xrightarrow{\, \beta \,} W; f ) \mapsto 
( \GG_a \times V \xleftarrow{ \, \, \id \times \alpha \, \,} \GG_a \times H \xrightarrow{\, \sigma \,}  \GG_a \times W),
$$
where $\sigma (x, h) = (x + f(h), \beta(h))$. 

\medskip

Next, let $\inv^{\GG_a, \exp} : \GG_a \on{-} \mathbf{rep} \to \DGCat$ be the functor of $(\GG_a, \exp)$-invariants. Recall the canonical equivalence 
$$
\Dmod(V) \simeq \inv^{\GG_a, \exp} \big( \Dmod( \GG_a \times V) \big) =: \Dmod(\GG_a \times V)^{\GG_a, \exp}
$$
under which $\oblv^{\GG_a,\exp}$ goes over to $ \exp \boxtimes -: \Dmod(V) \to \Dmod( \GG_a\times V)$.
We claim that our functor $\Dmod^{enh}$ is the composition 
$$
 \Dmod^{enh} \simeq \inv^{\GG_a, \exp} \circ \Dmod_{\GG_a} \circ \Xi.
$$
Indeed, $\sigma$ equals the composition $(m_{\GG_a} \times \id) \circ (\id \times f \times \beta ) \circ (\id \times \Delta)$ and the proof of the claim is routine.

\medskip

It remains to show that $\inv^{\GG_a, \exp}$ is symmetric monoidal, as a functor on $(\GG_a \rrep, \otimes_{\Dmod(\GG_a)})$. Apply $\inv^{\GG_a, \exp}$ to the natural equivalence of left $\GG_a \rrep$:
$$
\C \usotimes{\Dmod(\GG_a)} \E \simeq \Dmod(\GG_a) \usotimes{\Dmod(\GG_a) \otimes \Dmod(\GG_a)} (\C \otimes \C).
$$
To conclude, it is enough to note that the equivalence $\Vect_\exp \simeq \Vect_\exp \otimes \Vect_\exp$ is $(\GG_a, \GG_a)$-equivariant.
\end{proof}

\sssec{}
Let us exploit this formalism.
By construction, $\FT_V$ is the value of $\Dmod^{enh}$ on the following ``arrow" (which we also call $\FT_V$):
\begin{gather} \label{diag:decorated_corresp_FT}
\xy
(-20,10)*+{ V \overset{\FT_V}\dasharrow V^\vee \, \, \, \, := }="VVvee";
%
%
%
(0,0)*+{ V }="s";
(28,0)*+{ V^\vee. }="t";
%
(14,10)*+{ V \times V^\vee }="H";
(14,22)*+{ \GG_a }="Ga";
%
{\ar@{->}_{ p_1 } "H";"s"};
{\ar@{->}^{ p_2 } "H";"t"};
{\ar@{->}^{ Q} "H";"Ga"};
%
%
\endxy
\end{gather}


\begin{prop} \label{prop:FT_equiv_fin-dim}
For any finite dimensional vector space $V$, the functors $\FT_V$ and $\IFT_V$ are mutually inverse equivalences of categories.
\end{prop}

\begin{proof}
It suffices to exhibit a natural isomorphism $\IFT \circ \FT \simeq \id_V$. The functor $\IFT[2d_V]$ is the value of $\Dmod^{enh}$ on the arrow $\IFT' := (V^\vee \xleftarrow{\, p_2 \,} V^\vee \times V \xrightarrow{\, p_1 \,} V, -Q)$.

At the level of correspondences, the composition $\IFT'\circ \FT: V \dasharrow V^\vee \dasharrow V$ is given by
$$
(V \xleftarrow{\, \wh p_1 \,} V \times V \times V^\vee \xrightarrow{\, \wh p_2 \,} V, \xi ),
$$
where $\wh p_i : V\ \times V \times V^\vee \to V$ are the projections and $\xi : V \times V \times V^\vee \to \GG_a$ sends $(v,v',\phi) \mapsto \phi(v-v')$.
%
From (\ref{eqn:projection_exp}), one sees that $\xi^! (\exp) \simeq (\Delta_V \times i_{0 \to V^\vee})_*(\omega_V)[2d_V]$; the proof follows immediately.
\end{proof}
%%%%%%\begin{rem}
%%%%%%The proof of Lemma \ref{lem:projection_exp} shows that $\FT_V(\delta_v) \simeq v^!(exp)$. Here are two handy consequences:
%%%%%%\begin{equation} \label{eqn:FT_delta1,exp}
%%%%%%\FT_{\GG_a}(\delta_1) \simeq \exp, \hspace{1 cm} \FT_{\GG_a}(\exp) \simeq \delta_{-1}[2]. 
%%%%%%\end{equation}
%%%%%%\end{rem}
%
%
%
%%%%Let us now study the effect of Fourier transform on pushforwards and pullbacks along linear maps.

\begin{lem} \label{lem:FT-behavior-linear-maps}
Given a linear map of finite dimensional vector spaces $f: W \to V$ and its dual $\phi: V^\vee \to W^\vee$, the following diagrams are commutative:
%
%
\begin{gather} \label{eqn:i_*}
\xy
(00,00)*+{ \Dmod(V) }="00";
(0,15)*+{ \Dmod(W) }="10";
%
(35,00)*+{ \Dmod(V^\vee) }="01";
(35,15)*+{ \Dmod(W^\vee) }="11";
%
{\ar@{->}^{ \FT_V } "00";"01"};
{\ar@{<-}_{ f_*  } "00";"10"};
%
{\ar@{<-}_{\phi^! } "01";"11"};
{\ar@{->}^{ \FT_W} "10";"11"};
%
(60,0)*+{ \Dmod(V) }="20";
(60,15)*+{ \Dmod(W) }="30";
%
(97,0)*+{ \Dmod(V^\vee). }="21";
(97,15)*+{ \Dmod(W^\vee) }="31";
%
{\ar@{<-}^{ \FT_{V^\vee} } "20";"21"};
{\ar@{<-}_{ f_* [2d_V-2 d_W] } "20";"30"};
%
{\ar@{<-}_{(-\phi)^! } "21";"31"};
{\ar@{<-}^{ \FT_{W^\vee}} "30";"31"};
%
%
\endxy
\end{gather} 
%
\end{lem}

\begin{proof}
We only prove the first commutativity, the second follows by applying the inverse Fourier trasform.
Consider the arrows 
$$
W \overset {(id, f)} { \dasharrow}  V :=\left( W  \xleftarrow{ \, \id \,} W  \xrightarrow{\, f \,} V; 0 \right)
$$
and
$$
W^\vee \overset {(\phi, \id)} { \dasharrow}  V^\vee :=\left( W^\vee  \xleftarrow{ \, \phi \,} V^\vee  \xrightarrow{\, id \,} V^\vee; 0 \right).
$$
It suffices to prove that the following diagram is commutative:
%
\begin{gather} 
\xy
(0,0)*+{ V \, }="00";
(35,0)*+{ \, V^\vee. }="10";
%
(00,15)*+{  W  }="01";
(35,15)*+{   W^\vee }="11";
%
{\ar@{<--}^{ (\id, f) } "00";"01"};
{\ar@{-->}^{ \FT_V } "00";"10"}; % orizz bassa
%
{\ar@{-->}^{ \FT_W  } "01";"11"};
{\ar@{<--}_{ (\phi, \id) } "10";"11"};
%
%
\endxy
\end{gather}
%
We leave it to the reader to verify that both paths coincide with
$$
\left( W  \xleftarrow{ \, p_1 \,} W \times V^\vee  \xrightarrow{\, p_2 \,} V^\vee; h\right),
$$
where
$h = Q \circ (f \times \id_{V^\vee}) = Q \circ (\id_W \times \phi)$.
\end{proof}

\begin{cor}
$\FT_V$ is a monoidal equivalence between $(\Dmod(V), \star)$ and $(\Dmod(V^\vee), \otimes)$.
Analogously, $\FT_V$ is a comonoidal equivalence between $(\Dmod(V), \Delta_*)$ and $(\Dmod(V^\vee), m^!)$.
\end{cor}

Both statements follow at once from Lemma \ref{lem:FT-behavior-linear-maps}. Nevertheless, here is a direct proof.

\begin{proof}
We only prove the first claim, the second one is obtained by duality (or by a direct argument).
The convolution monoidal structure on $\Dmod(V)$ arises from the algebra structure on $V$ given by
$$
V \times V \overset m { \dasharrow}  V :=\left( V \times V \xleftarrow{ \, \id \,} V \times V  \xrightarrow{\, m \,} V; 0 \right);
$$
the pointwise monoidal structure on $\Dmod(V^\vee)$ from the algebra structure
$$
V^\vee \times V^\vee \overset \Delta { \dasharrow}  V^\vee := \left( V^\vee \times V^\vee \xleftarrow{ \, \Delta \,} V^\vee  \xrightarrow{\, \id \,} V^\vee; 0 \right).
$$
We just need to prove that $\FT: V \dasharrow V^\vee$  intertwines the two algebra structures, or equivalently that the following diagram in $\KL_{\GG_a}$ commutes:
%
%
\begin{gather} 
\xy
(0,0)*+{ V \, }="00";
(35,0)*+{ \, V^\vee. }="10";
%
(00,15)*+{  V \times V  }="01";
(35,15)*+{   V^\vee \times V^\vee }="11";
%
{\ar@{<--}^{ m } "00";"01"};
{\ar@{-->}^{ \FT } "00";"10"}; % orizz bassa
%
{\ar@{-->}^{ \FT \times \FT  } "01";"11"};
{\ar@{<--}_{ \Delta } "10";"11"};
%
%
\endxy
\end{gather}
%%
%
Again, it is routine to check that both paths coincide with
$$
\left( V \times V \xleftarrow{ \, p_{12} \,} V \times V \times V^\vee  \xrightarrow{\, m \times id_{V^\vee} \,} V^\vee ; f \right),
$$
where $f: V \times V \times V^\vee \to \GG_a$ sends $(v,w,\phi) \mapsto Q(\phi, w-v)$. 
\end{proof}

%%%\sssec{} \label{sec:FT_interpretation}
%%%
Hence, we have
$$
\FT_V : V\on{-}\mathbf{rep} := (\Dmod(V), \star) \mmod \xrightarrow{\, \, \, \simeq \, \, \, } (\Dmod(V^\vee), \otimes) \mmod;
$$
in other words, Fourier transform indentifies categorical representations of $V$ and \emph{crystals of categories} over $V^\vee$, that is, categories with an action of $(\Dmod(V^\vee), \otimes)$.\footnote{This statement uses $1$-affineness of $V^\vee_\dR$, estabilished in \cite{shvcat}.}
%%%%%%
%%%%%%\noindent
%%%%%%The action of $(\Dmod(V^\vee),\otimes)$ on $\C \in V \on{-}\mathbf{rep} $ is given by $P \otimes c \mapsto \IFT(P) \star c$.

\medskip 


%%%%%%%%%%%%
%%%%%%%%%%%%\sssec{}
%%%%%%%%%%%%
%%%%%%%%%%%%Our last computation in the finite dimensional case involves a character $\chi \in V^\vee$.
%%%%%%%%%%%%%
%%%%%%%%%%%%We first need a small result:
%%%%%%%%%%%%%
%%%%%%%%%%%%\begin{cor} \label{cor:FT of char}
%%%%%%%%%%%%$\FT_V( \chi^!(exp)) \simeq \delta_{-\chi, V^\vee}[2 d_V]$. In particular, by putting $\chi = 0$, we obtain $\FT_V( k_V) \simeq \delta_{0, V^\vee}$.
%%%%%%%%%%%%\end{cor}
%%%%%%%%%%%%
%%%%%%%%%%%%\begin{proof}
%%%%%%%%%%%%By (\ref{eqn:q_*}) and (\ref{eqn:FT_delta1,exp}), 
%%%%%%%%%%%%$$
%%%%%%%%%%%%\FT_V( \chi^!(exp)) \simeq (\chi^\vee)_* \FT_{\GG_a}(exp) [2 d_V - 2]
%%%%%%%%%%%%\simeq (\chi^\vee)_* \left( \delta_{-1, \GG_a} [2] \right) [2d_V-2]. 
%%%%%%%%%%%%%\simeq \left( \delta_{- \chi, V^\vee} [2 d_V] \right).
%%%%%%%%%%%%$$
%%%%%%%%%%%%Since $\chi^\vee : \GG_a \to V^\vee$ is the linear map sending $1$ to $\chi$, the conclusion follows.
%%%%%%%%%%%%\end{proof}
%%%%%%%%%%%%
%%%%%%%%%%%%\begin{prop} \label{prop:FT-char-subspace}
%%%%%%%%%%%%Let $i: W \hookrightarrow V$ be the embedding of a vector subspace and $\chi \in V^\vee$ a character. Let $\ol\chi: W \to \GG_a$ be its restriction to $W$ and $\chi_W := i_* \circ \ol\chi^! exp \in \Dmod(V)$. Then
%%%%%%%%%%%%$$
%%%%%%%%%%%%\FT_V( \chi_W )[-2 d_W] \simeq \omega_{-\chi + W^\perp},
%%%%%%%%%%%%$$
%%%%%%%%%%%%where $W^\perp$ is the annihilator of $W$ in $V^\vee$.
%%%%%%%%%%%%\end{prop}
%%%%%%%%%%%%
%%%%%%%%%%%%\begin{proof}
%%%%%%%%%%%%Using (\ref{eqn:i_*}) and the corollary above, we have
%%%%%%%%%%%%$$
%%%%%%%%%%%%\FT_V(\chi_W) \simeq \pi^! \big( \FT_W(\ol\chi^! exp) \big)
%%%%%%%%%%%%% 
%%%%%%%%%%%%\simeq \pi^! (\delta_{-\ol \chi, W^\vee}) [2 d_W],
%%%%%%%%%%%%$$
%%%%%%%%%%%%where $\pi: V^\vee \to W^\vee$ is the projection dual to $i$.
%%%%%%%%%%%%%
%%%%%%%%%%%%Noting that the fiber of the projection $\pi: V^\vee \to W^\vee$ at $\{-\ol{\chi}\}$ is precisely $-\chi + W^\perp$, we conclude
%%%%%%%%%%%%$$
%%%%%%%%%%%%\FT_V(\chi_W) = \omega_{-\chi + W^\perp} [2 d_W],
%%%%%%%%%%%%$$
%%%%%%%%%%%%as desired.
%%%%%%%%%%%%\end{proof}

\ssec{Fourier transform for Tate vector spaces}

\renewcommand{\bV}{\mathcal{V}}
\renewcommand{\bW}{\mathcal{W}}

We shall need the notion of Fourier transform for the loop group $\bA := \AA^n \ppart$ of the affine scheme $\AA^n$, thought of as a vector group. More generally, we define the Fourier transform functor for Tate vector spaces and estabilish its properties, parallel to the ones of the previous section.

\sssec{}
\nc{\PRO}{\mathsf{PRO}}
\nc{\TVect}{\mathsf{TopVect}}

In the sequel, we recall the notion of \emph{Tate vector space}. Our main references are \citep{Beil}, \citep{Dr}, \citep{FG2}. To fix the notation, let $\Vect^\heartsuit$ be the $1$-category of vector spaces, $\PRO(\Vect^\heartsuit)$ its pro-completion, $\Vect^{\ms{f.d.}}$ the $1$-category of finite dimensional vector spaces. 

\medskip

Consider the additive $1$-category $\TVect$ of complex \emph{topological vector spaces}, where $\CC$ is given the discrete topology. The topology of each object $V \in \TVect$ is supposed to be linear, complete and separated. Morphims in $\TVect$ are simply continuous linear maps.
%
There is an obvious adjunction
$$
\ms{\delta} : \Vect^\heartsuit \rightleftarrows \TVect : \oblv^{\ms{top}},
$$
where $\delta$ endows a vector space with the discrete topology and $\oblv^{\ms{top}}$ forgets the topology. Note that $\delta$ is fully faithful; its image is referred to as the subcategory of \emph{discrete vector spaces}. 

\sssec{}

$\TVect$ is the full subcategory of $\PRO(\Vect^\heartsuit)$ consisting of objects that can be realized as projective limits of discrete vector spaces along surjective (linear) maps (\citep{FG2}). 
Since $\TVect$ admits projective limits and filtered colimits, each set $\Hom_{\TVect}(V,W)$ is naturally a topological vector space that we shall denote by $\H om(V,W)$.
In particular, the \emph{topological dual} (henceforth simply called dual) of $V$ is the object $V^\vee := \H om(V, \CC) \in \TVect$.
A pro-finite dimensional vector space can be written as $\H om(V,\CC)$, where $V$ is discrete. 

\sssec{}

Let $\Tate$ be the full subcategory of $\TVect$ whose objects are the direct sums $P \oplus Q$, where $P$ is pro-finite dimensional and $Q$ is discrete. Obviously, the operation $\H om(-, \CC)$ transforms discrete into pro-finite and viceversa, hence $\Tate$ admits duals.

\medskip

Write $P = \underset{r, \p}\lim  P^r$ and $Q = \uscolim {n, \i} Q_n$ with finite dimensional $P^r$ and $Q_n$. Setting $V_n^r := P^r \oplus Q_n$, we have
\begin{equation} \label{def:tate}
P \oplus Q \simeq  \underset{r, \p}\lim \, \uscolim {n, \i} V_n^r  \simeq \uscolim {n, \i} \underset{r, \p}\lim V_n^r
\end{equation}
and
\begin{equation} \label{def:tate-dual}
(P \oplus Q)^\vee \simeq  \underset{r, \p^\vee}\colim \, \lim_{n, \i^\vee } (V_n^r)^\vee  \simeq \lim_ {n, \i^\vee} \underset{r, \p^\vee}\colim (V_n^r)^\vee.
\end{equation}
%
%
Thus, objects of $\Tate$ are filtered limits of discrete vector spaces along surjections of finite dimension, as well as filtered colimits of pro-finite dimensional vector spaces along injections of finite codimension. The latter characterization makes it clear that Tate vector spaces are the analogue of ind-pro-schemes in the linear algebra setting.

\sssec{}

More precisely, the extension of the natural functor $\Vect^{\ms{f.d.}} \to \Sch^{\fty}$ to $\PRO(\Vect^\heartsuit) \to \PreStk$ maps $\Tate$ to  $\IndSch^{\pro}$. We will henceforth view Tate vector spaces as ind-pro-schemes; in particular, for $\bV = \colim_n \lim_r  V_n^r$ as in (\ref{def:tate}), we have well-defined categories of $\fD$-modules:
$$
\Dmod^*(\bV) \simeq \uscolim{n, \i_*} \lim_{r, \p_*} \Dmod(V_n^r),
\hspace{.5 cm}
\Dmod^!(\bV^\vee) \simeq \lim_{r, (\p^\vee)^!} \uscolim{n, (\i^\vee)^!} \Dmod((V_n^r)^\vee).
$$
We wish to extend the Fourier transform equivalence to Tate vector spaces. This amounts to a combination of a left and a right Kan extension of the usual Fourier transform.

\sssec{}

Consider the usual finite dimensional Fourier transform $\FT_V$ as a functor of $V$:
%
\begin{equation} \label{eqn:FT-functor of V}
\FT : \Vect^{\mathsf{f. d.}} \to \Big(\DGCat^{\on{SymmMon},\simeq} \Big)^{\Delta^1}.
\end{equation}
%
Namely, at the level of objects, $\FT$ sends a finite dimensional vector space to the equivalence of symmetric monoidal categories categories $(\Dmod(V),\star) \to (\Dmod(V^\vee), \otimes)$. At the level of morphisms, $\FT$ sends the linear map $f: W \to V$ to the natural transformation
%
%
\begin{gather}
\xy
(0,0)*+{ \big(\Dmod(V), \star \big) }="00";
(35,0)*+{ \big(\Dmod(V^\vee), \otimes \big),  }="10";
%
(00,15)*+{ \big(\Dmod(W), \star \big)   }="01";
(35,15)*+{ \big(\Dmod(W^\vee), \otimes \big)   }="11";
%
{\ar@{<-}^{ f_*} "00";"01"};
{\ar@{->}^{ \FT_V } "00";"10"};
%
{\ar@{->}^{ \FT_W } "01";"11"};
{\ar@{<-}_{ \phi^! } "10";"11"};
%
%
\endxy
\end{gather}
% 
where $\phi = f^\vee : V^\vee \to W^\vee$. That this defines a functor is precisely the content of Lemma \ref{lem:FT-behavior-linear-maps}.
Note that $f_*$ and $\phi^!$ are compatible with the convolution and pointwise symmetric monoidal structures, respectively. 

\sssec{}

\renewcommand{\fd}{\ms{f.d.}}

Let $(\Vect^\fd)^\pro$ be the $1$-category of pro-finite dimensional vector spaces (thought of as pro-schemes). We define the functor
%
\begin{equation} \label{eqn:FT-functor of V-pro}
\FT : (\Vect^\fd)^\pro \to \Big( \DGCat^{\on{SymmMon},\simeq} \Big)^{\Delta^1}.
\end{equation}
%
by right Kan extension of (\ref{eqn:FT-functor of V}) along the inclusion $\Vect^{\mathsf{f. d.}} \hookrightarrow (\Vect^\fd)^\pro$.

\medskip
\nc{\indpro}{\ind \text{-} \pro}

To define $\FT$ at the level of Tate vector spaces, we left Kan extend (\ref{eqn:FT-functor of V-pro}) along $(\Vect^\fd)^\pro \hookrightarrow \Tate$:
%
\begin{equation} \label{eqn:FT-functor of V-indpro}
\FT :  \Tate \to \Big( \DGCat^{\on{SymmMon},\simeq} \Big)^{\Delta^1}.
\end{equation}
%

\sssec{}

Let us unravel the above constructions. If $V = \lim_{r, \pi} V^r \in (\Vect^{\ms{f.d.}})^\pro$ is a pro-finite dimensional vector space, then its dual $V^\vee$ is an ind-vector space of finite type. Denote by $\iota$ the injections dual to $\pi$ and by $(\iota_{r \to \infty})^! : \Dmod(V^\vee) \to \Dmod((V^r)^\vee)$ the tautological evaluation functors. Then, $\FT_V$ is defined by the requirement
%
\begin{equation} \label{eqn:FT-pro}
(\iota_{r \to \infty})^! \circ \FT_V  =  \FT_{V^r}  \circ (\pi_{\infty \to r})_*.
\end{equation}
%
Next, let $\bV$ be a Tate vector space, with presentation $\bV = \colim_{n, \i} V_n$ as an ind-scheme, where each $V_n$ is of pro-type. Its dual, $\bV^\vee$ is then written as $\lim_{n, \p} V_n^\vee$, where $\p$ are the projections dual to $\i$. 

\begin{lem} \label{lem:bravo}
Let $(\Vect^\fd)^{\ind }$ denote the $1$-category of vector spaces, considered as ind-schemes of finite type.
The functor $\Dmod^!: \Tate^\op \to \DGCat$ is the left Kan extension of $\Dmod : \big( (\Vect^\fd)^{\ind }\big)^\op \to \DGCat$ along the inclusion $(\Vect^\fd)^{\ind } \hookrightarrow \mathsf{Tate}$.
\end{lem}

\nc{\W}{\mathcal{W}}

\begin{proof}
Recall the presentation of $\W \in \Tate$ as in (\ref{def:tate}). We need to prove that the colimit and the limit in the formula $\Dmod^!(\W) \simeq \lim_{n, \i^!} \colim_{r, \p^!} \Dmod(W_n^r)$ can be switched:
\begin{equation} \label{tate-switch}
\lim_{n, \i^!} \uscolim{r, \p^!} \Dmod(W_n^r) \simeq  \uscolim{r, \p^!} \lim_{n, \i^!} \Dmod(W_n^r).
\end{equation}
Observe that the left adjoint of $\i^!$ is $\i_+$. Since the diagram $(n,r) \rightsquigarrow W_n^r$ obviously has Cartesian squares, the assertion follows from the $(\i_+, \p^!)$ base-change.
\end{proof}

This allows to write both $\Dmod^*(\bV)$ and $\Dmod^!(\bV^\vee)$ as colimits:
$$
\Dmod^*(\bV) \simeq \uscolim{n, \i_*} \Dmod^*(V_n),
\hspace*{.6cm}
\Dmod^!(\bV^\vee) \simeq \uscolim{n,\p^!} \Dmod(V_n^\vee).
$$
By construction, $\FT_{\bV}$ and $\FT_{V_n}$ are intertwined by the insertion functors:
\begin{equation} \label{eqn:FT-indpro}
\FT_\bV \circ (\iota_{n \to \infty})_*   =  (\p_{\infty \to n})^! \circ \FT_{V_n}.
\end{equation}

\begin{lem} \label{lem:importantFTtate}
Let $\bV$ and $\bW$ two Tate vector spaces. Given a linear map $f: \bW \to \bV$ and its dual $\phi : \bV^\vee \to \bW^\vee$, the following diagram is commutative:
%
\begin{gather}
\xy
(00,00)*+{ \Dmod^*(\bV) }="00";
(0,15)*+{ \Dmod^*(\bW) }="10";
%
(35,00)*+{ \Dmod^!(\bV^\vee). }="01";
(35,15)*+{ \Dmod^!(\bW^\vee) }="11";
%
{\ar@{->}^{ \FT_\bV } "00";"01"};
{\ar@{<-}_{ f_*  } "00";"10"};
%
{\ar@{<-}_{\phi^! } "01";"11"};
{\ar@{->}^{ \FT_\bW} "10";"11"};
%
\endxy
\end{gather} 
%%%%%%%%%%
%%%%%%%%%where
%%%%%%%%%$$
%%%%%%%%%(f_*)^\ren \simeq   \Lambda_\bV^{-1} \circ f_* \circ \Lambda_\bW
%%%%%%%%%\hspace{.6cm}
%%%%%%%%%(\phi^!)^\ren \simeq   \Lambda_\bV \circ \phi^! \circ \Lambda_\bW^{-1}.
%%%%%%%%%$$
Moreover, $(\FT_\V)^\vee \simeq \FT_{\V^\vee}$.
\end{lem}

\begin{proof}
Both statements are obtained by left-right Kan extending the corresponding statements in the finite dimensional case.
\end{proof}

\begin{rem} \label{rem:FT-ren-pro}
By left Kan extending (\ref{eqn:IFT-Q}), we deduce that, for a pro-scheme $V$ with dual $V^\vee$, we have $\FT^Q_V \circ \lambda_V \simeq \IFT^{-Q}_{V^\vee}$.
\end{rem}

As a corollary of Lemma \ref{lem:importantFTtate}, we obtain:

\begin{thm} \label{thm:mon-comon-Tate}
For any Tate vector space $\bV$, Fourier transform yields the monoidal equivalence
$$
\FT_{\bV}: (\Dmod^*(\bV), \star) \xrightarrow{\,\, \simeq \,\,} (\Dmod^!(\bV^\vee), \otimes),
$$
as well as the comonoidal equivalence
$$
\FT_{\bV}: (\Dmod^*(\bV), \Delta_*) \xrightarrow{\,\, \simeq \,\,} (\Dmod^!(\bV^\vee), m^!).
$$
\end{thm}


\ssec{Categories tensored over ind-pro-schemes}

The above theory motivates the study of \emph{categories tensored over} a Tate vector space $\bW$, i.e. objects of $(\Dmod^!(\bW), \otimes) \mmod$. More generally, we shall study modules for $(\Dmod^!(\X), \otimes)$ where $\X$ is an arbitrary ind-pro-scheme.
For clarity, given $\C$ tensored over $\X$, we indicate by $\diam$ the action $\Dmod^!(\X) \otimes \C \to \C$.

\sssec{}

Recall that, for any map $f: \Y \to \X$ of ind-pro-schemes, $f^!: \Dmod^!(\X) \to \Dmod^!(\Y)$ is $\Dmod^!(\X)$-linear. Thus, for any $\C \in \Dmod^!(\X) \mmod$, we define the \emph{corestriction} and the \emph{restriction} of $\C$ along $f$:
$$
\restr \C \Y := \Dmod^!(\Y) \usotimes{\Dmod^!(\X)} \C
\hspace{.6cm}
\uprestr \C \Y := \Hom_{\Dmod^!(\X)} (\Dmod^!(\Y), \C).
$$
(When $f: \pt \to \X$, we call the above categories \emph{cofiber} and \emph{fiber}, respectively.)
Applying the paradigm of Sect. \ref{paradigm}, we obtain the tautological functors 
$$
(f^!)_\C : \C \to \restr \C \Y
\text{\, \, \, and \, \, }
(f^!)^\C : \uprestr \C \Y \to \C.
$$
%
\begin{lem} \label{lem:useful}
Let $\iota: Y \hto X$ be a closed embedding of pro-schemes and $\C \in \Dmod^!(X)$. Then: 
\begin{itemize}

\item
there are adjuctions
$$
(\iota_+)_\C: \restr \C Y \rightleftarrows \C: (\iota^!)_\C
\hspace{.6cm}
(\iota^!)^\C : \uprestr \C Y \rightleftarrows \C : (\iota_+)^\C,
$$
with both left adjoints fully faithful;

\medskip

\item

$\iota_+(\omega_Y) \diam -: \C \to \C$ induces a functor that we name $\theta_{Y \hto X}: \restr \C Y \to \uprestr \C Y$;

\medskip

\item

$\theta_{Y \hto X} \simeq (\iota_+)^\C \circ (\iota_+)_\C$ is an equivalence with inverse $(\iota^!)_\C \circ (\iota^!)^\C: \uprestr \C Y \to \restr \C Y$.

\end{itemize}

\end{lem}

\begin{proof}
For the first claim, it suffices to note that $\iota_+$ is $\Dmod^!(X)$-linear (by base-change) and fully faithful. 
Secondly, the canonical equivalence of functors $\iota_+(\omega_Y) \otimes - \simeq \iota_+ \circ \iota^! : \Dmod^!(X) \to \Dmod^!(X)$ shows that $\iota_+(\omega_Y) \diam -$ factors as 
$$
\C \twoheadrightarrow \restr \C Y \xto{\, \, \theta_{Y \hto X} \, \,} \uprestr \C Y \hto \C.
$$
It remains to check that $\theta := \theta_{Y \hto X}$ is an equivalence. This can be done directly, by testing that the two compositions are the identity functors. 
\end{proof}

\sssec{}

Let now $i_x: \{x\} \hto \X$ be the inclusion of a point into an ind-pro-scheme $\X$. Assume that $\X$ has been given a dimension theory. This allows to define the functor $(i_x)_*^\ren := \Lambda^{-1} \circ (i_x)_*: \Vect \to \Dmod^!(\X)$ and the object $(\delta_x)^\ren := \Lambda^{-1}(\delta_x) \in \Dmod^!(\X)$. By the projection formula (Proposition \ref{prop:proj-formula-indschemes}), we obtain

\begin{lem}
The functor $(i_x)_*^\ren$ is (both left and right) $\Dmod^!(\X)$-linear.
(Informally, this means that tensoring $(\delta_x)^\ren$ with an object of $\Dmod^!(\X)$ is the same as tensoring $(\delta_x)^\ren$ with the $!$-fiber at $x$ of that object.) 
\end{lem}


Hence, $(\delta_x)^\ren \diam - :\C \to \C$ yields a functor
\begin{equation}
\label{eqn:Theta_x}
\Theta_{x \hto X} : \restr \C x \to \uprestr \C x.
\end{equation}


\begin{lem} \label{lem:FT-of-omega-ren}
For any Tate vector space $\bV$, and point $\m \in \bV^\vee$ and any pair of companion dimension theories\footcite{I.e., if $\Lambda_\bV$ corresponds $\dim (V_n^r) =0$ for some indices $n$ and $r$, then $\Lambda_{\bV^\vee}$ corresponds to $\dim ((V_n^r)^\vee) = 0$.} on $\bV$ and $\bV^\vee$, we have
\begin{equation}
\FT_\bV \big(\Lambda((-\m)^! exp ) \big) \simeq (\delta_{\m, \bV^\vee})^\ren \in \Dmod^!(\bV^\vee).
\end{equation}
\end{lem}

\begin{proof}
Let $\bV$ be presented as in (\ref{def:tate}).
By definition,
$$
\FT_\bV \big(\Lambda((-\m)^! exp ) \big)
\simeq
\colim_n \FT_{V_n} \circ \lambda_{V_n} \big( (-\mu)^! \exp \big) [2 \dim(V_n)].
$$
Thanks to Remark \ref{rem:FT-ren-pro} and $\FT (\delta_\m) \simeq \m^!(\exp)$, we know that 
$$
\FT_{V_n} \circ \lambda_{V_n} \big( (-\mu)^! \exp \big) \simeq \delta_{\chi, V_n^\vee}.
$$
The assertion follows, after tracing through the equivalence (\ref{tate-switch}) for $\W = \V^\vee$.
\end{proof}


\ssec{Invariants and coinvariants via Fourier transform}

Let $\bV \in \Tate$. Suppose that $\bV$ acts on $\C$; as usual, we indicate by $\star$ the action of $\Dmod^*(\bV)$ on $\C$.
We wish to express the invariant and coinvariant categories $\C^\bV$ and $\C_\bV$ in terms of the action of $(\Dmod^!(\bV^\vee),\otimes)$ on $\C$, which is given tautologically by
\begin{equation} \label{eqn:action_IFT}
\Dmod^!(\bV^\vee) \otimes \C \to \C, \, \, \, \, P \otimes c \mapsto \IFT_{\bV}(P) \star c.
\end{equation}


\begin{lem} \label{lem:coinv_FT-chi}
Let $\C$ be endowed with an action of $\Dmod^*(\V)$ and $\m: \V \to \GG_a$ be a character.
Under Fourier transform,
$$
\C_{\V, \chi} \simeq \restr \C {\chi}
\hspace*{0.6cm}
\C^{\V,\chi} \simeq \uprestr {\C}{\chi},
$$
where $\restr \C {\chi}$ and $\uprestr \C {\chi}$ denote the cofiber and fiber of $\C$ along $(i_\chi)^!: \Dmod^!(\V^\vee) \to \Dmod(\chi) \simeq \Vect$.
\end{lem}


\begin{proof}
It suffices to prove that the action of $\Dmod^*(\V)$ on $\Vect$ determined by the comonoidal functor
\begin{equation} \label{eqn:comocomo}
\Vect \to (\Dmod^!(\V), m^!), \hspace*{0.3cm} \CC \mapsto \chi^!(\exp)
\end{equation}
%
goes over to the action of $(\Dmod^!(\V^\vee), \Delta^!)$ on $\Vect$ given by $(i_\chi)^!$. 

By Fourier transform, (\ref{eqn:comocomo}) goes over to the comonoidal functor
\begin{equation} \label{eqn:comocomo1}
\Vect \to (\Dmod^*(\V^\vee), \Delta_*), \hspace*{0.3cm} \CC \mapsto \delta_{\chi, \V^\vee}.
\end{equation}
%
This is a quick consequence of Lemma \ref{lem:importantFTtate} and Theorem \ref{thm:mon-comon-Tate}, after recalling that $\FT_{\GG_a}(\delta_1) \simeq \exp$. To conclude, observe that the duality (\ref{eqn:eval-compact}) between $\Dmod^*(\V^\vee)$ and $\Dmod^!(\V^\vee)$ transforms (\ref{eqn:comocomo1}) into the desired monoidal functor $(i_\chi)^!$.
\end{proof}



\sssec{}

\renewcommand{\bV}{\mathcal{V}}
\renewcommand{\bW}{\mathcal{W}}

Consider a category $\C$ equipped with an action of a Tate vector space $\bV$. Let $\bW \hookrightarrow \bV$ be a Tate vector subspace. We wish to describe the procedure of taking twisted (co)invariants of $\C$ with respect to $\bW$ via Fourier transform. 
Let $\p: \bV^\vee \to \bW^\vee$ be the projection dual to $\bW \hookrightarrow \bV$ and $\bW^\perp \subset \bV^\vee$ the annihilator of $\bW$ in $\bV^\vee$. 

\begin{prop} \label{prop:FT_char_subspace_indpro}
Let $\chi \in V^\vee$ be a character. Under Fourier transform,
$$
\C_{\bW,\chi} \simeq \Dmod^!(\bW^\perp + \{ \chi \}) \underset{\Dmod^!(\V^\vee)}\otimes \C,
\, \, \, \, \, \, \,\,\,
\C^{\bW,\chi} \simeq \Hom_{\Dmod^!(V^\vee)} \big( \Dmod^!(\bW^\perp + \{ \chi \}), \C \big),
$$
where $\Dmod^!(V^\vee)$ acts on $\Dmod^!(\bW^\perp + \{\chi \})$ via $!$-pullback along the inclusion $\bW^\perp + \{\chi \} \subseteq V^\vee$. 
\end{prop}

\begin{proof}
Using the standard tensor-hom adjunction, we see that it suffices to prove the equivalence
$$
\Dmod^*(\bV)_{\bW, \chi} \simeq \Dmod^!(\{\chi \} +\bW ^\perp).
$$ 
By the lemma above, we have
$$
\Dmod^*(\bV)_{\bW, \chi} \simeq \Dmod(\chi) \underset{\Dmod^!(\bW^\vee)}\otimes \Dmod^!(\bV^\vee)
$$
and we conclude by Lemma \ref{lem:BN-ind}, since $\chi \times_{\bW^\vee} \bV^\vee \simeq \chi + \bW^\perp$.
\end{proof}

\renewcommand{\bV}{\mathbf{V}}
\renewcommand{\bW}{\mathbf{W}}

\section{Categories over quotient stacks} \label{SEC:trans_act}

Let $X/G$ be a quotient stack, where $X$ and $G$ are both of finite type. We have seen (Proposition \ref{prop:ShvCat(X/G)}) that a crystal of categories over $X/G$ is the same as a module category for $\Dmod(G) \ltimes \Dmod(X)$. The latter espression admits an obvious generalization when $G$ and $X$ are of ind-pro-finite type, which we introduce in \ref{ssec:6.1}. 

\medskip

A category over $X/G$ is a category tensored over $X$ in a $G$-equivariant way. For such $\C$, we analyze the relations between (co)invariants with respect to subgroups of $G$ and (co)restrictions along maps $Y \to X$.
In Sect. \ref{ssec:transitive-group-acts}, we specialize to the case where $G$ acts transitively on $X$. Then, the invariant category $\C^G$ can be understood in terms of its fiber at any point of $X$.


\ssec{The main definitions} \label{ssec:6.1}

Let $\G$ be an ind-pro-group and $\X$ an ind-pro-scheme endowed with an action of $\G$. Diagram (\ref{diag:GactsX}) shows that $\Dmod^!(\X)$ is algebra object in the $\infty$-category $\Dmod^!(\G) \ccomod$, hence we can form the crossed product monoidal category $\Dmod^*(\G) \ltimes \Dmod^!(\X)$.

\sssec{}

We say that $\C \in \DGCat$ is \emph{a category tensored over $\X/\G$} if it is endowed with the structure of a module category for $\Dmod^*(\G) \ltimes \Dmod^!(\X)$,
where we recall that
$$
\Dmod^*(\G) \ltimes \Dmod^!(\X) \mmod \simeq \Dmod^!(\X) \mod (\Dmod^!(\G) \ccomod).
$$

\sssec{}

Recall that we indicate by $M \diamond c$ the action of $M \in \Dmod^!(\X)$ on $c \in \C$, while the action of $\Dmod^*(\G)$ on $\C$ is the usual $\star$.
The construction $\Dmod^*(\G) \ltimes \Dmod^!(\X)$ entails the following compatibility between the actions of $\G$ and $\Dmod^!(\X)$ on $\C$: for each $M$ and $c$ as above, there is a canonical isomorphism
\begin{equation} \label{eqn:coact_compatib}
\coact_\G ( M \diamond c) \simeq \act_{\G,\X}^!(M) \diamond \coact_\G (c).
\end{equation}
To be precise, the symbol $\diamond$ in the RHS means $\otimes$ on the $\Dmod^!(\G)$-factor and action of $\Dmod^!(\X)$ on $\C$.

\begin{example} \label{example:theta-G-inv}
Let  $f:\Y \to \X$ be $\G$-equivariant. Then, $\Dmod^!(\Y)$, $\Dmod^!(\X)$ are tensored over $\X/\G$ in a natural way and $f^!: \Dmod^!(\X) \to \Dmod^!(\Y)$ is $\Dmod^*(\G) \ltimes \Dmod^!(\X)$-linear. 
\end{example}


\begin{lem} \label{lem:pullback-equivariant}
For $\C$ tensored over $\X/\G$, the morphisms $(f^!)_\C: \restr \C \X \to \restr \C \Y$ and $(f^!)^\C: \uprestr \C \Y \to \uprestr \C \X$ are $\G$-equivariant.
\end{lem}

\begin{proof}
For any $\M \in \Dmod^!(X) \mod (\Dmod^!(G) \ccomod)$, the functor $ \Hom_{\Dmod^!(X)}(\M, -)$ sends 
$$
\Dmod^!(X) \mod (\Dmod^!(G) \ccomod) \to  \Dmod^!(G) \ccomod,
$$
and similarly for the relative tensor product. Then, the assertions follows from the example above.
\end{proof}

\begin{cor} \label{cor:theta_inv}
Let $\iota: Y \hto X$ be a $G$-equivariant closed embedding of pro-schemes. For any $\C$ tensored over $X/G$, the functor $\theta_{Y \to X} : \restr \C Y \to \uprestr \C Y$ is $G$-equivariant.
\end{cor}

\begin{proof}
Lemma \ref{lem:useful} guarantees that the inverse of $\theta_{Y \to X}$ is the composition $(\iota^!)_\C \circ (\iota^!)^\C $, which is $G$-equivariant by Lemma \ref{lem:pullback-equivariant}.
\end{proof}



\begin{lem} \label{lem:stupid_cat_over_X/G}
Let $\C$ be a category tensored over $\X/\G$.
With the notation above:
%
\begin{itemize}
\item[(a)]
If $\G$ acts trivially on $\X$, then $\C^\G$ and $\C_\G$ are tensored over $\X$ compatibly with $\oblv^{\G}: \C^\G \to \C$ and $\pr_{\G}: \C \to \C_\G$.

\smallskip

\item[(b)]
If $\G = H$ is a pro-group acting trivially on $\X$ and $\Y$, there is a canonical equivalence $(\restr \C \Y)^H \simeq (\restr{\C^H)} \Y$.


\smallskip

\item[(c)]
Let $S$ be a pro-group mapping to $H$. If $S$ acts on $\Y$ and $f$ is $S$-equivariant, $\restr \C \Y$ is tensored over $\Y/S$ and there exists a dotted arrow making the following diagram commutative:
%
%
\begin{gather} \label{diag:X/G<---Y/K}
\xy
(0,18)*+{ \C^H }="00";
(00,00)*+{  \C }="10";
%
%
(33,18)*+{ (\restr \C \Y)^S}="01";
(33,00)*+{ \restr \C \Y.  }="11";
%
%
{\ar@{-->}^{} "00";"01"};
{\ar@{->}^{\oblv^H} "00";"10"};
%
{\ar@{->}^{ (f^!)_\C} "10";"11"};
{\ar@{->}^{\oblv^S} "01";"11"};
%
\endxy
\end{gather}

\end{itemize}
%
%
\end{lem}

\begin{proof}
To (co)simplicial category computing $\C^\G$ (resp., $\C_\G$) is easily seen to be a (co)simplicial object in $\Dmod^!(\X) \mmod$. This proves (a).
Item (b) is an immediate consequence of the fact that $\inv^H$ commutes with colimits.
As for (c), since $(f^!)_\C: \C \to \restr \C \Y$ is $S$-equivariant, the dotted arrow is simply the composition
$\rho := (f^!)_\C \circ \oblv^\rel :  \C^H \to \C^S \to (\restr \C \Y)^S$.
\end{proof}



\sssec{}

We pause to give one important example of category over a quotient stack.
Let $\G$ is an ind-pro-group acting on a Tate vector space $\V$. 
If $\G \ltimes \V$ acts on a category $\C$, then 

\begin{itemize}
\item
$\C$ fibers over $ \V^\vee$, by Fourier transform;
\item
$\G$ acts on $\V^\vee$, via the dual action;
\item
$\G$ acts on $\C$ via the embedding $\G \to \G \ltimes \V$.
\end{itemize}
%
The proposition below makes it precise that these three pieces of data are compatible:

\begin{prop} \label{prop:KV^vee-module}
With the above notation, Fourier transform induces a monoidal equivalence 
$$
\on{id}_{\Dmod^*(\G)} \otimes \FT_{\V} : \Dmod^*(\G \ltimes \V) \xrightarrow{ \, \, \simeq \, \,} \Dmod^*(\G) \ltimes \Dmod^!(\V^\vee).
$$
\end{prop}

\begin{proof}
It suffices to notice that $\Dmod^*$ preserves $\ltimes$, so that $\Dmod^*(\G \ltimes \V) \simeq \Dmod^*(\G) \ltimes \Dmod^*(\V)$.
\end{proof}

\begin{example}
Let $P \simeq GL_{n-1} \ltimes \AA^{n-1} \subset GL_n$ be the mirabolic group of $GL_n$. We obtain an equivalence between $P \ppart \mmod$ and the $\infty$-category of categories tensored over $\AA^{n-1} \ppart / GL_{n-1} \ppart$, where $GL_{n-1} \ppart$ acts on $\AA^{n-1} \ppart$ via the dual action.
\end{example}

\ssec{Interactions between invariants and corestrictions, I}

Let us analyze further the situation of Lemma \ref{lem:stupid_cat_over_X/G}, item (c).
In this section, we discuss the finite dimensional situation. 
Let $X$ be $G$-scheme of finite type and $\iota: Y \hookrightarrow X$ a closed embedding. Let $S \subset G$ be the closed subgroup preserving $Y$.

\medskip

Our goal is to inquire under what condition the dotted arrow $\rho: \C^G \to (\restr \C Y)^S$ of diagram (\ref{diag:X/G<---Y/K}) is an equivalence.

\sssec{}

By construction, we have the commutative diagram
%
\begin{gather}  \label{diag:amico}
\xy
(0,18)*+{ X_\dR/G_\dR }="00";
(00,00)*+{  X_\dR }="10";
%
%
(33,18)*+{ Y_\dR/S_\dR }="01";
(33,00)*+{ Y_\dR.  }="11";
%
%
{\ar@{<-}^{\wt\iota} "00";"01"};
{\ar@{<-}^{q^X} "00";"10"};
%
{\ar@{<-}_{ \iota } "10";"11"};
{\ar@{<-}^{q^Y} "01";"11"};
%
\endxy
\end{gather}
%
%

\begin{prop} \label{prop:X/G=Y/S}
For $\wt\C \in \ShvCat(X_\dR/G_\dR)$, let $\C := \bGamma(X_\dR, \wt\C)$.
Suppose that $\iota$ induces the isomorphism of stacks $\ul \iota: Y/S \xrightarrow{\, \simeq \, } X/G$. 
Then, there is an equivalence $\rho: \C^{G} \to (\restr \C Y)^{S}$ making the following diagram commutative:
%
\begin{gather} \label{diag:bello}
\xy
(0,18)*+{ \C^{G} }="00";
(00,00)*+{  \C }="10";
%
%
(33,18)*+{ (\restr \C Y)^{S}}="01";
(33,00)*+{ \restr \C Y.  }="11";
%
%
{\ar@{->}^{\simeq}_{\rho} "00";"01"};
{\ar@{->}^{\oblv^G} "00";"10"};
%
{\ar@{->}_{(\iota^!)_\C} "10";"11"};
{\ar@{->}^{\oblv^S} "01";"11"};
%
\endxy
\end{gather}
\end{prop}

\begin{rem}
Since $(\iota^!)_\C$ is $S$-equivariant, $\rho \simeq (\iota^!)_\C \circ \oblv^\rel$, where $\oblv^\rel: \C^{G,\m} \to \C^{S,\m}$ is the relative oblivion functor.
\end{rem}

\begin{proof}
Recall that, by Proposition \ref{prop:ShvCat(X/G)}, the functor $\ShvCat(X_\dR/G_\dR) \xrightarrow{\, \, q^! = \bGamma(X_\dR, -) \, \,} \DGCat$
enhances to the equivalence 
$$
\ShvCat(X_\dR/G_\dR) \to \Dmod(X) \mod ( \Dmod(G) \ccomod).
$$ 
%
Apply the contravariant functor $\bGamma(-, \wt\C)$ to (\ref{diag:amico}). Tautologically, $\bGamma((X/G)_\dR, \wt\C) \simeq \C^G$ and pull-back along $q^X$ (resp., $q^Y$) is $\oblv^G$ (resp., $\oblv^S$) by construction. On the other hand, pull-back along $\iota$ is the functor $(\iota^!)_\C : \C \to \Dmod(Y) \otimes_{\Dmod(X)} \C$, as claimed.
\end{proof}

\begin{rem} \label{rem:char-unnecessary}
Let $\mu: G \to \GG_a$ be a character. Then, $\C \otimes \Vect_{-\m}$ also fibers on $X/G$ (where $\Dmod(X)$ acts trivially on $\Vect_{-\m}$). The above proposition admits an obvious variant involving $\mu$-twisted invariants (Corollary \ref{cor:inverse_with_Av_!}). In every proof of Section \ref{SEC:trans_act}, it suffices to treat $\m = 0$ (the general case follows from the change $\C \rightsquigarrow \C \otimes \Vect_{-\m}$). We will tacitly assume that this step has been performed.
\end{rem}

To summarize the above discussion, we record:

\begin{cor} \label{cor:inverse_with_Av_!}
In the situation of Proposition \ref{prop:X/G=Y/S}, let $\mu: G \to \GG_a$ be a character.
The functor
$$
\Av_!^{G/S} \circ (\iota_*)_\C : (\restr \C Y)^{S, \mu} \to \C^{G, \mu} : \rho= (\iota^!)_\C \circ \oblv^\rel
$$
are mutually inverse equivalences of categories.
\end{cor}

Here and later in this section, $\Av_\heartsuit^{G/S} : \C^{S, \m} \to \C^{G, \m}$ is the relative $\heartsuit$-averaging functor for $\heartsuit=!$ or $\heartsuit=*$, whereas $\oblv^\rel: \C^{G,\m} \to \C^{S,\m}$ is the relative oblivion functor.

\begin{proof}
The functor from left to right is left adjoint (hence inverse) to the equivalence $\rho$.
\end{proof}

In particular, setting $Y = \{x\}$, we obtain:

\begin{cor} \label{cor:transitive_action_fd}
Let $\wt\C \in \ShvCat((X/G)_\dR)$ and suppose that $G$ acts transitively on $X$, with $S = \Stab(x \in X)$. Then, pull-back along $\iota: \{x\} \hookrightarrow X$ yields an equivalence 
$$
\rho: \C^{G, \mu}  \xto{\oblv^\rel} \C^{S, \m} \xto{(\iota^!)_\C} (\restr \C x)^{S, \mu}.
$$
\end{cor}

In other words, $\rho$ is an equivalence making diagram (\ref{diag:bello}) commutative.
The following result will be important in the sequel:

\begin{prop} \label{prop:inverse_with_Av_*-shift}
In the situation of Corollary \ref{cor:inverse_with_Av_!}, the functor
$$
\Av_*^{G/S} \circ (\iota_*)_\C [2 \codim(G:S)]: (\restr \C Y)^{S, \mu} \to \C^{G, \mu}
$$
is inverse to $\rho = (\iota^!)_\C \circ \oblv^\rel$. 
\end{prop}

Hence, we obtain a functorial identification 
\begin{equation} \label{eq:ident-shift}
\Av_!^{G/S} \circ (\iota_*)_\C \simeq \Av_*^{G/S} \circ (\iota_*)_\C [2 \codim(G:S)].
\end{equation}

\begin{proof}
Let us name $\tau$ the displayed functor and set $D := 2 \codim(G:S)$. It suffices to prove that $\tau \circ \rho$ is the identity of $\C^G$. By the compatibility between the $\Dmod(X)$-action and the $\Dmod(G)$-coaction, we have
$$
\tau \circ \rho 
\simeq
\Av_*^G  \circ (\iota_*)_\C \circ (\iota^!)_\C \circ \oblv^G [D]
\simeq
\Av_*^G \big(\iota_*(\omega_Y) \diam \oblv^G \big)[D]
\simeq
\Av_*^G (\iota_*(\omega_Y)) \diam \oblv^G [D].
$$
%%%%
%%%%%%$$
%%%%%%\simeq
%%%%%%\Av_*^G  \circ (\iota^!)^\C \circ (\iota_+)^\C \circ \oblv^G [D],
%%%%%%$$
%%%%%%thanks to $(\iota^!)^\C \circ (\iota_+)^\C \simeq \iota_+(\omega_Y) \diam -\simeq  (\iota_+)_\C \circ (\iota^!)_\C $.
%%%%%%
%%%%%%\medskip
%%%%%%
%%%%%%Notice now that each of the four functors on the RHS is obtained applying $\Hom_{\Dmod(G) \ltimes \Dmod(X)}(- , \C) $ to some other functor. 
%%%%%%Indeed, it is easy to check that $(\oblv^G, \Av_*^G)$ and $((\iota^!)^\C, (\iota_+)^\C)$
%%%%%%come from the adjunctions 
%%%%%%$$
%%%%%%\act_*: \Dmod(G) \ltimes \Dmod(X) \rightleftarrows \Dmod(X): \act^*
%%%%%%\hspace*{.6cm}
%%%%%%\iota_+: \Dmod(G) \ltimes \Dmod(Y) \rightleftarrows \Dmod(G) \ltimes \Dmod(X): \iota^!,
%%%%%%$$
%%%%%%respectively.
Hence, it suffices to prove the proposition for $\C = \Dmod(X)$, with its natural structure of category tensored over $X/G$.

\medskip

More generally, we will prove it for $\C = \Dmod(E)$, where $E$ is a scheme mapping $G$-equivariantly to $X$. Let $F:= Y \times_X E$ and (abusing notation) $\iota: F \hto E$ the closed embedding induced by $Y \hto X$. Consider the diagram
%
\begin{gather}
\xy
(00,0)*+{ E/G }="00";
(30,0)*+{ F/S. }="10";
%
(0,15)*+{ E }="01";
(30,15)*+{ F }="11";
%
{\ar@{<-}_{ q } "00";"01"};
{\ar@{<-}^{ \wt \iota  }_{\simeq} "00";"10"};
%
{\ar@{<-}^{ \iota } "01";"11"};
{\ar@{<-}_{  q' } "10";"11"};
%
%
\endxy
\end{gather}
%
Under the equivalence $\Dmod(E)^G \simeq \Dmod(E/G)$, the adjuncton $(\oblv^G, \Av_*^G)$ becomes $q^! : \Dmod(E/G) \rightleftarrows \Dmod(E) : q_*[-2d_G]$, while $\Av_!^G \simeq q_!$. Also, by smoothness,
$$
\iota^! \circ q^! 
\simeq
(q')^! \circ (\wt\iota)^!
 \simeq
(q')^* \circ (\wt\iota)^* [2 d_S]
 \simeq
 \iota^* \circ q^* [2 d_S]. 
$$
Hence, the original equivalence $\rho$ becomes
$$
\rho \simeq (q')_! \circ  \iota^* \circ q^* [2 d_S],
$$
whose right adjoint is evidently $\tau$.
\end{proof}


\ssec{Interactions between invariants and corestrictions, II}


Our goal now is to extend Proposition \ref{prop:X/G=Y/S} and its corollaries to the pro-finite dimensional setting. Let $X \in \Sch^{\pro}$ and $H$ a pro-group acting on it. We need to impose a condition on the $H$-action on $X$.

\sssec{} \label{sssec:quotient-wise}

Define the $1$-category $\Sch_H^\pro$ of \emph{pro-$H$-schemes} by repeating the construction $\Sch^{\fty} \rightsquigarrow \Sch^{\pro}$ starting from $\Sch_H^\fty$: objects of $\Sch^\pro_H$ are pro-schemes $X$ admitting a presentation $X = \lim_r X^r$ such that any $X^r$ is acted on by $H$ and the transition maps $\pi_{s \to r}$ are $H$-equivariant.

\begin{example} \label{ex:proHscheme}
If $X$ is itself a pro-group, equipped with a family of subgroups $X_r$ preserved by $H$ and shrinking to the identity element, then $X \simeq \lim_r X/X_r$ exhibits $X$ as a pro-$H$-scheme.
\end{example}

\sssec{} \label{sssec:setup}

Let us fix the following set-up:
\begin{itemize}
\item[(i)]
$H$ is a pro-group endowed with a character $\mu: H \to \GG_a$;
\item[(ii)]
$X= \lim_r X^r$ a pro-$H$-scheme;
\item[(iii)]
$\iota: Y \hookrightarrow X$ is a closed embedding;
\item[(iv)]
$S \subseteq H$ the subgroup preserving $Y$;
\item[(v)]
the natural map $Y/S \to X/H$ is an \emph{isomorphism}.
\end{itemize}

\medskip

Given the first four items, we may form the functor
$$
\rho_Y: \C^{H,\m} \xto{\, \oblv^{\rel} \,} \C^{S,\m} \xto { \, (\iota^!)_\C \,} (\restr \C Y)^{S,\m},
$$
where we have used that $(\iota^!)_\C$ is $S$-equivariant (Lemma \ref{lem:pullback-equivariant}).

\medskip

Item (v) implies that $S \hto H$ is a closed embedding of pro-groups, hence $\codim(H:S)$ is well-defined. We define the functor
$$
 \sigma_Y : 
(\restr \C Y)^{S,\m} \xto { \, (\iota_+)_\C \,} \C^{S,\m} \xto{\, \,\Av_*^{H/S} \, \,} \C^{H,\m}.
$$


\begin{prop} \label{prop:X/G=Y/S_pro}
In the set-up on Sect. \ref{sssec:setup}, $\rho_Y$ and $\sigma_Y[2 \codim(H:S)]$ are mutually inverse equivalences of categories.
\end{prop}


\begin{proof}
%
Let $D = 2\codim(H:S)$. By definition, $\iota$ is the pull-back of a closed embedding $\iota^r: Y^r \hookrightarrow X^r$. Thanks to Lemma \ref{lem:BN}, we have
$$
\restr \C Y \simeq \Dmod(Y^r) \usotimes{\Dmod(X^r)} \C,
$$
where $\Dmod(X^r)$ acts on $\C$ via the monoidal functor $(\pi_{\infty \to r})^!: \Dmod(X^r) \to \Dmod^!(X)$.
For any $r \in \R$, pick a normal subgroup $H^r \subseteq H$ such that the action of $H$ on $X^r$ factors through the finite dimensional quotient $H \twoheadrightarrow H/H^r$. Obviously, $H^r$ is normal in $S$ and $S/H^r$ acts on $Y^r$.

By Lemma \ref{lem:auxiliary} and Lemma \ref{lem:stupid_cat_over_X/G}b, we obtain
$$
(\restr \C Y)^S 
\simeq
\Big( \Dmod(Y^r) \usotimes{\Dmod(X^r)} \C \Big)^S
 \simeq
\Big(  \Dmod(Y^r) \usotimes{\Dmod(X^r)} \C^{H^r} \Big)^{S/H^r}.
$$
%
The hypothesis implies that $X^r / (H/H^r) \simeq Y^r/(S/H^r)$ and $\C^{H^r}$ fibers over $X^r / (S/H^r)$ (see Lemma \ref{lem:stupid_cat_over_X/G}a), whence we obtain the equivalence
$$
\rho^{\fty} : (\C^{H^r})^{H/H^r} \rightleftarrows \Big(  \Dmod(Y^r) \usotimes{\Dmod(X^r)} \C^{H^r} \Big)^{S/H^r}: \sigma^{\fty}[D],
$$
from Lemma \ref{prop:X/G=Y/S} and Proposition \ref{prop:inverse_with_Av_*-shift}.
Consequently, 
$$
\C^H \simeq  (\C^{H^r})^{H/H^r} 
\overset { \rho^{\fty} \, \, } {\underset{\sigma^{\fty}[D]} {\rightleftarrows} }
 \Big(  \Dmod(Y^r) \usotimes{\Dmod(X^r)} \C^{H^r} \Big)^{S/H^r}
\simeq
(\restr \C Y)^S.
$$
That these functors is exactly $\rho$ and $\sigma[D]$ can be traced immediately.
\end{proof}

From the isomorphism $(\iota^!)^\C \circ \theta_{Y \hto X} \simeq (\iota_+)_\C$, we deduce:

\begin{cor}
In the situation described in Sect. \ref{sssec:setup}, the three equivalences
%
\begin{gather} \label{triangle}
\xy
(25,25)*+{ \C^{H,\m} }="A";
(00,00)*+{ (\restr \C Y)^{S,\m} }="B";
%
(50,00)*+{ \big(\!\uprestr \C Y \! \big)^{S,\m} }="C";
%
{\ar@{->}_{ \rho_Y = (\iota^!)_\C \circ \, \oblv^\rel } "A";"B"};
{\ar@{->}^{ \theta_{Y \hto X}[2\codim(H:S)] } "B";"C"};
{\ar@{->}_{ \sigma_Y = \Av_*^{H/S} \circ \, (\iota^!)^\C } "C";"A"};
%
\endxy
\end{gather} 
%
form a commutative diagram.
\end{cor}

\ssec{Transitive groups actions} \label{ssec:transitive-group-acts}

We wish to generalize Corollary \ref{cor:transitive_action_fd} to the pro-finite dimensional setting. This is not implied by the above section as $x \hto X$ is not a closed embedding whenever $X$ is infinite dimensional. 
Instead, we will consider a family of triangles as above with $Y$ shrinking to $x$ and deal with a convergence problem.

\sssec{} \label{sssec:setup-transitive}

The set-up for the present subsection in the following:
\begin{itemize}
\item[(i)]
$H$ is a pro-group endowed with a character $\mu: H \to \GG_a$;
\item[(ii)]
$X= \lim_r X^r$ a pro-$H$-scheme;
\item[(iii)]
$\iota: \{x\} \hookrightarrow X$ be the inclusion of a point into $X$;
\item[(iv)]
$S \subseteq H$ is the stabilizer of $x$;
\item[(v)]
the $H$-action on $X$ is transitive, i.e., that the natural map $\{x\}/S \to X/H$ is an isomorphism. 
\end{itemize}
%
Our goal is to prove the following theorem:

\begin{thm}
In the situation described in Sect. \ref{sssec:setup-transitive}, the three functors
%
\begin{gather} \label{triangle-transitive}
\xy
(25,25)*+{ \C^{H,\m} }="A";
(00,00)*+{ (\restr \C x)^{S,\m} }="B";
%
(50,00)*+{ \big(\!\uprestr \C x \! \big)^{S,\m} }="C";
%
{\ar@{->}_{ \rho_x := (\iota^!)_\C \circ \, \oblv^\rel } "A";"B"};
{\ar@{->}^{ \Theta_{x \hto X} } "B";"C"};
{\ar@{->}_{ \sigma_x := \Av_*^{H/S} \circ \, (\iota^!)^\C } "C";"A"};
%
\endxy
\end{gather} 
%
form a commutative diagram of \emph{equivalences}. Here, $\Theta_{x \hto X}$ is the functor (\ref{eqn:Theta_x}), associated to the dimension theory of $X$ specified in Remark \ref{rem:dim_thry-chosen}.
\end{thm}

We shall proceed in several steps. The proof will be given in Propositions \ref{prop:X/H---x/S}, \ref{prop:sigma_equiv} and \ref{ZC}.

\sssec{}

We start with a general result concerning the inclusion of a point $x$ into a pro-scheme $X = \lim_r X^r$.

\medskip

For each $r \in \R$, define the pro-scheme $Y^r := \{x^r\} \times_{X^r} X$, where $x^r = \pi_{\infty \to r}(x)$. The induced map $\iota_r : Y^r \hto X$ is a closed embedding, as it is pulled back from $x^r \hto X^r$. 
For any arrow $r \to s$ in $\R$, there is a closed embedding $\iota_{r,s} : Y^r \hto Y^s$; the limit (i.e., intersection) of the $Y^r$'s along these maps is the singleton $\{x\}$.

\begin{lem} \label{lem:approx_x}
The inverse family of $\Dmod^!(X)$-linear functors $(\iota_{x \hto Y^r })^! : \Dmod^!(Y^r) \to \Dmod(x)$ gives rise to a $\Dmod^!(X)$-linear equivalence
$$
\uscolim{\R^{op}, (\iota_{r,s})^!} \Dmod^!(Y^r) \to \Dmod(x).
$$
\end{lem}

\begin{proof}
The identifications of Lemma \ref{lem:BN} give assemble into a $\Dmod^!(X)$-linear equivalence
$$
\uscolim{\R^{op}, (\iota_{r,s})^!} \Dmod^!(Y^r) 
\simeq
\uscolim{\R^{op}, \mathsf{proj}} \Dmod(x^r) \usotimes{\Dmod(X^r)} \Dmod^!(X).
$$
The RHS is itself equivalent to $\Dmod(x^r) \usotimes{\Dmod^!(X)} \Dmod^!(X) \simeq \Dmod(x)$.
The composition is the compatible family of pull-backs, as claimed.
\end{proof}
 
\begin{cor}
For any $\C \in \Dmod^!(X) \mmod$, the functor of the lemma yields equivalences
$$
\uscolim{\R^{op},( \iota^!)_\C} \! \restr  \C {Y^r} \xto{ \,\, \simeq \, \,} \restr \C x 
\hspace{.6cm}
\uprestr \C x \xto{ \,\, \simeq \, \,} \lim_{\R,( \iota^!)^\C} \uprestr \C {Y^r}.
$$
\end{cor}

\sssec{}

From now on, we reinstate the set-up of Sect. \ref{sssec:setup-transitive}.

\medskip

By hypothesis, $H$ acts on each $X^r$; we let $S^r$ be the stabilizer of $x^r \in X^r$. 
Equivalently, $S^r$ is the subgroup of $H$ preserving $Y^r$. 
%
The embeddings $\iota_{r,s}$ yield isomorphisms $Y^r/S^r \to Y^s/S^s$, which interpolate the given $\{x\}/S
\xto{\simeq} X/H$.

\begin{rem} \label{rem:dim_thry-chosen}
We equip $X$ with the dimension theory defined by $\dim(X^r) = \codim (H : S^r)$. Clearly, the latter integer equals $\codim(X : Y^r)$.
\end{rem}

\begin{lem} \label{lem:smooth-gen-in-action}
Consider the functor $\sQ : \R^\op \to \DGCat$ that sends $r$ to $(\restr \C {Y^r})^{S^r}$ and $r \to s$ to
$$
\rho_{r,s} := (\iota_{r,s}^!)_\C \circ \oblv^{S^s \to S^r}: (\restr \C {Y^s})^{S^s} \to (\restr \C {Y^r})^{S^r}.
$$
The compatible family of restrictions $(\restr \C {Y^r})^{S^r} \to (\restr \C {x})^{S}$ yields the equivalence
\begin{equation}
\label{eqn:zaaa}
\uscolim{\R^{\op}, \sQ } (\restr \C {Y^r})^{S^r} \simeq (\restr \C {x})^{S}.
\end{equation}
\end{lem}

\begin{proof}
Indeed,
%
\begin{eqnarray} \label{eqn:zbbb}
\uscolim{r \in \R^\op} (\restr \C{Y^r})^{S^r}
\simeq
\uscolim{r \in \R^\op} \,  \uscolim{s \in \R^\op_{r/}}  (\restr \C{Y^r})^{S^s}
\simeq
\Big(\,\uscolim{r \in \R^\op} (\restr \C{Y^r}) \Big) ^S
\simeq
(\restr \C{x}) ^S.
\end{eqnarray}
%
The first equivalence is a standard cofinality argument, the second one is ``smooth generation" for $S= \bigcap S^r$ (see Proposition \ref{prop:smooth-generation}), the third follows from Lemma \ref{lem:approx_x}.
\end{proof}

\begin{prop} \label{prop:X/H---x/S} 
The functor
$$
\rho_x :\C^{H, \mu} \to (\restr \C x)^{S, \mu}
$$
is an equivalence of categories.
\end{prop}

\begin{proof}
Our functor can be factored as
$$
\rho_x : \C^H \longrightarrow \uscolim{\R^{\op}, \sQ } (\restr \C {Y^r})^{S^r} \xto{\, \,\,\ref{eqn:zaaa} \,\, \,} (\restr \C {x})^{S},
$$
where the first arrow is the direct family of equivalences $\rho_{Y^r}$ of Proposition \ref{prop:X/G=Y/S_pro}.
\end{proof}


\begin{prop} \label{prop:sigma_equiv} 
The functor
$$
\sigma_x = \Av_*^{H/S} \circ (\iota^!)^\C :(\uprestr \C x)^{S, \mu} \to \C^{H, \mu} 
$$
is an equivalence.
\end{prop}

\begin{proof}
The proof is similar to the one above, using the equivalences $\sigma_Y$ and the obvious variant of Lemma \ref{lem:smooth-gen-in-action}.
\end{proof}


\begin{prop} \label{ZC}
The composition
$$
(\sigma_x)^{-1} \circ (\rho_x)^{-1}: (\restr \C x)_S \to (\uprestr \C x)^S
$$
coincides with the map $\Theta_{x \to X}$, which is, therefore, an equivalence.
\end{prop}

\begin{proof}
By Corollary \ref{triangle}, the composition in question comes from the colimit of the functors
$\theta_{Y^r \hto X} [2 \dim(H:S^r)]$. By the general paradigm of Section \ref{paradigm-colim-lim}, this functor is induced from the functor of action with
$$ 
\uscolim{r \in \R^\op} \Big( (\iota_{Y^r \hto X})_+ (\omega_{Y^r}) [2 \codim(H:S^r)] \Big) \in \Dmod^!(X).
$$
Thus, we need to show that the latter is isomorphic to $\delta_x^\ren$, for the dimension theory of Remark \ref{rem:dim_thry-chosen}.
It suffices to prove that $(\iota_{Y^r \hto X})_+ (\omega_{Y^r}) \simeq (\pi_{X \to X^r})^! (\delta_{x^r})$. Thanks to the $(\iota_+, p^!)$ base-change along
%
\begin{gather}
\xy
(00,0)*+{ \pt }="00";
(30,0)*+{ X^r, }="10";
%
(0,15)*+{ Y^r }="01";
(30,15)*+{ X  }="11";
%
{\ar@{<-}_{ p  } "00";"01"};
{\ar@{->}_{ {x^r} } "00";"10"};
%
{\ar@{->}^{ \iota } "01";"11"};
{\ar@{<-}_{  \pi_{\infty \to r}} "10";"11"};
%
%
\endxy
\end{gather} 
this is immediate.
\end{proof}

\ssec{Actions by semi-direct products} \label{ssec:semi-direct}

We conclude with a short discussion of action by semi-direct products.

\sssec{}

Let $H \ltimes S$ be a semidirect product of pro-groups and denote by $\q: H \ltimes S \to S$ the projection.
Since $\Dmod^*(H \ltimes S) \mmod \simeq \Dmod^*(S) \mod (\Dmod^*(H) \mmod)$, we obtain

\begin{lem} \label{lem:AvAv_commute}
For any $\C \in \Dmod^*(H \ltimes S) \mmod$, the functors $\oblv^S : \C^S \rightleftarrows \C: \Av_*^S$ are $\Dmod^*(H)$-linear.
\end{lem}

In particular, if $\m: H \ltimes S \to \GG_a$ is a character, we obtain a functorial identification 
$$
(-\m)_H \star (-\m)_S \simeq (-\m)_S \star (-\m)_H;
$$
in other words, $\Av_*^{H,\m}$ and $\Av_*^{S,\m}$ commute (when viewed as endofunctors of $\C$).
 
\sssec{} 

The action of $\Dmod^*(H)$ on $\C^S$ and $\C_S$ coincides with the action of $\Dmod^*((H \ltimes S)/S)$ guaranteed by Lemma \ref{lem:auxiliary}. This is obvious as $(H \ltimes S)/S$ and $H$ are isomorphic as groups. Hence,

\begin{lem} \label{lem:semidirect-pro}
There is an equivalence $(\C^S)^H \xto{\simeq} \C^{H \ltimes S}$ compatible with $\oblv^H$ and $\oblv^{H \ltimes S}$. Likewise, there is an equivalence $(\C_S)_H \xleftarrow{\simeq} \C_{{H \ltimes S}}$ compatible with $\pr_S$ and $\pr_{H \ltimes S}$. In other words, the following diagrams are commutative:
%
\begin{gather}
\xy
(00,00)*+{ \C^S }="00";
(35,00)*+{ \C }="10";
%
(00,15)*+{ (\C^S)^H }="01";
(35,15)*+{ \C^{H \ltimes S} }="11";
%
{\ar@{<-}^{ \oblv^H } "00";"01"};
{\ar@{->}_{ \oblv^S  } "00";"10"};
%
{\ar@{->}_{\simeq } "01";"11"};
{\ar@{<-}^{ \oblv^{H \ltimes S}} "10";"11"};
%
(60,0)*+{ \C_{H \ltimes S} }="20";
(60,15)*+{ \C }="30";
%
(97,0)*+{ (\C_S)_H. }="21";
(97,15)*+{ \C_S }="31";
%
{\ar@{->}^{ \simeq } "20";"21"};
{\ar@{<-}_{ \pr_{H \ltimes S} } "20";"30"};
%
{\ar@{<-}_{ \pr_{H} } "21";"31"};
{\ar@{->}^{ \pr_S} "30";"31"};
%
%
\endxy
\end{gather} 
%
\end{lem}

\begin{rem}
By adjunction, the above equivalence $(\C^S)^H \xto{\simeq} \C^{H \ltimes S}$ is also compatible with $\Av_*^H$ and $\Av_*^{H \ltimes S}$.
\end{rem}

\begin{rem}
We will also need a version of the above lemma for ind-pro-groups. Let $\H = \colim_\I H_n$, $\S = \colim_\I S_n$ be filtered colimits of pro-groups under closed embeddings and assume that $H_n$ acts on $S_n$ for each $n \in \I$. For instance, $\H = \bN'$ and $\S = \bA$ (see Sect. \ref{combinat}).
The statement of Lemma \ref{lem:semidirect-pro} applies verbatim to $\H \ltimes \S$. The proof is straightforward.
\end{rem}

We shall apply these statements repeatedly (rather, their versions involving characters) in the next section.


\section{Actions by the loop group of $GL_n$} \label{SEC:GL_n}

We assume from now on that $G = GL_n$. Let $B$ be the Borel subgroup of upper triangular matrices and $N \subset B$ its unipotent radical: upper triangular matrices with $1$'s on the diagonal. Further, let $P \subseteq G$ be the mirabolic subgroup, $T \subseteq B$ be the maximal torus of diagonal matrices, $B_-$, $N_-$ be the opposite Borel and its unipotent radical.
The character $\chi$ on $\bN$ simply computes the sum of the residues of the entries in the diagonal $(i,i+1)$.
To approximate $\bN$, we shall use an explicit sequence of group schemes $\bN_k$: consider the diagonal element $\gamma:= \on{diag}\left( t^{nk}, t^{(n-1)k}, \ldots, t^{k} \right) \in \bT$. We let $\bN_k := \gamma^{-1} \cdot N[[t]] \cdot \gamma$. 

We will also need analogous notations for subgroups of $G' := GL_{n-1}$, its loop group and so on. Thus $B', N', \bG', \bN', \chi'$ have their obvious meanings.




\ssec{Statement of the main theorem}


Let $\C$ be a category acted on by $\bP := P \ppart$. 
Let $k \geq 1$ and fix once and for all the trivialization of the dimension torsor of $\bN$ defined by $\dim(\bN_k) = 0$; we indicate by $\Lambda$ the corresponding equivalence $\Lambda: \Dmod^!(\bN) \to \Dmod^*(\bN)$. 
Recall the functor $\Theta := \Theta_{\Lambda} : \C_{\bN, \chi} \to \C^{\bN, \chi}$, defined in Conjecture \ref{conj:Tequiv}:
$$
\Theta \simeq \uscolim{\ell \geq k} \, \Av_*^{\bN_\ell, \chi} [2 \codim(\bN_\ell : \bN_k)].
$$
%
Our goal is to prove:

\begin{thm} \label{thm:main}
For any $\C \in \bP \rrep$, the above functor $\Theta: \C_{\bN,\chi} \to \C^{\bN,\chi}$ is an equivalence of categories.
\end{thm}

In the remainder of the text, we shall define a pro-unipotent group scheme $\bH_k \subset GL_n \ppart$ and estabilish the following two results, whose combination implies the above theorem.

\begin{thm} \label{thm:refinement}
For any $k \geq 1$, the functors
$$
\Phi:= \Av_*^{\bH_k, \chi} \circ \, \oblv^{\bN,\chi}: \C^{\bN,\chi} \longrightarrow  \C^{\bH_k, \chi}
$$
$$
\Psi:= \pr_{\bN, \chi} \circ \, \oblv^{\bH_k,\chi}: \C^{\bH_k, \chi} \longrightarrow \C_{\bN,\chi}
$$
are equivalences of categories.
\end{thm}

\begin{prop} \label{prop:comp=T}
For $k$ and $\Lambda$ chosen as above, we have a commutative triangle
%
\begin{gather} \label{trianleZ}
\xy
(20,20)*+{ \C^{\bH_k,\chi} }="A";
(00,00)*+{ \C_{\bN,\chi} }="B";
%
(40,00)*+{ \C^{\bN,\chi}. }="C";
%
{\ar@{->}_{ \Psi } "A";"B"};
{\ar@{->}^{ \Theta } "B";"C"};
{\ar@{->}_{ \Phi } "C";"A"};
%
\endxy
\end{gather} 
%
\end{prop}
\smallskip

\ssec{Some combinatorics of $GL_n$} \label{combinat}

Let us introduce the long-awaited group ${\bH}_k$.
For $k \geq 1$, consider the product $\bB_-^k \cdot N(\O) \subset \GO$. To show it is a group, notice that it is the preimage of $N[t]/t^n$ under the group epimorphism $\p: \GO \to G[t]/t^k$.

\sssec{}

Let $\gamma':= \on{diag}\left( t^{(n-1)k}, \ldots, t^{k} \right) \in \bT'$. We first define the group
$$
\bG_k := \Ad_{\gamma'^{-1}} \left( (\bB_-')^k \cdot N'(\O) \right) = \Ad_{\gamma'^{-1}} \left( (\bN_-')^k \right)  \cdot (\bT')^{k} \cdot \bN'_k.
$$
%
For example, when $n=2$ and $n=3$ and $n=4$, we have
$$
\bG^{(2)}_k = (1+ t^k \O ),
\, \, \, 
\bG^{(3)}_k = \left( \begin{array}{cc}
1+t^k \O & t^{-k}\O  \\
t^{2k}\O & 1+t^k\O \end{array} \right),
\, \, \,
\bG_k^{(4)} = \left( \begin{array}{ccc}
1+t^k\O & t^{-k}\O & t^{-2k}\O \\
t^{2k}\O & 1+t^k\O & t^{-k}\O \\
t^{3k}\O & t^{2k}\O & 1+t^k\O \end{array} \right).
$$
For higher $n$, the structure of $\bG_k$ follows the evident pattern. The first important feature of $\bG_k$ is the following:

\begin{lem}
The group $\bG_k$ is endowed with a character $\chi_g$ that extends the character $\chi'$ on $\Ad_{\gamma'^{-1}} \left( N'(\O) \right) = \bN'_k$ and that is trivial on $ \Ad_{\gamma'^{-1}} \left( (\bB_-')^k \right)$. In other words, $\chi_g$ computes the sums of the residues of the entries $(i,i+1)$.
\end{lem}

\begin{proof}
Each element of $\bG_k$ can be written uniquely as $\gamma'^{-1} \cdot y \cdot \gamma'$. We set
$$
\chi_g( \gamma'^{-1} \cdot y \cdot \gamma' ) := \chi'(\gamma'^{-1} \cdot \wh\p(y) \cdot \gamma'),
$$
where $\wh\p: (\bB_-')^k \cdot N'(\O) \to N'[t]/t^k$ is the projection. Since $\wh\p$ is multiplicative at sight, $\chi_g$ is a character. The other required properties are obvious.
\end{proof}

In view of this, we henceforth denote the character $\chi_g$ simply by $\chi'$.

\sssec{}

Consider now, for any $\ell \in \mathbb N$ the vector group
$$
\bA_\ell := \bA^{n-1}_\ell := \left(  \begin{array}{l}
t^{-(n-1)\ell}\O \\
t^{-(n-2)\ell}\O \\
\vdots \\
t^{-\ell }\O \end{array} \right)
$$
%
Let $\chi_a$ be the character on $\bA_\ell$ that computes the residue of the \emph{last entry}.
%
The second important feature of $\bG_k$ is that it acts on $\bA_k$. We thus form the semidirect product
$$
\bH_k :=\bG_k \ltimes \bA_k =  
\left( \begin{array}{cc}
\bG_k &  \bA_k \\
0 & 1 \end{array} \right).
$$
It is evident that $\chi_a$ in $\bG_k$-invariant; hence the sum $\chi' + \chi_a$ defines a characte on $\bH_k$, also indicated by $\chi$.

\sssec{}

Let $\bA := \mathbb A^{n-1} \ppart$, identified with its dual via the residue pairing. The annihilator of $\bA_k$ in $\bA ^\vee \simeq \bA$ is obviously
$$
\bA_k^\perp =
\left(  \begin{array}{l}
t^{(n-1)k}\O \\
t^{(n-2)k}\O \\
\vdots \\
 t^{k}\O \end{array} \right).
$$
%
%
Let $\bL_k := \{e_{n-1}\} + \bA_k^\perp \subset \bA^\vee$.
Notice the third important feature of $\bG_k$, which is actually the main motivation for the theory of Section \ref{SEC:trans_act}.

\begin{lem} \label{lem:last_row_e_n-1}
The dual $\bG_k$-action on $\bA^\vee$ makes $\bL_k$ into a pro-$\bG_k$-scheme  (see \ref{sssec:quotient-wise}), with $\bG_k$ acting transitively. The stabilizer of  $e_{n-1} \in \bL_k$ is exactly $\bH'_k$.
\end{lem}

\begin{proof}
The action in question preserves $\bL_k$, it is transitive at sight and it fits the paradigm of Example \ref{ex:proHscheme}.
To show the last claim, it suffices to notice that $\bH'_k$ is obtained from $\bG_k$ by setting the last row of the latter to be $(0, \ldots,0,1)$.
\end{proof}


\ssec{Proof of the main theorem}


We have introduced all the necessary tools to prove Theorem \ref{thm:refinement}, which we have split in two: Propositions \ref{lem:PHI} and \ref{lem:PSI}.

\begin{prop} \label{lem:PHI}
For any category equipped with an action of $\bP$, the functor
$$
\Phi:= \Av_*^{\bH_k, \chi} : \C^{\bN,\chi} \longrightarrow \C^{\bH_k, \chi}
$$
is an equivalence.
\end{prop}

\begin{proof}
We proceed by induction on $n$, the claim being tautologically true for $n=1$. 
First off, $\C^{\bA, \chi}$ retains an action of $\bP' \subseteq \bG'$, so that $\Phi'$ can be applied to $(\C^{\bA, \chi})^{\bN', \chi}$. Secondly, the equivalence
$\sigma : (\uprestr \C {\chi_a} )^{\bH'_k, \chi'}
\xrightarrow{ \, \,}
(\uprestr \C {\bL_k} )^{\bG_k, \chi} \simeq \C^{\bH_k}
$
of Proposition \ref{prop:sigma_equiv}, goes over via Fourier transform to 
$$
\Av_*^{\bH_k, \chi} : \C^{\bH'_k \ltimes \bA, \chi} \to \C^{\bH_k}.
$$
Hence, the composition
$$
\C^{\bN, \chi} \simeq (\uprestr \C {\chi_a })^{\bN', \chi'} \xrightarrow{ \, \, \Phi' =\Av_*^{\bH'_k, \chi'} \, \,}
(\uprestr \C {\chi_a} )^{\bH'_k, \chi'}
\xrightarrow{ \, \, \sigma \,\,}
(\uprestr \C {\bL_k} )^{\bG_k, \chi} \simeq \C^{\bH_k}
$$
is manifestly isomorphic to $\Phi$ and it is an equivalence by the induction hypothesis.
\end{proof}

This concludes the proof the first part of Theorem \ref{thm:refinement}. Before proceeding with the analysis of $\Theta$, let us observe that

\begin{rem} \label{rem:leftleftadjoints}
For any $k \geq 1$, the functor $\Phi^{-1}$ equals
$$
\Av_!^{\bN, \chi} \simeq \uscolim{\ell \geq k} \Av_!^{\bN_\ell, \chi}: \C^{\bH_k,\chi} \longrightarrow \C^{\bN, \chi}.
$$
Likewise, $\sigma^{-1} \simeq \Av_!^{\bA, \chi_a} : \C^{\bH_k, \chi} \to \C^{\bH'_k \ltimes \bA, \chi}$.
\end{rem}


\begin{prop}  \label{lem:PSI}
For any category equipped with an action of $\bP$, the functor
$$
\Psi := \pr_{\bN, \chi}: \C^{\bH_k, \psi} \to \C_{\bN ,\chi}
$$
is an equivalence.
\end{prop}


\begin{proof}
Since $\C_{\bA, \chi}$ retains an action of $\bG'$, the functor 
$$
\Psi': (\C_{\bA, \chi})^{\bH_k', \chi'} \to (\C_{\bA, \chi})_{\bN', \chi'} \simeq \C_{\bN, \chi}
$$
is an equivalence by induction. Proposition \ref{prop:X/H---x/S} provides the equivalence
$$
\rho: 
(\uprestr \C {\bL_k})^{\bG_k, \chi} \to (\restr \C {\chi_a})^{\bH'_k},
$$
which goes over via Fourier transform to 
$$
\pr_{\bA, \chi} : \C^{\bH_k, \chi} \xto{\, \, \simeq \, \,} (\C_{\bA, \chi})^{\bH_k', \chi}.
$$
The resulting equivalence 
$$
\C^{\bH_k, \chi} 
\xrightarrow{\, \, \pr_{\bA, \chi}}
\big( \C_{\bA, \chi}\big)^{\bH'_k, \chi'} 
\xrightarrow{\, \, \Psi'}
\big( \C_{\bA, \chi}\big)_{\bN', \chi'}
\simeq
\C_{\bN, \chi}
$$
is evidently equivalent to $\Psi$.
\end{proof}


\sssec{}

Proposition \ref{prop:comp=T} is what remains to be proven. We have the semi-direct product decomposition $\bN_\ell \simeq \bN_\ell' \ltimes \bA_\ell$, for any $\ell \geq 1$. Fix the dimension theory of $\bA$ determined by $\dim( \bA_k) = 0$ and let $\Theta_\bA : \C_{\bA,\chi} \to \C^{\bA, \chi}$ be the corresponding renormalized $*$-averaging functor.
With these choices,
\begin{equation} \label{blah}
\Theta_\bN \simeq \Theta_{\bN'} \circ \Theta_{\bA}.
\end{equation}
%
%
Recall now that $\sigma \circ \rho \simeq \Theta_{\chi_a}$ and that, under Fourier transform, $\Theta_{\chi_a}$ goes over to $\Theta_\bA$ (Lemma \ref{lem:FT-of-omega-ren}).

\medskip

Hence, Proposition \ref{prop:comp=T} is a consequence of the following statement:

\begin{lem}
The square
%
\begin{gather}
\xy
(0,0)*+{ \C^{\bN,\chi}  }="00";
(32,0)*+{  \C_{\bN,\chi} }="10";
%
%
(00,15)*+{ (\C^{\bA,\chi})^{\bH'_k,\chi}}="01";
(32,15)*+{  (\C_{\bA,\chi})^{\bH_k',\chi} }="11";
%
{\ar@{->}_{ \Phi'   } "00";"01"};
{\ar@{<-}^{ \Theta_{\bN} } "00";"10"};
%
{\ar@{<-}_{ \Psi'  } "10";"11"};
{\ar@{<-}^{ \Theta_{\bA} } "01";"11"};
%
%
\endxy
\end{gather} 
is commutative.
\end{lem}

\begin{proof}
We may assume by induction that $\Theta_{\bN'} \simeq \Phi^{-1} \circ \Psi^{-1}$. Thanks to (\ref{blah}), it suffices to prove that $\Phi'$ (viewed as an endofunctor of $\C$) commutes with $\Theta_{\bA}$ (also viewed as an endofunctor of $\C$). This follows from Lemma \ref{lem:AvAv_commute}. In order to apply the Lemma, one has to write $\bA$ as a colimit of vector spaces on which $\bH_k'$ acts; this can always be done.
\end{proof}



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\newcommand{\eprint}[1]{Preprint {\tt #1}}\newcommand{\available}[1]{Available
  from \url{#1}}
\begin{thebibliography}{BIK08}

\bibitem[AG]{AG}
D. Arinkin, D. Gaitsgory.
\newblock Singular support of coherent sheaves, and the geometric Langlands conjecture. \newline
\newblock \eprint{arXiv:1201.6343}.

\bibitem[AB]{AB}
S. Arkhipov, R. Bezrukavnikov.
\newblock Perverse sheaves on affine flags and Langlands dual group. \newline
\newblock \eprint{arXiv:0201073}.


\bibitem[BZFN]{BFN}
D.~Ben-Zvi, J.~Francis, D.~Nadler.
\newblock Integral transforms and {D}rinfeld centers in derived algebraic geometry.
\newblock {\em J. Amer. Math. Soc.}, 23(4):909--966, 2010.

\bibitem[BZN]{BN}
D.~Ben-Zvi, D.~Nadler.
\newblock The character theory of a complex group. 
\newblock \eprint{arXiv:0904.1247}.

\bibitem[Ba]{Ba}
J. Barlev.
\newblock Notes from a seminar.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Apr15(LimsofCats).pdf}.

\bibitem[Bei]{Beil}
A. Beilinson.
\newblock Remarks on topological algebras.
\newblock \eprint{arXiv:0711.2527}.


\bibitem[Bez]{Bez1}
R. Bezrukavnikov.
\newblock On two geometric realizations of an affine Hecke algebra
\newblock \eprint{arXiv:1209.0403}.

%%%%\bibitem[B2]{B2}
%%%%R. Bezrukavnikov.
%%%%\newblock Noncommutative counterparts of the Springer resolution.
%%%%\newblock \eprint{arXiv:0604445}.

\bibitem[D]{Dr}
V. Drinfeld.
\newblock Infinite-dimensional vector bundles in algebraic geometry. 
\newblock \eprint{arXiv:0309155}.

\bibitem[DG]{DGa}
V. Drinfeld, D. Gaitsgory.
\newblock On some finiteness questions for algebraic stacks. 
\newblock \eprint{arXiv:1108.5351}.

%\bibitem[DG1]{DrGa1}
%V.~Drinfeld, D.~Gaitsgory.
%\newblock Compact generation of the category of {D}-modules on the stack of {G}-bundles on a curve.
%\newblock \eprint{arXiv:1112.2402}.
%
%\bibitem[Fr]{Fr}
%J. Francis.
%\newblock The tangent complex and Hochschild cohomology of $\on{E}_n$-rings
%\newblock \eprint{arXiv:1104.0181}.

\bibitem[F]{F}
E. Frenkel.
\newblock Langlands Correspondence for Loop Groups.
\newblock Cambridge Studies in Advanced Mathematics
\textit{Cambridge University Press},  June 2007.

\bibitem[FB]{FB}
E. Frenkel, D. Ben-Zvi.
\newblock Vertex Algebras and Algebraic Curves.
\newblock Mathematical Surveys and Monographs (vol. 88). \textit{American Mathematical Society}, 2004.

\bibitem[FGV]{FGV}
E. Frenkel, D. Gaitsgory, K. Vilonen.
\newblock Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves.
%\newblock \eprint{arXiv:1111.2087}.

\bibitem[FG0]{FG2}
E. Frenkel, D. Gaitsgory.
\newblock Local geometric Langlands correspondence and affine Kac-Moody algebras.
\newblock \eprint{arXiv:0508382}.

\bibitem[FG1]{FG3}
E. Frenkel, D. Gaitsgory.
\newblock Local geometric Langlands correspondence: the spherical case.
\newblock \eprint{arXiv:0711.1132}.

\bibitem[FG2]{FG6}
E. Frenkel, D. Gaitsgory.
\newblock D-modules on the affine flag variety and representations of affine Kac-Moody algebras.
\newblock \eprint{arXiv:0712.0788}.

\bibitem[G]{Ga}
D. Gaitsgory.
\newblock Twisted Whittaker model and factorizable sheaves.
\newblock \eprint{arXiv:0705.4571}.

\bibitem[G0]{DG}
Notes on Geometric Langlands:
\newblock Generalities on {D}{G} categories.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G1]{Stacks}
Notes on Geometric Langlands:
\newblock Stacks. %\newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G2]{QCoh}
Notes on Geometric Langlands:
\newblock Quasi-coherent sheaves on stacks. %\newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G3]{IndCoh}
Notes on Geometric Langlands:
\newblock Ind-coherent sheaves. %\newline
\newblock \eprint{arXiv:1105.4857}.

\bibitem[G4]{Contr}
Notes on Geometric Langlands:
\newblock The contractibility of the space of rational maps. \newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G5]{Whit}
Notes on Geometric Langlands:
\newblock The extended Whittaker category.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G6]{Ran}
Notes on Geometric Langlands:
\newblock Categories over the Ran space.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G7]{shvcat}
D. Gaitsgory.
\newblock Sheaves of categories on prestacks.


\bibitem[GR0]{Crys}
D.~Gaitsgory, N. Rozenblyum.
\newblock Crystals and {D}-modules.
\newblock \eprint{arXiv:1111.2087}.

\bibitem[GR1]{indschemes}
D.~Gaitsgory, N. Rozenblyum.
\newblock DG indschemes.
\newblock \eprint{arXiv:1108.1738}.

\bibitem[GR2]{GR_corresp}
D.~Gaitsgory, N. Rozenblyum.
\newblock Private communication.
%\newblock \eprint{arXiv:1108.1738}.

\bibitem[KV]{KV}
M. Kapranov, E. Vasserot.
\newblock Vertex algebras and the formal loop space. \newline
\newblock \eprint{arXiv:0107143}.


\bibitem[La]{Laumon}
G. Laumon.
\newblock {\em Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil}.
\newblock {\em Publications Mathematiques de l'IHES}, (65): 131-210.

\bibitem[L0]{L0}
J.~Lurie.
\newblock {\em Higher topos theory}, volume 170 of {\em Annals of Mathematics
  Studies}.
\newblock Princeton University Press, Princeton, NJ, 2009.

\bibitem[L1]{L1}
J.~Lurie.
\newblock Higher algebra.
\newblock \available{http://www.math.harvard.edu/~lurie}.

\bibitem[L2]{L2}
J.~Lurie.
\newblock DAG-XI.
\newblock \available{http://www.math.harvard.edu/~lurie}.

\bibitem[MV]{MV}
I.~Mirkovic and K.~Vilonen.
\newblock Geometric Langlands duality and representations of algebraic groups
  over commutative rings.
\newblock {\em Ann. of Math. (2)}, 166(1):95--143, 2007.

\bibitem[R]{Sam:Dmod}
S.~Raskin.
\newblock PhD thesis.


\bibitem[T]{T}
C.~Teleman.
\newblock Borel-Weil-Bott Theory on the moduli of G-bundles over a curve.
\newblock {\em Invent. Math.},  134 (1998), 1–57.

\bibitem[TT]{TT}
R.~W. Thomason and T.~Trobaugh.
\newblock Higher algebraic {$K$}-theory of schemes and of derived categories.
\newblock In {\em The {G}rothendieck {F}estschrift, {V}ol.\ {III}}, volume~88
  of {\em Progr. Math.}, pages 247--435. Birkh\"auser Boston, Boston, MA, 1990.

\end{thebibliography}



\end{document}



1310.5127.extracted_bib
\begin{thebibliography}{BIK08}

\bibitem[AG]{AG}
D. Arinkin, D. Gaitsgory.
\newblock Singular support of coherent sheaves, and the geometric Langlands conjecture. \newline
\newblock \eprint{arXiv:1201.6343}.

\bibitem[AB]{AB}
S. Arkhipov, R. Bezrukavnikov.
\newblock Perverse sheaves on affine flags and Langlands dual group. \newline
\newblock \eprint{arXiv:0201073}.


\bibitem[BZFN]{BFN}
D.~Ben-Zvi, J.~Francis, D.~Nadler.
\newblock Integral transforms and {D}rinfeld centers in derived algebraic geometry.
\newblock {\em J. Amer. Math. Soc.}, 23(4):909--966, 2010.

\bibitem[BZN]{BN}
D.~Ben-Zvi, D.~Nadler.
\newblock The character theory of a complex group. 
\newblock \eprint{arXiv:0904.1247}.

\bibitem[Ba]{Ba}
J. Barlev.
\newblock Notes from a seminar.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Apr15(LimsofCats).pdf}.

\bibitem[Bei]{Beil}
A. Beilinson.
\newblock Remarks on topological algebras.
\newblock \eprint{arXiv:0711.2527}.


\bibitem[Bez]{Bez1}
R. Bezrukavnikov.
\newblock On two geometric realizations of an affine Hecke algebra
\newblock \eprint{arXiv:1209.0403}.

%%%%\bibitem[B2]{B2}
%%%%R. Bezrukavnikov.
%%%%\newblock Noncommutative counterparts of the Springer resolution.
%%%%\newblock \eprint{arXiv:0604445}.

\bibitem[D]{Dr}
V. Drinfeld.
\newblock Infinite-dimensional vector bundles in algebraic geometry. 
\newblock \eprint{arXiv:0309155}.

\bibitem[DG]{DGa}
V. Drinfeld, D. Gaitsgory.
\newblock On some finiteness questions for algebraic stacks. 
\newblock \eprint{arXiv:1108.5351}.

%\bibitem[DG1]{DrGa1}
%V.~Drinfeld, D.~Gaitsgory.
%\newblock Compact generation of the category of {D}-modules on the stack of {G}-bundles on a curve.
%\newblock \eprint{arXiv:1112.2402}.
%
%\bibitem[Fr]{Fr}
%J. Francis.
%\newblock The tangent complex and Hochschild cohomology of $\on{E}_n$-rings
%\newblock \eprint{arXiv:1104.0181}.

\bibitem[F]{F}
E. Frenkel.
\newblock Langlands Correspondence for Loop Groups.
\newblock Cambridge Studies in Advanced Mathematics
\textit{Cambridge University Press},  June 2007.

\bibitem[FB]{FB}
E. Frenkel, D. Ben-Zvi.
\newblock Vertex Algebras and Algebraic Curves.
\newblock Mathematical Surveys and Monographs (vol. 88). \textit{American Mathematical Society}, 2004.

\bibitem[FGV]{FGV}
E. Frenkel, D. Gaitsgory, K. Vilonen.
\newblock Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves.
%\newblock \eprint{arXiv:1111.2087}.

\bibitem[FG0]{FG2}
E. Frenkel, D. Gaitsgory.
\newblock Local geometric Langlands correspondence and affine Kac-Moody algebras.
\newblock \eprint{arXiv:0508382}.

\bibitem[FG1]{FG3}
E. Frenkel, D. Gaitsgory.
\newblock Local geometric Langlands correspondence: the spherical case.
\newblock \eprint{arXiv:0711.1132}.

\bibitem[FG2]{FG6}
E. Frenkel, D. Gaitsgory.
\newblock D-modules on the affine flag variety and representations of affine Kac-Moody algebras.
\newblock \eprint{arXiv:0712.0788}.

\bibitem[G]{Ga}
D. Gaitsgory.
\newblock Twisted Whittaker model and factorizable sheaves.
\newblock \eprint{arXiv:0705.4571}.

\bibitem[G0]{DG}
Notes on Geometric Langlands:
\newblock Generalities on {D}{G} categories.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G1]{Stacks}
Notes on Geometric Langlands:
\newblock Stacks. %\newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G2]{QCoh}
Notes on Geometric Langlands:
\newblock Quasi-coherent sheaves on stacks. %\newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G3]{IndCoh}
Notes on Geometric Langlands:
\newblock Ind-coherent sheaves. %\newline
\newblock \eprint{arXiv:1105.4857}.

\bibitem[G4]{Contr}
Notes on Geometric Langlands:
\newblock The contractibility of the space of rational maps. \newline
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G5]{Whit}
Notes on Geometric Langlands:
\newblock The extended Whittaker category.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G6]{Ran}
Notes on Geometric Langlands:
\newblock Categories over the Ran space.
\newblock \available{http://www.math.harvard.edu/~gaitsgde/GL/}.

\bibitem[G7]{shvcat}
D. Gaitsgory.
\newblock Sheaves of categories on prestacks.


\bibitem[GR0]{Crys}
D.~Gaitsgory, N. Rozenblyum.
\newblock Crystals and {D}-modules.
\newblock \eprint{arXiv:1111.2087}.

\bibitem[GR1]{indschemes}
D.~Gaitsgory, N. Rozenblyum.
\newblock DG indschemes.
\newblock \eprint{arXiv:1108.1738}.

\bibitem[GR2]{GR_corresp}
D.~Gaitsgory, N. Rozenblyum.
\newblock Private communication.
%\newblock \eprint{arXiv:1108.1738}.

\bibitem[KV]{KV}
M. Kapranov, E. Vasserot.
\newblock Vertex algebras and the formal loop space. \newline
\newblock \eprint{arXiv:0107143}.


\bibitem[La]{Laumon}
G. Laumon.
\newblock {\em Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil}.
\newblock {\em Publications Mathematiques de l'IHES}, (65): 131-210.

\bibitem[L0]{L0}
J.~Lurie.
\newblock {\em Higher topos theory}, volume 170 of {\em Annals of Mathematics
  Studies}.
\newblock Princeton University Press, Princeton, NJ, 2009.

\bibitem[L1]{L1}
J.~Lurie.
\newblock Higher algebra.
\newblock \available{http://www.math.harvard.edu/~lurie}.

\bibitem[L2]{L2}
J.~Lurie.
\newblock DAG-XI.
\newblock \available{http://www.math.harvard.edu/~lurie}.

\bibitem[MV]{MV}
I.~Mirkovic and K.~Vilonen.
\newblock Geometric Langlands duality and representations of algebraic groups
  over commutative rings.
\newblock {\em Ann. of Math. (2)}, 166(1):95--143, 2007.

\bibitem[R]{Sam:Dmod}
S.~Raskin.
\newblock PhD thesis.


\bibitem[T]{T}
C.~Teleman.
\newblock Borel-Weil-Bott Theory on the moduli of G-bundles over a curve.
\newblock {\em Invent. Math.},  134 (1998), 1–57.

\bibitem[TT]{TT}
R.~W. Thomason and T.~Trobaugh.
\newblock Higher algebraic {$K$}-theory of schemes and of derived categories.
\newblock In {\em The {G}rothendieck {F}estschrift, {V}ol.\ {III}}, volume~88
  of {\em Progr. Math.}, pages 247--435. Birkh\"auser Boston, Boston, MA, 1990.

\end{thebibliography}