Generalized Hasse invariants for Shimura varieties of Hodge type
Jean-Stefan Koskivirta\footnote{Dept. of Mathematics
University of Paderborn
Warburger Str. 100
D-33098 Paderborn
Germany
{\tt jsk@math.uni-paderborn.de
1406.2178.tex
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%=====================================================

\begin{document}

\title{Generalized Hasse invariants for Shimura varieties of Hodge type}

\author{Jean-Stefan Koskivirta\footnote{Dept. of Mathematics, University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany, {\tt jsk@math.uni-paderborn.de}}\ \ and Torsten Wedhorn\footnote{Dept. of Mathematics, University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany, {\tt wedhorn@math.uni-paderborn.de}}}

\maketitle

%------------------------------------------------------------------

\noindent{\scshape Abstract.\ } We construct canonical Hasse invariants for arbitrary Shimura varieties of Hodge type.

\bigskip

\noindent{\scshape MSC 2010.\ } 14G35, 14K10, 20G40, 11G18


%==================================================================

\section*{Introduction}

The Hasse invariant is a modular form which has been an important tool for constructing congruences between automorphic forms. It is defined for for certain Shimura varieties that are endowed with a universal abelian scheme (like Siegel Shimura varieties, Hilbert-Blumenthal varieties, or more generelly Shimura varieties of Hodge type). Its non-vanishing locus on the special fiber in positive characteristic $p$ is the locus where the universal abelian scheme is ordinary. It is a section of the $(p-1)$-th power of the Hodge line bundle of the universal abelian scheme.

In this paper we consider the following situation. Let $(\Gbf,\Xbf)$ be a Shimura datum of Hodge type, $K = K_pK^p \subseteq \Gbf(\AA_f)$ an open compact subgroup such that $K_p \subseteq \Gbf(\QQ_p)$ is hyperspecial and such that $K^p \subseteq \Gbf(\AA^p_f)$ is sufficiently small. Let $S = S_K(\Gbf,\Xbf)$ be the special fiber of the canonical integral model of a Shimura variety of Hodge type at a place $v$ of the reflex field $E$ lying over $p$. Then usually the ordinary locus is empty, hence the classical Hasse invariant vanishes (in fact, in the PEL case the ordinary locus is non-empty if and only if $E_v = \QQ_p$ by \cite{Wd_Ord}~Theorem~1.6.3). Instead, the generic Newton stratum for good reductions of Shimura varieties of Hodge type is the so-called $\mu$-ordinary locus $S^{\mu}$, introduced by the second author in \cite{Wd_Ord}, which has been shown to be open and dense by Wortmann (\cite{Wort_MuOrd}). In this paper we define an integer $N \geq 1$, the Hasse number (Definition~\ref{DefHasseNumber}). It depends only on the special fiber $G$ of the reductive group scheme over $\ZZ_p$ defined by $K_p$ and on the conjugacy class of cocharacters defining the reflex field. Then our main result is the following.

\begin{intro-theorem}\label{IntroMain}
There exists a canonical section $H$ of the $N$-th power of the Hodge line bundle $\omega$ on $S$ such that the non-vanishing locus of $H$ is $S^\mu$.
\end{intro-theorem}

We refer to the main text (Theorem~\ref{ExistHasse}) for the precise meaning of ``canonical''. If $\Gbf$ is split over $\QQ_p$ and its derived group is simply connected, then the ordinary locus is non-empty, we obtain $N = p-1$ (Corollary~\ref{HasseSplit}) and $H$ is the classical Hasse invariant.

All other generalizations of Hasse invariants in positive characteristic that we are aware of are special cases of our construction. We give two examples: For Hilbert-Blumenthal varieties partial Hasse invariants have been constructed by Goren in \cite{Goren_Hasse}. This is rephrased in our language in Example~\ref{ExampleHB}. For Shimura varieties of unitary PEL type, Hasse invariants have been considered by Goldring and Nicole in \cite{GoNi_Hasse1} and \cite{GoNi_Hasse2}. We consider a special case in Example~\ref{ExampleUnitary} if $\Gbf_{\QQ_p}$ is non-split; here the Hasse number is $p^2-1$.

A formal argument using the definition of the minimal compactification and its normality shows that the Hasse invariant $H$ can be extended to the minimal compactification and that some power of $H$ can be lifted to characteristic $0$. We define the $\mu$-ordinary locus $S^{\min,\mu}$ of the minimal compactification $S^{\min}$ as the non-vanishing locus of the extension of $H$. As the Hodge line bundle on the minimal compactification is ample, we deduce the following corollary.

\begin{intro-corollary}\label{IntroMuOrdAffine}
The $\mu$-ordinary locus $S^{\min,\mu}$ of the minimal compactification is affine.
\end{intro-corollary}

The idea behind our construction is to consider the Ekedahl-Oort stratification of $S$ which is given by a smooth morphism $\zeta\colon S \to \GZip^{\chi}$ constructed by Zhang (\cite{Zhang_EO}, see also \cite{Wort_MuOrd}). Here $G$ is a reductive reduction of $\Gbf$ over $\FF_p$, $\chi$ is a certain cocharacter of $G$, defined over a finite extension of $\FF_p$, given by the Shimura datum and $\GZip^{\chi}$ is the algebraic stack of $G$-zips of type $\chi$ introduced by Pink, Ziegler, and the second author in \cite{PWZ2} (see Subsection~\ref{EOStrat} for details). By a result of Wortmann (\cite{Wort_MuOrd}), the inverse image of the generic point of $\GZip^{\chi}$ is the $\mu$-ordinary locus $S^{\mu}$.

We then construct a line bundle $\omega^{\flat}$ on $\GZip^{\chi}$ whose pull back via $\zeta$ is the Hodge line bundle on $S$. Moreover we show that the space of global sections of any power $(\omega^{\flat})^{\otimes k}$ over $\GZip^{\chi}$ has dimension at most $1$ and that there is a non-vanishing section $H^{\flat}$ for $k = N$. Then our main technical result (Theorem~\ref{PositivityThm}) shows that its vanishing locus is precisely the complement of the generic point of $\GZip^{\chi}$. Hence $H := \zeta^*(H^{\flat})$ is the desired Hasse invariant.

\bigskip

We now give an overview of the paper. As line bundles on quotient stacks (like $\GZip^{\chi}$) are just line bundles with equivariant structures, we recall in Section~1 several results  on equivariant Picard groups. None of them are probably new but for many of them we did not find a reference or only references with too restrictive hypotheses (such as that the ground field is of characteristic $0$). Thus we included also some proofs.

In Section~2 we examine the Picard group of the stack of $G$-zips and of related stacks like the Bruhat stack. We then state in Section~3 a positivity conjecture and prove the conjecture in a special case which is sufficient for the applications to Shimura varieties stated above.  This is the technical heart of our paper.

In the last section we deduce the above applications to Shimura varieties of Hodge type.

\bigskip

\textbf{Notation and terminolgy}: A linear algebraic group $G$ over a field $k$ is an affine smooth group scheme over $k$. We denote by $X^*(G)$ its group of $k$-rational characters.

%=====================================================================

\tableofcontents

%==================================================================

\section{Equivariant Picard groups}

In this section, $G$ denotes an arbitrary connected linear algebraic
group over an algebraically closed field $k$. Products of schemes are fiber products over $k$. A variety is an integral $k$-scheme of finite type. A $G$-scheme is a $k$-scheme endowed with a $G$-action $G \times X \to X$.

We always consider left actions. If $G$ is an algebraic group and $H$ is an algebraic subgroup we denote by $G/H$ the quotient of $G$ by the left action $(h,g) \sends gh^{-1}$.

\subsection{$G$-linearization of a line bundle}

Let $X$ be a $G$-scheme, where the action is given by a map $a\colon G\times X\to X$. Let $\mathscr{L}$ be a line bundle on $X$. Define the projections $p_{23}\colon G\times G\times X\to G\times X$, $(g,h,x)\mapsto (h,x)$ and $p_2:G\times X\to X$, $(g,x)\mapsto x$. Finally, write $\mu_G$ for the multiplication map $G\times G\to G$.

\begin{definition}
A $G$-linearization of $\mathscr{L}$ is an isomorphism $\phi : a^*(\mathscr{L})\to p_{2}^*(\mathscr{L})$ satisfying the cocycle condition
$$ p_{23}^*(\phi) \circ (\id_G \times a)^*(\phi) =(\mu_G \times \id_X)^*(\phi) $$
\end{definition}

If $\pi\colon L\to X$ is the corresponding geometric line bundle, a linearization of $L$ is an action $a_L : G\times L\to L$ such that :
\begin{itemize} \itemsep1pt \parskip0pt
\parsep0pt
\item[$-$] The map $\pi : L\to X$ is $G$-invariant.
\item[$-$] The induced isomorphism $G\times L \to a^*(L)$ is a morphism of geometric line bundles. 
\end{itemize}
The first condition ensures that we have a cartesian diagram :
$$\xymatrix@1{
G\times L \ar[r]^-{a_L} \ar[d]_{\id_G \times \pi} & L \ar[d]^{\pi} \\
G\times X \ar[r]_-{a} & X 
}$$
The second condition says that $G\times L \simeq a^*(L)$ as geometric line bundles.
We denote by $\Pic^{G}(X)$ the group of isomorphism classes of $G$-linearized line bundles on $X$. There is a natural map
\[
\Pic^{G}(X) \to \Pic(X)
\]
whose image is the subgroup of $G$-linearizable line bundles, denoted by $\Pic_{G}(X)$.

%Assume that the quotient $X//G$ exists as
%a $k$-variety. By this we mean, there is a $k$-variety $Y$ and
%an affine map $\pi\colon X\to Y$ such that $\pi$ is constant on $G$-orbits,
%and such that for $U\subset Y$ affine, the natural map 
%\[
%\mathcal{O}_{Y}(U)\to\mathcal{O}_{X}(\pi^{-1}(U))^{G}
%\]
%is an isomorphism of $k$-algebras. If it exists, the quotient is
%unique. For example, when $H\subset G$ is a subgroup of an algebraic
%group, the quotient $G/H$ exists.
%It is affine if and only if $H$ is reductive (NOT TRUE IN GENERAL).
%There is a natural map
%\[
%\Pic(X//G)\to\Pic^{G}(X)
%\]
%defined by pulling back. It is always injective. When the $G$-action
%is free, it is an isomorphism.
%

The group $\Pic^{G}(X)$ can be identified with the Picard group of the quotient stack $\left[G \bs X\right]$. Then $\Pic^G(X) \to \Pic(X)$ is the homomorphism given by pull back by the projection $X \to [G \bs X]$. Thus, if $H\subset G$ are linear algebraic groups, there is an isomorphism $\Pic(G/H)\simeq \Pic^H(G)$. Notice that in this case, $[G/H]=G/H$ is a scheme.

%which is defined as the stack associated to the pair of maps
%\[
%X\times G\overset{\pr_{X}}{\underset{a}{\rightrightarrows}}X
%\]
%where $a$ is the $G$-action and $\pr_{X}$ the projection
%on $X$.
%
%Let us recall briefly the definitions. A quasicoherent sheaf
%on a stack $\mathcal{X}$ is the following data:
%\begin{itemize}
%\item a family of quasicoherent sheaves $\mathcal{F}_{U}$ for all atlases
%$U\to\mathcal{X}$.
%\item for each morphism of atlases $\phi\colon U\to V$, an isomorphism $a_{\phi}\colon \mathcal{F}_{U}\to\phi^{*}\mathcal{F}_{V}$
%which satisfy the natural cocycle condition.
%\end{itemize}
%We may define in the same manner locally free sheaves on $\mathcal{X}$.
%The Picard group of $\mathcal{X}$ is then the set of isomorphism
%classes of line bundles (i.e locally free sheaves of rank one) on
%$\mathcal{X}$.
%

%------------------------------------------------------------------


\subsection{General results on equivariant Picard groups}

Let us first recall the following result by Rosenlicht (\cite{FoIv}~Corollary 2.2).

\begin{thm}[Rosenlicht]\label{Rosenlicht}
Let $G$ be a connected linear algebraic group and let $f\colon G \to \GG_m$ be a morphism of $k$-schemes such that $f(1) = 1$. Then $f$ is a character.
\end{thm}

For every $k$-scheme $X$, we define :
\[
\Ecal(X) := \GG_m(X) / k^{\times}.
\]
where $\GG_m(X)=\mathcal{O}(X)^{\times}$. Recall the following result from \cite{KKV}~1.3.

\begin{lemma}\label{EXfree}
If $X$ is an integral $k$-scheme of finite type, then $\Ecal(X)$ is a finitely generated free abelian group. 
\end{lemma}

This implies that if $X$ carries an algebraic action by an algebraic group $G$, then the induced action of $G$ on $\Ecal(X)$ factors over $\pi_0(G)$, the finite group of components of $G$. Note that if $H$ is an algebraic group, then the inclusion $X^{*}(H)\subset \GG_m(H)$ induces an isomorphism
\[
X^{*}(H) \cong \Ecal(H)
\]
by Rosenlicht's theorem.

%Let $X$ be a $G$-variety. A geometric quotient of $X$ by $G$
%is a variety $Y$ together with a map $\pi:X\to Y$ satisfying the
%following conditions:
%\begin{itemize}
%\item The map $\pi$ is $G$-equivariant (for the trivial action on $Y$).
%\item For any affine open subset $U\subset Y$, the set $\pi^{-1}(U)$ is
%affine and $\Gamma(U,\mathcal{O}_{Y})\to\Gamma(\pi^{-1}(U),\mathcal{O}_{X})^{G}$
%is an isomorphism. 
%\end{itemize}
%When $X$ admits a quotient and if $G$ acts freely
%on $X$, then $\mbox{Pic}^{G}(X)\simeq\mbox{Pic}(G\backslash\backslash X)$.
%We will need the following proposition :

In the statement of the next proposition we denote by $H^1(\ ,\ )$ group cohomology and by $H^1_{\rm alg}(G,\GG_m(X))$ the subgroup of classes of algebraic cocycles, i.e., of maps $c\colon G(k) \to \GG_m(X)$ corresponding to a morphism $G \times X \to \GG_m$.

\begin{prop}\label{exseqgen}
Let $G$ be a smooth algebraic group, and let $X$ be an irreducible $G$-variety. Then
there are exact sequences:
\[
0\to \frac{\GG_m(X)^G}{k^\times}\to \Ecal(X)^{G}\to X^{*}(G)\to H^1_{\rm alg}(G,\GG_m(X)) \to H^{1}(\pi_0(G)(k),\Ecal(X))
\]
\[
0\to H^1_{\rm alg}(G,\GG_m(X)) \to \Pic^{G}(X) \to \Pic(X)
\]
If $X$ is normal and $G$ connected, the second exact sequence has an extension by a map $\Pic(X) \to \Pic(G)$.
\end{prop}

\begin{proof}
This is \cite{KKV}, Proposition 2.3 and Lemma~2.2. The general assumption that $k$ is of characterstic $0$ is not needed in the proof there, only the smoothness of the algebraic group $G$ is used.
\end{proof}

If $G$ is connected, the two exact sequences of Proposition~\ref{exseqgen} combine into a longer one. We obtain the following result.

\begin{thm}\label{exseq}
Let $G$ be a smooth connected algebraic group and let $X$ be a normal irreducible $G$-variety over $k$. Then we have an exact sequence of abelian groups
\[
1 \to k^{\times} \to \GG_m(X)^G \to \Ecal(X) \to X^{*}(G) \to \Pic^{G}(X) \to \Pic(X) \to \Pic(G)
\]
\end{thm}

For later use let us give the explicit construction of these maps.

\begin{itemize}
\item[$-$]
If $f\in \GG_m(X)$, then for any $g\in G$, there is
$\lambda_f(g)\in k^{\times}$ such that
\[
g\cdot f=\lambda_f(g)f.
\]
Then $\lambda_f$ is a character of $G$. This defines the map $\Ecal(X)\to X^{*}(G)$, $f \sends \lambda_f$.
\item[$-$]
Then, if $\lambda \in X^{*}(G)$, we define a $G$-linearization of the
trivial line bundle on $X$
\[
G\times X\times \AA^1_k \to X \times \AA^1_k
\]
by $(g,x,s)\mapsto(g\cdot x,\lambda(g)s)$. This yields a map
$X^{*}(G)\to\Pic^{G}(X)$.
%Actually, giving a linearization
%of $X\times k$ is the same as giving a map $\gamma\colon G\times X\to k^{\times}$
%such that 
%\[
%\gamma(g,g'\cdot x)\gamma(g,x)=\gamma(gg'\cdot x)
%\]
%In turn, this boils down to giving a (group-theoretic) 1-cocycle $G\to\GG_m(X)$,
%such that the associated map $G\times X\to k$ is algebraic. The group
%of classes of such cocycles is often denoted by $H_{\rm alg}^{1}(G,\GG_m(X))$.
%The theorem of Rosenlicht shows that any such cocycle comes from a
%character of $G$ (i.e has values in $k^{\times}\subset\GG_m(X)$).
\item[$-$] Finally, the map $\Pic(X) \to \Pic(G)$ which terminates
the sequence depends on the choice of an element $x_{0} \in X$. An
element $L\in\Pic(X)$ is sent to $a^{*}(L)_{|G\times\left\{ x_{0}\right\} }$.
\end{itemize}

\begin{cor}\label{ExactGroupQuot}
Let $G$ be a smooth connected algebraic group, let $H$ be a smooth connected algebraic subgroup of $G$, and let $\pi\colon G \to G/H$ be the quotient by left action $(h,g) \sends gh^{-1}$. Then there is an exact sequence
\[
1 \to X^*(G)^H \to X^*(G) \to X^*(H) \ltoover{\xi} \Pic(G/H) \to \Pic(G) \to \Pic(H).
\]
\end{cor}

%Here $\rho$ is given by $\lambda \sends \lambda\rstr{H}$ (it is the negative of the map $E(G) \to X^*(H)$ of Theorem~\ref{exseq}).
The homomorphism $\xi$ attaches to $\lambda \in X^*(H)$ the invertible $\Ocal_{G/H}$-module $\Lcal_{\lambda}$ whose sections over $U \subseteq G/H$ are given by
\begin{equation}\label{EqSectionLB}
\Gamma(U,\Lcal_{\lambda}) = \sett{f\colon \pi^{-1}(U) \to k}{$f(gh^{-1}) = \lambda(h)f(g)$ for $g \in \pi^{-1}(U)$, $h \in H$}.
\end{equation}
The corresponding geometric bundle is $(G\times \AA^1_k)/H$, where $H$ acts on $\AA^1_k$ via $\lambda : H\to \GG_{m}$.
%Finally $i\colon H \to G$ is the inclusion (then $i^*$ is the composition of the map $\Pic(G) \to \Pic(H)$ for the choice of $x_0 = 1$ of Theorem~\ref{exseq} and the isomorphism $\Pic(H) \iso \Pic(H)$ given by pullback with $h \sends h^{-1}$).


\begin{rem}
If $\pi\colon G \to G/H$ has locally for the Zariski topology a section, then $i^*\colon \Pic(G) \to \Pic(H)$ is surjective (\cite{Sansuc}~Prop.~6.10).
\end{rem}

If the group acting is not connected, we do not have in general a long exact sequence as in Theorem \ref{exseq}. However, this is true in the case of $G/H$ where $H\subset G$ is a subgroup (or more generally when the group acts trivially on $\Ecal(X)$). 

\begin{prop}\label{exactseqs}
Let $G$ be a smooth connected algebraic group and $H$ a smooth subgroup of $G$. Then there is an exact sequence:
\[
0 \to \Ecal(G/H) \to X^*(G) \to X^{*}(H) \to \Pic^{H}(G) \to \Pic(G).
\]
\end{prop}

\begin{proof}
We apply Proposition~\ref{exseqgen} for $X = G$ and $G = H$. The group $H$ acts trivially on $\Ecal(G)=X^{*}(G)$, since $\chi(gh^{-1})=\chi(g)\chi(h)^{-1}$
for all $g\in G$, $h\in H$, $\chi\in X^{*}(G)$, so $h\cdot\chi=\chi$
in $\Ecal(G)$. It follows that $H^{1}(\pi_0(H),\Ecal(G)) = \Hom_{Grp}(\pi_0(H),\Ecal(G)) = 0$
because $\pi_0(H)$ is finite and $\Ecal(G)$ is torsion-free.
\end{proof}

We will also need the following proposition.

\begin{proposition} \label{charpic}
Let $H \subset G$ be algebraic groups. Then there is a natural isomorphism:
\[
\Pic^G(G/H)\simeq X^*(H).
\]
\end{proposition}

\begin{proof}
One has $\Pic^G(G/H)\simeq \Pic([G\backslash (G/H)])\simeq \Pic([1/H])\simeq \Pic^H(1)\simeq X^*(H)$.
\end{proof}

%------------------------------------------------------------------


\subsection{Functoriality of the equivariant Picard group}

Let $G$, $G'$ be algebraic groups and $f\colon G\to G'$ a morphism of algebraic groups. Let $X$ be a $G$-scheme, let $X'$ be a $G'$-scheme, and $\pi\colon X\to X'$ be an $f$-equivariant morphism, i.e. a morphism of $k$-schemes such that
\begin{equation}
\pi(g\cdot x)=f(g)\cdot\pi(x)\quad\forall g\in G,x\in X\label{compcond}
\end{equation}
Then $\pi$ induces naturally a homomorphism
\begin{equation}\label{EqPullBack}
\pi^*\colon \Pic^{G'}(X')\to\Pic^{G}(X)
\end{equation}
by interpreting $\pi^*$ as the pull-back by the induced map of stacks
\[
\left[G\backslash X\right]\to\left[G'\backslash X'\right].
\]
A more concrete description of $\pi^*$ is the following. Let $L'\in\Pic^{G'}(X')$
be a (geometric) $G'$-linearized line bundle. Then define a $G$-action on $L = X \times_{X' }L'$ by 
\[
(g,(x,l')) \sends (gx,(gx,f(g)l')).
\]
%
%considering the natural diagram
%\[ \xymatrix{
%G\times L \ar[dd] \ar[rd] \ar[rr] & & G'\times L' \ar'[d][dd] \ar[rd] \\
%& L \ar[dd] \ar[rr] & & L' \ar[dd] \\
%G\times X \ar'[r][rr] \ar[rd] & & G'\times X' \ar[rd] \\
%& X \ar[rr] & & X'.
%} \]
%

If $X$ and $X'$ are smooth varieties and if $G$, $G'$ are smooth and connected, we obtain a natural morphism between the exact sequences provided by Theorem~\ref{exseq}.

%$$\xymatrix{
% 0\ar[r] &k^{\times} \ar[r] \ar@{=}[d] & \left(\GG_m(X')\right)^{G'} \ar[r]\ar[d]^{\pi^{*}} & E(X') \ar[r] \ar[d]^{\pi^*} & X^{*}(G') \ar[r] \ar%[d]^{f^*} & \Pic^{G'}(X') \ar[r]\ar[d]^{\pi^*} & \Pic(X') \ar[r]\ar[d]^{\pi^*} &  \Pic(G') \ar[d]^{f^*} \\
%0\ar[r] &k^{\times} \ar[r]  & \left(\GG_m(X)\right)^{G} \ar[r] & E(X) \ar[r]  & X^{*}(G) \ar[r]  & \Pic^{G}(X) \ar[r] & \Pic(X) \ar[r] &  \Pic%%%(G) 
%}$$

%------------------------------------------------------------------

\subsection{The case of algebraic groups defined over finite fields}

In this section we assume that $k$ is an algebraic closure of a finite field $\FF_q$ with $q$ elements. We denote by $\sigma \in \Gal(k/\FF_q)$ the arithmetic Frobenius $a \sends a^q$. We begin with the well-known theorem of Lang (e.g., \cite{Steinberg_Endo}):

\begin{thm}[Lang]\label{thmLangSteinberg}
Let $G$ be a connected algebraic group defined over $\mathbb{F}_{q}$. Let $\varphi \colon G\to G$ be the $q$-th power Frobenius. Then the Lang map
\[
L\colon G\to G, \qquad x\mapsto \varphi(x)x^{-1}
\]
is finite \'etale and surjective. 
\end{thm}

\begin{cor}\label{CorLang}
Let $G$ be a connected algebraic group defined over $\mathbb{F}_{q}$. Then the Lang map $L$ of Theorem~\ref{thmLangSteinberg} induces an isomorphism of varieties $G/G(\mathbb{F}_{q}) \iso G$.
\end{cor}

\begin{proof}
Clearly $L$ induces an injective map $G/G(\mathbb{F}_{q})\to G$ which is surjective and finite \'etale by Theorem~\ref{thmLangSteinberg} and hence an isomorphism.
%To see
%that it is an isomorphism, it suffices to prove that $g:G\to G$ is
%finite �tale. It is clearly finite and flat, so we need only show
%that the induced map on tangent spaces at $1$ is bijective. For this,
%we may assume that $G=GL_{n}$. An easy computation shows that the
%differential of $g$ at $1$ is $h\mapsto-h$, so $g$ is �tale.
\end{proof}


\begin{prop}\label{surject}
Let $G$ be a connected linear algebraic group defined over
$\mathbb{F}_{q}$. Assume that $\Pic(G)=0$. Then there is an exact sequence :
\[
0 \to X^{*}(G)\to X^{*}(G)\to \Hom(G(\mathbb{F}_{q}),k^{\times}) \to 0
\]
where the first map is $\chi \mapsto \sigma \cdot \chi-\chi$.
\end{prop}

\begin{proof}
If $\sigma\cdot\chi = \chi$, the image of $\chi$ is contained in the finite group $\mu_{q-1}(k)$. As $G$ is connected, this implies that $\chi$ is trivial. Hence the first map is injective. The sequence is clearly exact in the middle. Write $H=G(\mathbb{F}_{q})$. The Lang map $L\colon G\to G$, $x\mapsto \varphi(x)x^{-1}$ induces an isomorphism of varieties $G/H \to G$, so we deduce that $\Pic^H(G) = \Pic(G/H) =0$. Hence $X^{*}(G) \to X^{*}(H)$ is surjective by Proposition~\ref{exactseqs}.
\end{proof}


\begin{rem}\label{NeedPicZero}
This result is false without the condition $\Pic(G) = 0$. Consider $G = \PGL_{n}$ over $\mathbb{F}_{q}$. Then $G(\mathbb{F}_{q}) = \GL_{n}(\mathbb{F}_{q})/\mathbb{F}_{q}^{\times}$ because $H^{1}(\mbox{Gal}(k/\mathbb{F}_{q}),k^{\times}) = 0$ by Hilbert 90. Then the derived group of $G(\mathbb{F}_{q})$ is $\SL_{n}(\mathbb{F}_{q})/\mu_{n}(\mathbb{F}_{q})$ and the abelianization of $G(\mathbb{F}_{q})$ is $\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times n}\simeq\mathbb{Z}/d\mathbb{Z}$ where $d=\textrm{gcd}(q-1, n)$. It follows that $\mbox{Hom}(G(\mathbb{F}_{q}),k^{\times})\simeq\mathbb{Z}/d\mathbb{Z}$, whereas $X^{*}(G)=0$.

Note that $\Pic(G) \cong \ZZ/n\ZZ$ by Proposition~\ref{PicardReductive} below. 
\end{rem}

\begin{example}\label{CharSLn}
For example, if $G=SL_n$, then the above proposition implies that there are no nontrivial group homomorphisms $SL_n(\mathbb{F}_q)\to k^{\times}$.
%Evariste Galois claimed without proof in 1831 that the group $PSL_{n}(\mathbb{F}_{p})$ is simple for all $p>3$. This was proved for any finite %field of characteristic $>3$ some time later by Jordan, Moore, Dickson.
\end{example}

%---------------------------------------------------------------------

\subsection{$G$-varieties and divisors}

Let $X$ be a $k$-variety. As usual we denote the free group generated by the irreducible subvarieties of codimension $i\geq0$ by $Z^{i}(X)$. Elements of $Z^{1}(X)$ are the Weil divisors on $X$.

If $X$ is a $G$-variety, we define an action of $G$ on $Z^{i}(X)$ in the obvious way: For $D=\sum n_{C}\left[C\right]\in Z^{i}(X)$ we set
\[
g\cdot D=\sum n_{C}\left[g\cdot C\right].
\]
Let $Z^{i}(X)^{G}$ be the subgroup of $G$-invariant elements of $Z^{i}(X)$.

Assume that $X$ is regular. Recall that the locally free $\mathcal{O}_{X}$-module $\mathcal{L}(D)$ of rank $1$ associated to $D = \sum_C n_C[C] \in Z^{1}(X)$ is defined by
\[
\mathcal{L}(D)(U) = \sett{f\in K(X)}{$v_C(f) + n_C \geq0$ for all $C$ intersecting $U$}. 
\]
Thus we see that when $D\in Z^{1}(X)^{G}$ and $g\in G$, there is
a natural isomorphism $a_{g}\colon g^{*}\mathcal{L}(D)\to\mathcal{L}(D)$
defined by $f\mapsto g\cdot f$ (notice that $\div(g\cdot f) = g \cdot\div(f)$).
These isomorphisms $\left(a_{g}\right)_{g\in G}$ satisfy the cocycle
condition $a_{gh}=a_{h}\circ\left(h^{*}a_{g}\right)$ so in turn,
they define a $G$-linearization of $\mathcal{L}(D)$. We have constructed
a commutative diagram
$$\xymatrix{
Z^1(X)^G\ar[d]^{\mathcal{L}} \ar@{^{(}->}[r]  & Z^1(X)\ar[d] \\
\Pic^G(X) \ar[r] &  \Pic(X)
}$$
Of course this is a very special instance of the theory of equivariant Chow groups. Assume from now on that $X$ is a smooth $G$-variety.

\begin{proposition}\label{inject}
Let $G$ be a smooth algebraic group and let $X$ be an irreducible smooth $G$-variety. There is a commutative diagram with exact rows and columns:
$$
\xymatrix{
& & & 0 \ar[d]& \\
& 0 \ar[d] & & H^1_{\rm alg}(G,\GG_m(X)) \ar[d] \ar@{^{(}->}[r]& H^1(G,\GG_m(X)) \ar[d] \\
0 \ar[r]& \frac{K(X)^{\times G}}{\GG_m(X)^G} \ar[r]\ar[d] & Z^1(X)^G
 \ar[d]^= \ar[r]^{\mathcal{L}} & \Pic^G(X) \ar[d] \ar[r] & H^1(G,K(X)^\times) \ar[d] \\
0 \ar[r] & \left( \frac{K(X)^\times }{\GG_m(X) }\right)^G \ar[r] \ar[d]& Z^1(X)^G \ar[r] &  \Pic_G(X) \ar[d] \ar[r] & H^1 \left(G,\frac{K(X)^\times}{\GG_m(X)} \right) \\
& H^{1}_{\rm alg}(G,\GG_m(X)) \ar[d] && 0 & \\
& H^{1}(G,K(X)^{\times}) & & &
}$$
\end{proposition}

\begin{proof}
For us only the central square of this diagram will be important. First, let us define the maps. Let $\mathcal{L}\in \textrm{Pic}_{G}(X)$ be a $G$-linearizable line bundle on $X$. We may assume that $\mathcal{L}=\mathcal{L}(D)$ for some Weil divisor $D$ on $X$. Then $g\cdot D=D+\textrm{div}(f_g)$ for some $f_g \in \frac{K(X)^\times}{\GG_m(X)}$. This defines a cocycle, independant of the choice of $D$. We get a map $\textrm{Pic}_G(X) \to H^1 \left(G,\frac{K(X)^\times}{\GG_m(X)} \right)$.

Now, let $\tilde{\mathcal{L}}\in \textrm{Pic}^G(X)$ and let $\Lcal\in \Pic(X)$ be the underlying line bundle. Again, we may assume that $\Lcal=\mathcal{L}(D)$ for some Weyl divisor $D$. The $G$-linearization of $\mathcal{L}$ gives an isomorphism $\mathcal{L}(gU)\to \mathcal{L}(U)$ for all open subsets $U$ and all $g\in G$. Choose $U\subset X-|D|$. Then $\mathcal{L}(U)=\mathcal{O}_X(U)$, so the element $1\in \mathcal{L}(U)$ is mapped to some $\lambda_g \in \mathcal{L}(gU)\subset K(X)$. This defines a 1-cocycle $g\mapsto \lambda_g$ in $H^{1}(G,K(X)^{\times})$.

Finally, we define also in the above diagram the two maps with target $Z^1(X)^G$ by $f\mapsto \div(f)$. The commutativity of the diagram is an easy exercise.

First, we prove the exactness of the upper horizontal sequence at the term $ Z^1(X)^G $. So let $D$ be a $G$-invariant Weil divisor, mapping to zero in $\textrm{Pic}^G(X)$. In particular, we can write $D=\textrm{div}(f)$ with some $f\in K(X)^\times$. Then the map $$\gamma : g\mapsto \frac{g\cdot f}{f}$$ is an algebraic cocycle mapping to zero in $\textrm{Pic}^G(X)$. It follows that this cocycle is a coboundary, so there is a function $h\in \GG_m(X)$ such that $\gamma(g)=\frac{g\cdot h}{h}$ for all $g\in G$. It follows that $\frac{f}{h} \in K(X)^{\times G}$ and this shows the first part.

Secondly, we prove exactness of the lower horizontal sequence at the term $\textrm{Pic}_G(X)$. Let $\mathcal{L}=\mathcal{L}(D)$ be a $G$-linearizable line bundle on $X$, such that the associated cocycle of  $\frac{K(X)^\times}{\GG_m(X)}$ is a coboundary. By definition, there is $f\in K(X)^\times$ such that $g\cdot D=D+\textrm{div}\left( \frac{g\cdot f}{f} \right)$ and it follows that $D-\textrm{div}(f)$ is a $G$-invariant Weil divisor mapping to $\mathcal{L}$. We will not need the rest of the diagram, so we leave the proofs to the reader.


%Thirdly, let $f\in \left( \frac{K(X)^\times }{\mathcal{O}(X)^\times }\right)^G$. Then for all $x\in X$, there exists $\lambda_{g} \in \mathcal{O}(X)^\times$ such that $$f(g^{-1}\cdot x)=\lambda_g(x)f(x)$$ for all $g\in G$ and $x\in X$. We have to show that the induced function $G\times X\to k^\times$, $(g,x)\mapsto \lambda_g(x)$ is algebraic. Consider the function $$\phi(g,x)=\frac{f(g^{-1}\cdot x)}{f(x)}\in K(G\times X)$$ For all $g\in G$, $\phi(g,-)$ extends to a non vanishing function on ${g}\times X$, namely $\lambda_g$. This proves that $\phi$ does not have any poles or zeroes in $G\times X$, and thus extends to a non-vanishing function $\phi \in \mathcal{O}(G\times X)^\times$. This concludes the proof.

\end{proof}


\subsection{$G$-varieties with open orbit}\label{GVarOpen}

In this subsection we assume that $X$ is a smooth irreducible $G$-variety which has an open $G$-orbit $U\subset X$. In this case it is clear that $K(X)^G=k$, so the map $\mathcal{L}\colon Z^1(X)^G \to \Pic^G(X)$ is injective by Proposition~\ref{inject}. We assume further that $G$ is connected. Then $G$ necessarily acts on the finitely many irreducible components of $X \setminus U$ trivially. Therefore the group $Z^1(X)^G$ is the free group generated by those irreducible components in $X \setminus U$ which are of codimension $1$ in $X$.

\begin{proposition}\label{PicardDiagramOpenOrbit}
There is the following commutative diagram with exact rows and columns.
\begin{equation}\label{diag}
\begin{aligned}
\xymatrix{ &  &  &  & 0\ar[d]\\
 &  &  & 0 \ar[d] & \frac{\Ecal(U)}{\Ecal(X)}\ar@{^{(}->}[d]\\
 &  0 \ar[d] &  & Z^{1}(X)^{G}\ar@{^{(}->}[d]^{\mathcal{L}}\ar@{=}[r] & Z^{1}(X)^{G}\ar[d]^{\mathcal{L}}\\
0\ar[r] & \Ecal(X)\ar[r]\ar@{^{(}->}[d] & X^{*}(G)\ar[r]\ar@{=}[d] & \Pic^{G}(X)\ar[r]\ar[d] & \Pic_{G}(X)\ar[r]\ar[d] & 0\\
0\ar[r] & \Ecal(U)\ar[r] & X^{*}(G)\ar[r] & \Pic^{G}(U)\ar[d]\ar[r] & \Pic_{G}(U)\ar[d]\ar[r] & 0\\
 &  &  & 0 & 0
}
\end{aligned}
\end{equation}
\end{proposition}


% We deduce the following corollary :
%\begin{cor}\label{PicRkBounded}
%Let $c$ be the number of irreducible components of $X \setminus U$. Then
%\[
%\dim_{\QQ}\left(\Pic^{G}(X)_{\QQ}\right) \geq c.
%\]
%\end{cor}

%\begin{proof}
%This follows from Proposition~\ref{inject} because $Z_{n-1}(X \setminus U) \cong \ZZ^c$.
%\end{proof}

\begin{proof}
The commutativity is clear. The exactness of the two rows is a simple application of Theorem~\ref{exseq} using the fact that $\mathcal{O}(X)^{G}=\mathcal{O}(U)^{G}=k$. Let us prove the surjectivity of the last two vertical maps. Note that we only need to prove the surjectivity of $\Pic^{G}(X)\to\Pic^{G}(U)$. From Theorem~\ref{exseq} and functoriality we obtain a commutative diagram
$$\xymatrix{
 0\ar[r] &\Ecal(X) \ar[r] \ar@{^{(}->}[d] & X^{*}(G) \ar[r]\ar@{=}[d] & \Pic^G(X) \ar[r]\ar[d] & \Pic(X) \ar[r]\ar@{->>}[d] &  \Pic(G) \ar@{=}[d]\\
0\ar[r] &\Ecal(U) \ar[r] & X^{*}(G) \ar[r] & \Pic^G(U) \ar[r] & \Pic(U) \ar[r] &  \Pic(G)
}$$
where the rightmost horizontal maps have to be defined using the same
choice of an element in $U$. As $X$ is smooth, $\Pic(X) \to \Pic(U)$ is surjective and a simple diagram chase shows the surjectivity of $\Pic^{G}(X)\to\Pic^{G}(U)$.

%Now choose a $G$-linearizable line
%bundle $L$ on $U$, and a preimage $L'$ in $\Pic(X)$. Using
%the exactness of the extended diagram, we see that $L'$ is mapped
%to zero in $\Pic(G)$, so it is $G$-linearizable.
%
An element in the kernel of $\Pic_{G}(X)\to\Pic_{G}(U)$
comes from a Weil divisor $D$ with support in $X \setminus U$, which is $G$-stable since $Z^{1}(X)^{G} = Z_{n-1}(X-U)$. This proves the exactness of the rightmost column. It remains to show that the kernel
of $\Pic^{G}(X)\to\Pic^{G}(U)$ is $Z^{1}(X)^{G}$. Let
$\tilde{\Lcal} \in\Pic^{G}(X)$ be mapped to zero in $\Pic^{G}(U)$. Since
we proved the exactness of the rightmost column,
we may assume further that the underlying line bundle $\Lcal$ is the trivial on $X$, so
it comes from a charater of $G$. Since it is trivial in $\Pic^{G}(U)$,
this character is attached to a function $f\in \Ecal(U)$. Then the opposite of the divisor
of $f$ is a preimage of $\tilde{\Lcal}$ in $Z^{1}(X)^{G}$.
\end{proof}


\begin{remark} \label{minusdiv}
From the above diagram, we get a natural map $\Ecal(U)\to X^*(G)\to \textrm{Pic}^G(X)$, whose image is contained in $Z^1(X)^G$. As we mentionned in the proof, this map is $f\mapsto -\textrm{div}(f)$.
\end{remark}

%\begin{example}
%Let $G=GL_{n,k}$, $T\subset G$ the maximal torus of diagonal matrices
%and $B\subset G$ the Borel subgroup of upper-triangular matrices. Let $W = S_n$ be the Weyl group of $(G,T)$ and let $I \subseteq W$ be the set of simple reflections given by $B$, i.e., $I$ consists of the transpositions which commute $i$ and $i+1$ for $1 \leq i \leq n-1$.

%Consider the action of $B\times B$ on $G$ by $(b,b')\cdot x=bxb'^{-1}$.
%The orbits are given by the permutation matrices $E_w$ for $w \in S_{n}=W(G,T)$. The orbit corresonding to $w \in S_n$ has dimension $\ell(w) + \dim(B)$, where $\ell$ is the length function. Hence the open orbit $U\subset G$ corresponds to the element 
%\[
%w_0 = (1\ n)(2\ n-1)\cdots.
%\]
%Exactly $n-1 = \#I$ permutations correspond to orbits of codimension one, namely those elements $w \in S_n$ of the form $sw_0$ for $s \in I$.
%Now $\Pic^{B\times B}(G)$ is a free abelian group (this is well known and also follows from the results below). Hence from the diagram we deduce
%\[
%\rk(\Pic^{B\times B}(G)) = 2\rk(X^{*}(B)) - \rk(X^{*}(G)) = 2n-1.
%\]
%We conclude that $\Pic^{B\times B}(U)$ and $E(U)$ have rank $n$. Analogous assertions hold for every reductive group $G$.
%\end{example}

The choice of a point $x_0 \in U$ yields an isomorphism $G/G_{x_0} \cong U$, where $G_{x_0}$ is the (scheme-theoretic) stabilizer of $x_0$. If $G_{x_0}$ is finite (equivalently, $\textrm{dim}(G)=\textrm{dim}(X)$), the orbit map
\[
u\colon G \sends U, \qquad g \sends g\cdot x_0
\]
is a finite flat $G$-equivariant morphism whose degree is $N:=\dim_k \Ocal(G_{x_0})$. It is finite �tale if and only if $G_{x_0}$ is reduced.

%\begin{proposition}\label{rankirrcomp}
%In the above situation $\Pic^G(U)$ is a torsion group killed by $N$ and $\dim_{\QQ}(\Pic^{G}(X)_{\QQ})$ is the number of irreducible components of $X-U$.
%\end{proposition}

%\begin{proof}
%As $u$ is $G$-equivariant, the norm map $\Ncal_u\colon \Pic(G) \to \Pic(U)$ attached to $u$ induces a homomorphism $\Ncal_u\colon \Pic^G(G) \to %\Pic^G(U)$ such that $\Ncal_u \circ u^*$ is the multiplication by $N$. As $\Pic^G(G) = 0$, the first assertion follows.

%The second assertion follows then from the exactness of~\eqref{diag} and Proposition~\ref{inject}.
%\end{proof}

The morphism $u$ induces a group homomorphism $\Ecal(U)\to X^*(G)$ whose image is the subgroup of $X^{*}(G)$ of characters which vanish on $G_{x_0}$. The natural isomorphism $X^{*}(G_{x_0})\simeq \Pic^G(U)$ from Proposition \ref{charpic} induces an isomorphism of exact sequences
\begin{equation}\label{DiagDenseOrbit}
\begin{aligned}\xymatrix{
0\ar[r] & \Ecal(U) \ar[r] \ar[d]^{=} & X^{*}(G)\ar[d]^{=} \ar[r] & X^{*}(G_{x_0}) \ar[d]^{\simeq} \ar[r] & \frac{X^{*}(G_{x_0})}{\textrm{im}(X^{*}(G)\to X^{*}(G_{x_0}))} \ar[d]^{\simeq} \ar[r] &  0 \\
0\ar[r] &\Ecal(U) \ar[r]  & X^{*}(G) \ar[r]  & \textrm{Pic}^G(U) \ar[r]  & \textrm{Pic}_G(U) \ar[r] &  0 
}\end{aligned}
\end{equation}

For example, assume that $G$ is defined over $\mathbb{F}_{q}$, acting on itself via Frobenius-conjugation. Then $U=G$ and one has $\mbox{Pic}^{G}(U)\simeq\mbox{Hom}(G(\mathbb{F}_{q}),k^{\times})$.

%---------------------------------------------------------------------

\subsection{The space of global sections}\label{SubGlobalSec}

\begin{prop}\label{GlobalSec}
Let $G$ be an algebraic group and let $X$ be an irreducible $G$-variety
with an open orbit $U\subset X$. Denote by $\pi:X\to\left[X/G\right]$
the projection. Let $\mathscr{L}$ be a line bundle on the stack $\left[X/G\right]$
and write $L=\pi^{*}\mathscr{L}$.
\begin{assertionlist}
\item
$H^{0}(\left[X/G\right],\mathscr{L})$ identifies with $H^{0}(X,L)^{G}$.
\item
The $k$-vector space $H^{0}(\left[X/G\right],\mathscr{L})$
has dimension less than 1.
\item
If $H^{0}(\left[X/G\right],\mathscr{L})\neq0$ then $\mathscr{L}$
restricts to the trivial line bundle on $\left[U/G\right]$.
\item
If $\mathscr{L}$ is trivial, $H^{0}(\left[X/G\right],\mathscr{L})=k$.
\end{assertionlist}
\end{prop}

\begin{proof}
A global section $s\in H^{0}(\left[X/G\right],\mathscr{L})$ is the
same as a morphism of $\mathcal{O}_{\left[X/G\right]}$-modules $\mathcal{O}_{\left[X/G\right]}\to\mathscr{L}$,
and this is the same as a morphism of $\mathcal{O}_{X}$-modules $\mathcal{O}_{X}\to L$
which commutes with the $G$-action (with respect to the trivial action
on $\mathcal{O}_{X}$). This proves the first assertion.

The map $H^{0}(X,L)\to H^{0}(U,L)$ is injective and $G$-equivariant,
so in order to prove the rest of the proposition, we may assume $X=U$
and show that only the trivial line bundle has global sections. Now
let $H\subset G$ be the stabilizer of some element in $X$. Then
$X$ identifies with $G/H$. Now, one has $\mbox{Pic}^{G}(G/H)\simeq X^{*}(H)$.
If $\chi\in X^{*}(H)$ is a character, we denote by $\mathscr{L}(\chi)$
and $L(\chi)$ the corresponding line bundles on $\left[G\backslash G/H\right]$
and $G/H$ respectively. The global sections of $L(\chi)$ on $G/H$
is the set :
\[
H^{0}(G/H,L(\chi))\simeq\left\{ f:G\to k,\ f(gh)=\chi(h)f(g),\ \forall g\in G,\forall h\in H\right\} 
\]
and the $G$-invariant global sections are zero unless $\chi=1$.
\end{proof}
Let $A\subset\mbox{Pic}(\left[X/G\right])$ be the set of line bundles
admitting global sections. We have seen that the set $A$ is contained in 
\[
Z^{1}(X)^{G}=\Ker\left(\Pic(\left[X/G\right])\to\Pic(\left[U/G\right])\right)
\]
Then $A$ is equal to the cone of effective divisors in $Z^{1}(X)^{G}$.
Indeed, if $C\in Z^{1}(X)^{G}$ and $\mathscr{L}(C)\in \Pic^G(X)$ is the associated
line bundle, then the global sections of $\mathscr{L}(C)$ are zero
or the constant functions $k\subset K(X)$ if $C\geq0$.

On the other hand, if $\chi\in X^{*}(G)$ and $\mathscr{L}(\chi)\in \Pic^G(X)$
is the associated line bundle, then $\mathscr{L}(\chi)\in A$ if and
only if there is a nonzero function $f:X\to k$ such that 
\[
f(g\cdot x)=\chi(g)f(x),\quad\forall g\in G,\forall x\in X
\]
In this representation of $\mathscr{L}(\chi)$, the global sections
of $\mathscr{L}(\chi)$ identify with the space (of dimension $1$)
of all such functions $f$. This difference in the interpretation
of the space of global functions lies in the fact that we have two
separate ways to define an element of $\Pic(\left[X/G\right])$.

%===========================================================================

\section{The Picard group of the stack of $G$-zips}

In this section we will compute the Picard group of the stack $[E\backslash G]$ which was studied in detail in \cite{PWZ1} and \cite{PWZ2}. All schemes and in particular all algebraic groups are schemes over an algebraically closed field $k$.

\subsection{The Picard group of a flag variety}

First recall some general facts about the Picard group of a linear group:

\begin{proposition}\label{PicardReductive}
Let $G$ be a connected linear algebraic group over $k$. Let $G^{\rm red} = G/R_u(G)$ be its maximal reductive quotient, let $G^{\rm der}$ the derived group of $G^{\rm red}$ and let $\pi\colon \Gtilde \to G^{\rm der}$ be the simply connected cover of $G^{\rm der}$. Then :
$$\Pic(G)=\Pic(G^{\rm red})=\Pic(G^{\rm der})=X^*(\Ker(\pi)).$$

In particular $\Pic(G)$ is a finite group.
\end{proposition}

\begin{proof}
%Clearly $\Pic(G) = \Pic(G/R_u(G)) = \Pic(G^{\rm red})$. Hence we may assume that $G$ is reductive. Then the Picard groups can be explicitly described in terms of the root datum (\cite{FoIv}~Prop.~5.1, see also~\eqref{EqPicReductive} below).
This follows from the description of the Picard group in terms of root datum (\cite{FoIv}~Prop.~5.1).%, see also~\eqref{EqPicReductive} below).
\end{proof}

%\begin{remark}\label{DescribePicRed}
%The isomorphism $\iota\colon X^*(\Ker(\pi)) \iso \Pic(G^{\rm der})$ can be described as follows. Let $\lambda\colon \Ker(\pi) \to \GG_m$ be a character. The central isogeny $\Gtilde \to G^{\rm der}$ is a $\Ker(\pi)$-torsor over $G^{\rm der}$ and hence yields an element in $H^1(G^{\rm der},\Ker(\pi))$. Then $\iota(\lambda)$ is the image of this element in $H^1(G^{\rm der},\GG_m) = \Pic(G^{\rm der})$ under the map induced by $\lambda$.
%\end{remark}


\begin{cor}\label{PicRedTrivial}
Let $G$ be a reductive group. Then the following assertions are equivalent.
\begin{equivlist}
\item
$\Pic(G) = 0$.
\item
The derived group $G^{\rm der}$ is simply connected.
\item
For any parabolic subgroup $P$ and any Levi subgroup $L \subseteq P$ one has $\Pic(L) = 0$.
\end{equivlist}
\end{cor}

\begin{proof}
The equivalence of (i) and (ii) is clear by Proposition~\ref{PicardReductive} (because $\Ker(\pi)$ is diagonalizable). Moreover, if the derived group of $G$ is simply connected, then this holds for every Levi subgroup $L$ (e.g., \cite{SGA3}~Exp.~XXI, 6.5.11).
\end{proof}


%Fix a Borel pair $(B,T)$ where $T\subseteq B$ is a maximal torus and a Borel subgroup of $G$. Let $(X,\Phi,X\vdual,\Phi\vdual,\Delta)$ be the corresponding based root datum. %Recall (\cite{SGA3}~ADD) that $G$ is semisimple if and only if $\langle \Phi \rangle_{\ZZ}$ (the subgroup of $X$ generated by $\Phi$) has the same rank as $X$. Moreover, $G$ is adjoint (resp.~simply connected) if and only if $\langle\Phi\rangle_{\ZZ} = X$ (resp.~$\langle \Phi\vdual\rangle_{\ZZ} = X\vdual$).
%Let $V(\Phi) = \langle\Phi\rangle_{\QQ}$, $V(\Phi\vdual) = \langle\Phi\vdual\rangle_{\QQ}$. We denote by

%The derived group $G^{\rm der}$ has based root datum $(X',\Phi',X'\vdual,\Phi'\vdual)$ with $X'\vdual = X\vdual \cap V(\Phi\vdual)$, $\Phi'\vdual = \Phi\vdual$, $X'$ the $\ZZ$-dual of $X'\vdual$ and $\Phi'$ the image of $\Phi$ in $X'$ under the transpose of the inclusion $X'\vdual \mono X\vdual$. If $P$ is a parabolic subgroup of $G$ containing $T$, then $P' := P \cap G^{\rm der}$ is a parabolic subgroup of $G$ containing $T' := T \cap G^{\rm der}$. Let $G^{\rm ab} := G/G^{\rm der}$ be the maximal abelian quotient. Then $G^{\rm ab}$ is a torus. The exact sequence $0 \to P' \to P \to G^{\rm ab} \to 0$ yields an exact sequence of free abelian groups
%\begin{equation}\label{EqCharDer}
%0 \lto X^*(G^{\rm ab}) \lto X^*(P) \lto X^*(P') \lto 0.
%\end{equation}
%Moreover, Proposition~\ref{PicardReductive} shows that the inclusion $P' \to P$ yields an isomorphism
%\begin{equation}\label{EqPicDer}
%\Pic(P) \iso \Pic(P').
%\end{equation}

%\[
%\Lambda := \sett{x \in V(\Phi)}{$\langle \alpha\vdual, x \rangle \in \ZZ$ for all $\alpha\vdual \in \Phi\vdual$}
%\]
%the weight lattice. It is a free abelian group and a basis is given by the fundamental weights. %Let $\Gtilde$ be the simply connected cover of the derived group of $G$ and let $\Ttilde$ and $\Btilde$ be the inverse images of $T$ and $B$ in $\Gtilde$.Then the map $X^*(T) \to X^*(\Ttilde)$ identifies $\Lambda$ with $X^*(\Ttilde)$.
%The duality of $V(\Phi\vdual)$ and $V(\Phi)$ induces a perfect pairing
%\[
%(V(\Phi\vdual) \cap X\vdual)/\langle \Phi\vdual\rangle_{\ZZ}\ \times\ \Lambda/(V(\Phi) \cap X) \lto \QQ/\ZZ.
%\]
%In particular the derived group of $G$ is simply connected if and only if $\Lambda \subseteq X$ and in this case $\Lambda$ is a direct summand on $X$.

%As $(V(\Phi\vdual) \cap X\vdual)/\langle \Phi\vdual\rangle_{\ZZ} = (X_*/\langle \Phi\vdual \rangle_{\ZZ})_{\rm tors}$ one obtains by Proposition~\ref{PicardReductive}
%\begin{equation}\label{EqPicReductive}
%\Pic(G) \cong \Lambda/(V(\Phi) \cap X) \cong ((X\vdual/\langle \Phi\vdual \rangle_{\ZZ})_{\rm tors})^*,
%\end{equation}
%where $(\cdot)^*$ denotes the Pontryagin duality $\Hom_{\ZZ}(\cdot,\QQ/\ZZ)$ of finite abelian groups.

%\begin{remark}\label{PicAdjoint}
%If $G^{\rm der}$ is an adjoint group, then the minuscule fundamental weights form a system of representatives of the nonzero elements in $\Lambda/(V(\Phi) \cap X) \cong \Pic(G)$ (\cite{Bou_Lie} VIII, \S7, Prop.~8). In particular, the order of the Picard group is the connectivity index of the root system attached to $(G,T)$.
%\end{remark}

%Recall the following result (\cite{CTS}~Lemme~11).

%\begin{proposition}\label{PicProduct}
%Let $X$, $Y$ be smooth varieties over $k$. Assume that $X$ is rational. Then the natural map
%\[
%\Pic(X) \oplus \Pic(Y) \to \Pic(X\times Y)
%\]
%is an isomorphism.
%\end{proposition}

%\begin{proof}
%For the surjectivity, the technique is the same as in the proof of
%lemma 4.2 in \cite{greenbook}. Let us prove the injectivity. Let
%$D,D'$ be two Weyl divisors respectively on $X,Y$, such that $\pr_{X}^{*}(D)\otimes\pr_{Y}^{*}(D')=\mbox{div}(f)$
%where $f$ is a function on some open subset of $X\times Y$. Since
%its divisor has this form, the regular locus of $f$ contains an open
%subset of the form $U\times V$, and we may assume that $f$ does
%not vanish on it. Then by the theorem of Rosenlicht, there are non
%vanishing functions $u,v$ respectively on $U$ and $V$, such that
%$\forall(x,y)\in U\times V$, $f(x,y)=u(x)v(y)$. Then it is clear
%that $D=\mbox{div}(u)$ and $D'=\mbox{div}(v)$.
%\end{proof}

%\begin{example}\label{ExampleRational}
%\begin{assertionlist}
%\item
%Every linear algebraic group is rational. One can reduce to the reductive case, using the isomorphism $G\simeq R_u(G) \times G/R_u
%(G)$. Then, choose a Borel pair $(B,T)$ and let $B^-$ be the opposite Borel. The multiplication map $B^-\times U \to G$ defines an isomorphism onto the open Bruhat stratum ("the big cell").
%\item
%If $G$ is a reductive group and $P$ a parabolic subgroup, then $G/P$ is rational. Indeed, choose a Levi subgroup $L$ of $P$, let $P^-$ be the opposite parabolic of $P$ with respect to $L$, and let $U^-$ be its unipotent radical. Then $U^- \to G/P$, $u \sends uP$ is an open immersion.
%\end{assertionlist}
%\end{example}

For a linear algebraic group $G$ and a parabolic subgroup $P\subset G$, we define the following integer:
\begin{equation}\label{EqDefinemP}
m_{P} := \rk_{\mathbb{Z}}\left(X^{*}(P)\right) - \rk_{\mathbb{Z}}\left(X^{*}(G)\right).
\end{equation}

%$= \#(I \setminus J)$
%In particular, $m_P = 1$ if and only if $P$ is a maximal parabolic subgroup of $G$.

\begin{proposition}
Let $G$ be a linear algebraic group over $k$ and let $P\subset G$ be a parabolic subgroup. Then $\Pic(G/P)$ is a free group of rank $m_P$.
\end{proposition}

\begin{proof}
Replacing $G$ by $G/R_u(G)$, we may assume $G$ reductive. Let $\Gtilde$ be the simply connected cover of the derived group of $G$. Let $\phi \colon \Gtilde \to G$ be the canonical homomorphism and let $\Ptilde := \phi^{-1}(P)$. Then $\Ptilde$ is a parabolic subgroup of $\Gtilde$ and $\phi$ induces an isomorphism $\Gtilde / \Ptilde \simeq G/P$. As $\Pic(\Gtilde) = 0$ (Corollary~\ref{PicRedTrivial}) and $X^*(\Gtilde) = 0$, Corollary~\ref{ExactGroupQuot} yields an isomorphism
\[
X^*(\Ptilde) \iso \Pic(\Gtilde/\Ptilde) = \Pic(G/P).
\]
so $\Pic(G/P)$ is free. Finally, we have an exact sequence:
\[
0\to X^*(G)\to X^*(P)\to \Pic(G/P)\to \Pic(G),
\]
which shows that $\Pic(G/P)$ has rank $m_P$.
\end{proof}


%\begin{rem}\label{PicGP}
%Let $\Gtilde$ be the simply connected cover of the derived group of $G$. Let $\varphi\colon \Gtilde \to G$ be the canonical homomorphism and let $\Qtilde := \varphi^{-1}(Q)$. Then $\Qtilde$ is a parabolic subgroup of $\Gtilde$ with $m_{\Qtilde} = m_Q$, and $\varphi$ induces an isomorphism $\Gtilde/\Qtilde \iso G/Q$ and hence an isomorphism $\Pic^Q(G) \iso \Pic^{\Qtilde}(\Gtilde)$.

%A basis of $\Pic(G/Q)$ is given by $(\Lcal_{\omega})_{\omega \in A}$.
%\end{rem}

Now let $G$ be a reductive group over $k$ and let $P$ be a parabolic subgroup of $G$. Fix a Borel pair $(B,T)$ of $G$ such that $T\subset B \subset P$. Let $(X,\Phi,X\vdual,\Phi\vdual,\Delta)$ be the corresponding based root datum. Denote by $W = W(G,T) := N_{G}(T)/T$ the Weyl group and by $I \subseteq W$ the set of simple reflections defined by $B$. For $\alpha \in \Phi$ we denote by $s_{\alpha} \in W$ the corresponding reflection. We obtain a bijection
\begin{equation}\label{EqBaseSimple}
\Delta \liso I, \qquad \alpha \sends s_{\alpha}.
\end{equation}
There are natural bijections between the powerset of $I$ and the set of parabolic subgroups of $G$ containing $B$ (these are called standard). If $J \subseteq I$
is a subset, the corresponding standard parabolic will be denoted
$P_{J}$. Let $M_{J}$ is the unique Levi subgroup of $P_{J}$ containing
$T$. Then we get an inclusion $W_J := W(M_{J},T)\hookrightarrow W(G,T)$
such that $(W_J, J)$ is a Coxeter system and
\[
J = W_J \cap I.
\]
Every parabolic subgroup $Q$ of $G$ is conjugate to a unique standard parabolic subgroup $P_J$ and $J \subseteq I$ is called the \emph{type of $P$}.

\begin{prop}
The integer $m_P$ is equal to the cardinality of $I\backslash J$, where $J$ is the type of $P$.
\end{prop}

\begin{proof}
Again, we may assume that $G$ is simply connected. The set of fundamental weights corresponding to $I \setminus J$ is a basis of $X^*(P)_{\mathbb{Q}}$ and the result follows.
\end{proof}

%Recall (\cite{SGA3}~???) that $G$ is semisimple if and only if $\langle \Phi \rangle_{\ZZ}$ (the subgroup of $X$ generated by $\Phi$) has the same rank as $X$. Moreover, $G$ is adjoint (resp.~simply connected) if and only if $\langle\Phi\rangle_{\ZZ} = X$ (resp.~$\langle \Phi\vdual\rangle_{\ZZ} = X\vdual$). Let $V(\Phi) = \langle\Phi\rangle_{\QQ}$, $V(\Phi\vdual) = \langle\Phi\vdual\rangle_{\QQ}$.

%The derived group $G^{\rm der}$ has based root datum $(X',\Phi',X'\vdual,\Phi'\vdual)$ with $X'\vdual = X\vdual \cap V(\Phi\vdual)$, $\Phi'\vdual = \Phi\vdual$, $X'$ the $\ZZ$-dual of $X'\vdual$ and $\Phi'$ the image of $\Phi$ in $X'$ under the transpose of the inclusion $X'\vdual \mono X\vdual$. If $P$ is a parabolic subgroup of $G$ containing $T$, then $P' := P \cap G^{\rm der}$ is a parabolic subgroup of $G$ containing $T' := T \cap G^{\rm der}$. Let $G^{\rm ab} := G/G^{\rm der}$ be the maximal abelian quotient. Then $G^{\rm ab}$ is a torus. The exact sequence $0 \to P' \to P \to G^{\rm ab} \to 0$ yields an exact sequence of free abelian groups
%\begin{equation}\label{EqCharDer}
%0 \lto X^*(G^{\rm ab}) \lto X^*(P) \lto X^*(P') \lto 0.
%\end{equation}
%Moreover, Proposition~\ref{PicardReductive} shows that the inclusion $P' \to P$ yields an isomorphism
%\begin{equation}\label{EqPicDer}
%\Pic(P) \iso \Pic(P').
%\end{equation}

%We denote by
%\[
%\Lambda := \sett{x \in V(\Phi)}{$\langle \alpha\vdual, x \rangle \in \ZZ$ for all $\alpha\vdual \in \Phi\vdual$}
%\]
%the weight lattice. It is a free abelian group and a basis is given by the fundamental weights. Let $\Gtilde$ be the simply connected cover of the derived group of $G$ and let $\Ttilde$ and $\Btilde$ be the inverse images of $T$ and $B$ in $\Gtilde$. Then the map $X^*(T) \to X^*(\Ttilde)$ identifies $\Lambda$ with $X^*(\Ttilde)$.

%The duality of $V(\Phi\vdual)$ and $V(\Phi)$ induces a perfect pairing
%\[
%(V(\Phi\vdual) \cap X\vdual)/\langle \Phi\vdual\rangle_{\ZZ}\ \times\ \Lambda/(V(\Phi) \cap X) \lto \QQ/\ZZ.
%\]
%In particular the derived group of $G$ is simply connected if and only if $\Lambda \subseteq X$ and in this case $\Lambda$ is a direct summand on $X$.

%As $(V(\Phi\vdual) \cap X\vdual)/\langle \Phi\vdual\rangle_{\ZZ} = (X_*/\langle \Phi\vdual \rangle_{\ZZ})_{\rm tors}$ one obtains by Proposition~\ref{PicardReductive}
%\begin{equation}\label{EqPicReductive}
%\Pic(G) \cong \Lambda/(V(\Phi) \cap X) \cong ((X\vdual/\langle \Phi\vdual \rangle_{\ZZ})_{\rm tors})^*,
%\end{equation}
%where $(\cdot)^*$ denotes the Pontryagin duality $\Hom_{\ZZ}(\cdot,\QQ/\ZZ)$ of finite abelian groups.

%\begin{remark}\label{PicAdjoint}
%If $G^{\rm der}$ is an adjoint group, then the minuscule fundamental weights form a system of representatives of the nonzero elements in $\Lambda/(V(\Phi) \cap X) \cong \Pic(G)$ (\cite{Bou_Lie} VIII, \S7, Prop.~8). In particular, the order of the Picard group is the connectivity index of the root system attached to $(G,T)$.
%\end{remark}

%Let $\chi\colon \GG_{m} \to T$ be a cocharacter. We denote by $L(\chi)$ be the centralizer of $\chi$ in $G$ and by $P_{\pm}(\chi) = L(\chi) \ltimes U_{\pm}(\chi)$ the unique pair of opposite parabolic subgroups with common Levi subgroup $L$ and unipotent radical $U_{\pm} = U_{\pm}(\chi)$ such that $\Lie U_+$ is the sum of the weight spaces of weights $>0$, and $\Lie U_-$ is the sum of the weight spaces of weights $<0$ in $\Lie G$ under ${\rm Ad} \circ \chi$. Then $\chi$ is $B$-dominant if and only if $B \subseteq P_+(\chi)$.

%--------------------------------------------------------------------- 

\subsection{The Picard group of the Bruhat stack}

In this section, we fix two parabolic subgroups $P$ and $Q$ of a linear algebraic group $G$. We consider the quotient stack $\left[P\backslash G/Q\right]$, which we call the Bruhat stack. It is the quotient stack associated to the action of $P\times Q$ on $G$ defined by
\[
(p,q)\cdot x=pxq^{-1}
\]
%We define $X := \left(G/P\right) \times \left(G/Q\right)$. Then $G$ acts on $X$ by left multiplication on both factors. The map $G \to X$ %defined by $g \mapsto (P,gQ)$ yields an isomorphism of stacks
%\begin{equation}\label{EqDoubleQuot}
%\left[P\backslash G/Q\right]\cong\left[G\backslash X\right].
%\end{equation}
%We obtain isomorphisms $\Pic(\left[P\backslash G/Q\right]) = \Pic^{P \times Q}(G) \cong \Pic^G(X)$.


\begin{lemma}\label{picGH}
Let $X$ be a $G\times H$-variety, where
$G,H$ are two linear algebraic groups. Assume that $\GG_m(X)^G=\GG_m(X)^H = k^\times$. Then there is an exact sequence
\begin{equation}\label{EqExactPicGH}
0\to \Ecal(X)\to \Ecal(X)\oplus \Ecal(X)\to\Pic^{G\times H}(X)\to\Pic^{G}(X)\oplus\Pic^{H}(X).
\end{equation}
If $\Pic(X)=0$, the last map is surjective. If $\Pic^{G}(X)$ and $\Pic^{H}(X)$ are free, then $\Pic^{G\times H}(X)$ is also free.
\end{lemma}

\begin{proof}
The first map is the diagonal map. The second one is the composition of the natural maps
\[
\Ecal(X)\oplus \Ecal(X)\to X^{*}(G)\oplus X^{*}(H)=X^{*}(G\times H)\to\Pic^{G\times H}(X),
\]
where the first homomorphism is injective by our hypothesis (Theorem~\ref{exseq}). The third one is given by forgetting the action on one side. Then all but the last assertion follow from a diagram chasing in the following commutative diagram with exact lines
$$\xymatrix{
 \Ecal(X) \ar@{^{(}->}[r] \ar@{^{(}->}[d] & X^*(G\times H) \ar[r] \ar[d]^{\simeq} & \textrm{Pic}^{G\times H}(X) \ar[d] \ar[r] & \textrm{Pic}_{G\times H}(X) \ar@{^{(}->}[d] \\
\Ecal(X)\oplus \Ecal(X) \ar@{^{(}->}[r] & X^*(G)\oplus X^*(H) \ar[r] & \textrm{Pic}^G(X)\oplus \textrm{Pic}^H(X) \ar[r] & \textrm{Pic}_{G}(X)\oplus \textrm{Pic}_{H}(X)
}$$
The last assertion then follows from the exactness of \eqref{EqExactPicGH} and because $\Ecal(X)$ is a free abelian group (Lemma~\ref{EXfree}).
\end{proof}

\begin{rem}
The image of $\Pic^{G\times H}(X)\to\Pic^{G}(X)\oplus\Pic^{H}(X)$ consists of elements of the form $(L,L)$, where $L$ is a line bundle on $X$ endowed with a $G$-action and an $H$-action that commute with each other.
\end{rem}



%\begin{proposition}\label{rankGP}
%Let $G$ be a reductive group over an algebraically closed field $k$ and $P$ a parabolic subgroup. Then
%\[
%\Pic(G/P) = \Pic^{P}(G) \cong \mathbb{Z}^{m_{P}}.
%\]
%\end{proposition}
%
%This is well known, see for instance ???, but it also follows immediately from the results above.
%
%\begin{proof}
%By Remark~\ref{PassToSimplyConn} we may assume that $G$ is semisimple simply connected. We apply Corollary~\ref{ExactGroupQuot}. As $G$ is simply connected, we have $\Pic(G) = 0$ and $E(G) = X^*(G) = 0$ and we obtain an isomorphism $\Pic^P(G) \cong X^*(P) \cong \mathbb{Z}^{m_{P}}$.
%
%Notice that a $P$-invariant function $P\to k$ induces a function
%on $G/P$, which is proper, so it must be constant. Thus we have an
%exact sequence
%\[
%0\to X^{*}(G)\to X^{*}(P)\to\Pic^{P}(G)\to\Pic(G)
%\]
%So the rank of the free part of $\Pic^{P}(G)$ is $d$. To prove
%that this group is free, we may assume $G$ semisimple by moding out
%the radical of $G$ (using $\Pic^{P}(G)\cong\Pic(G/P)$).
%Also, the Picard group of $G/P$ does not change if we consider a
%finite cover $\pi\colon G'\to G$ with $P'=\pi^{-1}(P)$. So we may assume
%that $G$ is semisimple and simply connected. In this case the groups
%$\Pic(G)$ and $X^{*}(G)$ are trivial, so we get an isomorphism
%\[
%X^{*}(P)\cong\Pic^{P}(G)
%\]
%Now, when $P$ is maximal, it can be proven in this simpler setting
%that $X^{*}(P)$ has indeed rank one (proof or reference needed).
%Reciprocally, if $P$ is not maximal, choose a maximal parabolic $Q$
%such that $P\subset Q$. But moding out the unipotent radical of $Q$
%gives a parabolic subgroup $P'=P/R_{u}(Q)$ in a reductive group $Q'=Q/R_{u}(Q)$
%such that $X^{*}(P')$ and $X^{*}(Q')$ have the same rank. This is
%absurd by proposition \ref{rankGP}.
%\end{proof}


%\begin{rem}\label{PicardMaxPsgp}
%Assume that $P$ is a maximal parabolic of $G$. If $G$ is semisimple simply connected group one can give a generator of $\Pic(G/P) = \Pic^{P}(G)$ as follows. As $P$ is maximal, its type is of the form $I \setminus \{s\}$ for some simple reflection $s$. Let $\alpha \in \Delta$ be the corresponding simple root~\eqref{EqBaseSimple}. Then the fundamental weight $\omega = \omega_P$ corresponding to $\alpha$ is a character of $T$ (since we assumed $G$ to be simply connected), and extends to a character of $P$. The construction of Corollary~\ref{ExactGroupQuot} then yields an element $\Lcal_\omega \in \Pic^{P}(G)$ which generates this group. It can even be shown that $\Lcal_\omega$ is ample (???).
%\end{rem}
%

%We now describe $\Pic(P\bs G/Q) = \Pic^P(G/Q)$. Set $X := G/Q$. If $G$ is simply connected, then $\Pic(P) = 0$ (Corollary~\ref{PicRedTrivial}) and Theorem~\ref{exseq} yields an exact sequence
%\begin{equation}\label{EqPicDoubleSC}
%0 \lto X^*(P) \lto \Pic^P(X) \lto \Pic(X) \lto 0.
%\end{equation}
%In general, denote the intersection of $P$ and $Q$ with $G^{\rm der}$ by $P'$ and $G'$ and denote the inverse image of $P'$ and $Q'$ in the simply connected cover $\Gtilde$ of $G^{\rm der}$ by $\Ptilde$ and $\Qtilde$. Then we may identify $X = G^{\rm der}/Q' = \Gtilde/\Qtilde$. 

%\begin{lemma}\label{PicDoubleDer}
%There are exact sequences
%\begin{equation}\label{EqPicDouble1}
%0 \lto X^*(G^{\rm ab}) \lto \Pic^P(X) \lto \Pic^{P'}(X) \lto 0
%\end{equation}
%\begin{equation}\label{EqPicDouble2}
%0 \lto \Pic^{P'}(X) \lto \Pic^{\Ptilde}(X) \lto X^*(\Ptilde)/X^*(P') \to 0.
%\end{equation}
%\end{lemma}

%Note that $X^*(G^{\rm ab}) = X^*(G)$ is a free abelian group and that $X^*(\Ptilde)/X^*(P')$ is a finite abelian group which is trivial if $G^{\rm der}$ is simply connected.

%\begin{proof}
%Both follow from the functoriality of the exact sequence in Theorem~\ref{exseq}. For the first exact sequence one also uses~\eqref{EqPicDer} and~\eqref{EqCharDer} and for the second exact sequence one uses $\Pic(\Ptilde) = 0$.
%\end{proof}

\begin{proposition}
Let $P,Q\subset G$ be parabolic subgroups. Then $\Pic^{P\times Q}(G)$ is free of rank $\rk_{\mathbb{Z}}(X^*(P))+\rk_{\mathbb{Z}}(X^*(Q))-\rk_{\mathbb{Z}}(X^*(G))$.
%\[
%\Pic^{P\times Q}(G)\cong\mathbb{Z}^{m_{P}+m_{Q}+r}.
%\]
%When $P$ and $Q$ are maximal, this group has rank is $r+2$.
\end{proposition}

\begin{proof}
%Lemma~\ref{PicDoubleDer} and~\eqref{EqPicDoubleSC} show that

%$\dim_{\QQ}(\Pic^{P\times Q}(G)_{\QQ}) = m_{P}+m_{Q}+r$.
Theorem \ref{exseq} gives an exact sequence
\[
0\to X^{*}(G)_{\mathbb{Q}}\to X^{*}(P\times Q)_{\mathbb{Q}}\to\Pic^{P\times Q}(G)_{\mathbb{Q}}\to 0
\]
and Lemma~\ref{picGH} shows that $\Pic^{P\times Q}(G)$ is free, so we are done.
%(using that $\left(\mathcal{O}(G)\right)^{P\times Q}=k$), hence $\dim_{\QQ}(\Pic^{P\times Q}(G)_{\QQ}) = m_{P}+m_{Q}+r$ because $\Pic(G)$ is finite. But Corollary~\ref{picGHfree} shows that $\Pic^{P\times Q}(G)$ is free, so we are done.
\end{proof}


%\begin{example}\label{GSpHodge}
%Consider the example $G = \GSp(V,\lrangle)$ for a symplectic space $(V,\lrangle)$ of dimension $2g$, $g \geq 1$. Then $G$ is of Dynkin type $C_g$, $G^{\rm der} = \Sp(V,\lrangle)$ is simply connected and hence $\Pic(G) = 0$ (Corollary~\ref{PicRedTrivial}). Moreover, $X^*(G)$ is the free abelian group of rank $1$ generated by the multiplier character.
%
%Let $Y, Z \subseteq V$ be Lagrangian subspaces (i.e., maximal totally isotropic subspaces) and let $P$ and $Q$ be the stabilizer of $Y$ and $Z$ in $G$, respectively. The types of $P$ and $Q$ are equal and are given by the set of those simple reflections that correspond to the vertices $\alpha_1, \dots, \alpha_{g-1}$ in Bourbaki's notation (\cite{Bou_Lie} Chap.~VI, Planche~III). In particular $m_P = m_Q = 1$.
%
%Let $\Fcal$ be the moduli space of Lagrangian subspaces of $(V,\lrangle)$. Then
%\[
%X := G/Q \liso \Fcal, \qquad gQ \sends g(Z)
%\]
%is an isomorphism of smooth projective varieties.
%
%Let $\Xscr$ be the category fibered in groupoids over the category of $k$-schemes such that $\Xscr(S)$ is the category of triples $\underline{\Mscr}$ consisting of a symplectic $\Oscr_S$-module $\Mscr$ of rank $2g$ and Lagrangian submodules $\Escr$ and $\Fscr$ of $\Mscr$. Morphisms are isomorphisms of such triples. This is a stack with respect to the fpqc-topology as fpqc-descent data are effective for finite locally free modules.
%
%Let $S$ be a $k$-scheme. Zariski locally on $S$, every triple in $\Xscr(S)$ is isomorphic to $\underline{\Mscr}_g := ((V,\lrangle) \otimes_k \Oscr_S, Y \otimes_k \Oscr_S, g(Z \otimes_k \Oscr_S))$ for some $g \in G(S)$. Hence $g \sends \underline{\Mscr}_g$ yields a morphism $G \to \Xscr$ and it is straight forward to check that this morphism induces an isomorphism
%\[
%[P \bs G/Q] \liso \Xscr.
%\]
%The canonical projection $\pi\colon \Fcal \cong X \to [P \bs G/Q] \cong \Xscr$ is given by
%\[
%\Fscr \sends ((V,\lrangle) \otimes_k \Oscr_S, Y \otimes_k \Oscr_S, \Fscr).
%\]
%Now let $(\Mscr,\Escr,\Fscr)$ be the universal triple over $\Xscr$ and let $\Lscr := \det(\Escr)$. Then $\pi^*\Escr = Y \otimes_k \Oscr_{\Fcal}$ is a trivial, hence the image of $\Lscr \in \Pic(\Xscr)$ in $\Pic(X)$ is trivial. Hence there is a unique character $\lambda \in X^*(P)$ such that $\Lscr \cong \Lscr_{\lambda}$. 
%
%Le $\omega \in \Lambda$ be the unique minuscule fundamental weight. Then $X^*(P) = \ZZ\omega \oplus X^*(G)$. The image of $\lambda$ in $\ZZ\omega$ is $-\omega$ (CHECK! -- might be $\omega$) and the image in $X^*(G)$ is the multiplier character (CHECK!).
%
%Let $\omega$ be the fundamental weight corresponding to $\alpha_g$ which defines a character of $T$ (because $G^{\rm der}$ is simply connected) and this character can be extended to a character of $P$. Let $\Lcal_{\omega}$ be the corresponding line bundle which generates $\Pic(G/P)$.
%
%Then we have the following description of $\Lcal_{\omega}$. The parabolic $P$ is the stabilizer of a Lagrangian of $(V,\lrangle)$ and $G/P$ can be identified with the moduli space of Lagrangian subspaces of $(V,\lrangle)$. Let $\Ucal \subset V \otimes \Ocal_{G/P}$ be the universal Lagrangian submodule. Then
%\[
%\Lcal_{\omega} \cong \det((V \otimes \Ocal_{G/P})/\Ucal) \cong \det(\Ucal)\vdual. 
%\]
%\end{example}


%We now make use of Theorem~\ref{exseq} for the $G$-variety $X = G/P \times G/Q$. Note $\Pic(X) = \Pic(G/P) \oplus \Pic(G/Q)$ by Proposition~\ref{PicProduct} (applicable because of Example~\ref{ExampleRational}). Since $X$ is proper, one has $\mathcal{O}(X)=k$, so we get an exact
%sequence
%\[
%0 \to X^{*}(G) \to \Pic^{G}(X) \to \Pic(X).
%\]
%It is possible to relate this sequence to the ones we have already
%seen. There is a commutative diagram
%\[
%\xymatrix{ & 0\ar[r] & X^{*}(G)\ar@{^{(}->}[d]\ar[r] & \Pic^{G}(X)\ar[d]^{\cong}\ar[r] & \Pic(X)\ar[d]^{\cong}\\
%0\ar[r] & X^{*}(G)\ar[r] & X^{*}(G)\oplus X^{*}(G)\ar[r] & \Pic^{P\times Q}(G)\ar[r] & \Pic^{P}(G)\oplus\Pic^{Q}(G)
%}
%\]
%The third vertical map is the natural one. However, the other two
%depend on the choice we made of the isomorphism of stacks $\left[P\backslash G/Q\right]\cong\left[G\backslash X\right]$,
%namely $g\mapsto(P,gQ)$ (we could have chosen $g\mapsto(gP,Q)$).
%The second map is the induced isomorphism by indentifying the Picard
%groups of these stacks and the groups $\Pic^{P\times Q}(G)$
%and $\Pic^{G}(X)$.
%
%The first vertical map is then given by
%\[
%\begin{array}{ccc}
%X^{*}(G) & \to & X^{*}(P)\oplus X^{*}(Q)\\
%\alpha & \mapsto & \left(\alpha_{|P},0\right)
%\end{array}
%\]
%whereas the horizontal map $X^{*}(G)\to X^{*}(P)\oplus X^{*}(Q)$
%is the diagonal embedding $\alpha\mapsto\left(\alpha_{|P},\alpha_{|Q}\right)$.
%

%---------------------------------------------------------------------

\subsection{Frobenius zip datum}\label{FrobZipDatum}

In the sequel we will be mainly consider the following situation: Let $k$ be an algebraically closed extension of a finite field $\FF_q$ with $q = p^m$. Let $G$ be a reductive group over $k$ defined over $\FF_q$. Denote by $\sigma$ the $q$-th power Frobenius of $k$. For every $k$-scheme or morphism of $k$-schemes we denote its pullback under $\sigma$ by $(\ )^{(q)}$. We fix a Borel pair $(B,T)$ of $G$ defined over $\FF_q$. Let $B^-$ be the opposite Borel subgroup of $B$ with respect to $T$. Let $P$ be a parabolic of $G$ defined over $k$ containing $B^{-}$, let $L$ be the unique Levi subgroup of $P$ containing $T$, and let $P^-$ be the opposite parabolic subgroup with respect to $L$. Define $Q = (P^{-})^{(q)}$. Let $\varphi\colon G\to G = G^{(q)}$ be the relative $q$-th power Frobenius isogeny. We also denote its restriction $L \to M := L^{(q)}$ again by $\varphi$. Note that $M$ is the unique Levi subgroup of $Q$ containing $T$. Then $(G,P,Q,\varphi)$ is an algebraic zip datum in the sense of \cite{PWZ1}~10.1.


For $x\in P$, we denote by $\overline{x}$ the image of $x$ in $P/R_{u}(P) = L$, similarly for the image of $y \in Q$ in $Q/R_u(Q) = L^{(q)}$. The associated \emph{zip group} is defined by
\[
E := \set{(x,y)\in P\times Q}{\varphi(\overline{x})=\overline{y}} 
\]
and $E$ acts on $G$ by restricting the action of $P\times Q$ to $E$. Note that $\dim(E) = \dim(G)$. The quotient stack $[E\backslash G]$ is called the \emph{stack of $G$-zips}. One has
\[
E = \set{(u\ell,v\varphi(\ell))}{u \in R_u(P), v \in R_u(Q), \ell \in L}.
\]
The subgroup $\widetilde{E} = \set{(x,y)\in L\times M}{\varphi(x)=y}$ is a Levi subgroup of $E$, isomorphic to $L$.

By \cite{PWZ1}~Proposition~7.3, $E$ acts with finitely many orbits on the variety $G$. These orbits are parametrized as follows (\cite{PWZ1}~Theorem~7.5). Let $(W,I)$ be the Weyl group of $(G,T)$ with its set of simple reflections given by $B$. As $B$ and $T$ are defined over $\FF_q$, the relative Frobenius $\varphi\colon G \to G^{(q)} = G$ induces an isomorphism $\bar\varphi$ of the Coxeter system $(W,I)$.

Let $J \subseteq I$ and $K \subseteq I$ be the type of $P$ and $Q$, respectively. Then $K = \varphi(J)^{\rm opp}$. For every $w \in W$ we choose a representative $\dot w \in \Norm_G(T)$ such that $(w_1w_2)^{\cdot} = \dot w_1\dot w_2$ if $\ell(w_1w_2) = \ell(w_1) + \ell(w_2)$. Let $w_{0,J} \in W_J$ and $w_0 \in W$ the longest elements. In particular $w_0^2 = w_{0,J}^2 = 1$.

Set $g_0 := (w_{0} w_{0,J})^{\cdot}$. Hence $g_0 \in \Norm_G(T)$ lifts $w_{0,J}w_0 \in W$. Then $(B,T,g_0)$ is a frame in the sense of \cite{PWZ1}~Definition~3.6: As $(w_{0}w_{0,J}w_{0})^{\cdot} \in L \subseteq P$ and $\leftexp{w_0}{B} = B_-$, we have
\begin{gather*}
B = B^{(q)} \subseteq Q, \qquad \leftexp{g_0}{B} = \leftexp{w_{0}w_{0,J}w_{0}}(B_-) \subseteq P,\\
\varphi(L \cap \leftexp{g_0}{B}) = \varphi(\leftexp{w_{0}w_{0,J}w_{0}}{(L \cap B_-)}) = \varphi(L \cap B) = M \cap B,\\
\varphi(\leftexp{g_0}{T}) =  T.
\end{gather*}
By \cite{PWZ1}~Theorem~5.12 and Theorem~7.5 we obtain a bijection
\begin{equation}\label{EqEOrbits}
\leftexp{J}{W} \liso \{\text{$E$-orbits on $G$}\}, \qquad w \sends O^w := E\cdot (g_0\dot{w})
\end{equation}
such that $\dim O^w = \ell(w) + \dim(P)$.


%-------------------------------------------------------------------------

\subsection{The Picard group of the stack of a Frobenius zip datum}


\begin{lem}\label{TrivChar}
One has $X^{*}(G)^{E}=0$.
\end{lem}

\begin{proof}
Let $\chi$ be an $E$-invariant character of $G$. In particular, for all $x\in L$, one
has $\chi(x)=\chi(\varphi(x))$, so $\chi = \chi \circ \varphi$ because the restriction $X^*(G) \to X^*(L)$ is injective. But $\chi$
is defined over some finite field, so for some $r\geq1$, $\chi=\chi\circ\varphi^{r}=\sigma^{r}\circ\chi$. We conclude that the image of $\chi$ is finite, so $\chi=1$.
\end{proof}

Using that $\Ecal(G) = X^*(G)$ by Rosenlicht's theorem we deduce from Theorem~\ref{exseq} an exact sequence
$$\xymatrix{
 0 \ar[r] & X^{*}(G) \ar[r] & X^{*}(E)\ar[r] & \Pic^{E}(G)\ar[r] & \Pic_{E}(G) \ar[r] & 0.
}$$

\begin{cor}
\label{rankPic}
One has $\dim_{\QQ}\left(\Pic^E(G)_{\QQ}\right) = m_P = \#(I \setminus J)$, where $m_P$ was defined in~\eqref{EqDefinemP} and where $J$ is the type of $P$.
%\textup{$m_{P}=\mbox{rank}_{\mathbb{Z}}(X^{*}(P))-\mbox{rank}_{\mathbb{Z}}(X^{*}(G))$}.
%It is the number of elements of $^{I}W$ of length $\ell(\eta)-1$.
%When $P$ and $Q$ are maximal, the Picard group of $\left[E\backslash G\right]$
%has free rank one, and $G-G^{\eta}$ is irreducible.
\end{cor}

\begin{proof}
As $E/R_{u}(E) \cong P/R_{u}(P) \cong L$, we have $X^{*}(E) \cong X^{*}(P)$.
\end{proof}

Note that the inclusion $E \subseteq P \times Q$ yields a morphism of quotient stacks $\left[E\backslash G\right]\to\left[P\backslash G/Q\right]$ and thus a homomorphism $\beta : \Pic^{P\times Q}(G)\to  \Pic^{E}(G)$. We have a morphism of exact sequences :
\begin{equation} \label{EqPicEG}
\begin{gathered}
\xymatrix{
 X^{*}(G)\ar@{^{(}->}[r]\ar@{=}[d] & X^{*}(P)\oplus X^{*}(Q)\ar[r]\ar[d]^{\alpha} & \Pic^{P\times Q}(G)\ar[r]\ar[d]^{\beta} & \Pic_{P\times Q}(G)\ar@{^{(}->}[d]^{\gamma} \ar[r] & 0 \\
 X^{*}(G)\ar@{^{(}->}[r] & X^{*}(E)\ar[r] & \Pic^{E}(G)\ar[r] & \Pic_{E}(G) \ar[r] & 0.
}
\end{gathered}
\end{equation}

%For the rest of the section, we study only algebraic zip data arising as follows. Let $k$ be an algebraically closed field of characteristic $p > 0$. Assume that $G$ is already defined over a finite field $\FF_q$ with $q = p^r$ elements contained in $k$. We fix a Borel pair $(B,T)$ of $G$ defined over $\FF_q$. Let $\kappa$ be a finite field extension of $\FF_q$ and let $\chi\colon \GG_{m,\kappa} \to T_{\kappa}$ be a $B$-dominant cocharacter over $\kappa$. Let $P_{\pm} = P_{\pm}(\chi)$ the attached pair of opposite parabolic subgroups with common Levi subgroup $L$ the centralizer of $\chi$. Let $U_{\pm}$ be the unipotent radical of $P_{\pm}$. Then $P_{\pm}$ and $L$ are defined over $\kappa$. We obtain an algebraic zip datum $\Zcal_{G,\chi} := (G,P_+,P^{(q)}_-,\varphi)$, where $(\ )^{(q)}$ denotes the pullback under absolute Frobenius $x \sends x^q$ and where $\varphi\colon L \to L^{(q)}$ is the relative Frobenius. In particular we obtain an attached zip group $E = E_{G,\chi}$. We set $P := P_+$ and $Q := P^{(q)}_-$.

It can be checked immediately that this diagram induces isomorphisms $\Ker(\alpha) \cong \Ker(\beta)$ and $\Coker(\beta)\cong\Coker(\gamma)$. In particular, the cokernel of $\beta$ is finite. If $\Pic(G)=0$, the map $\beta$ is surjective. In this case, any line bundle on the stack of $G$-zips arises as the pull-back of a line bundle on the Bruhat stack.


%For a $G$-variety $X$,
%let us denote by $\Pic^{G}(X)^{\rm triv}$ the kernel of
%the map $\Pic^{G}(X)\to\Pic(X)$. To sum up, we have an isomorphism of
%exact sequences
%
%$$\xymatrix{
%0 \ar[r] & \Ker(\beta) \ar[r] \ar[d]^{\cong} & \Pic^{P\times Q}(G)^{\rm triv} \ar[r]^\beta \ar[d]^{\cong} & \Pic^{E}(G)^{\rm triv} \ar[r]\ar[d]^{\cong} & 0 \\
%0 \ar[r] & X^{*}(L) \ar[r] &  \frac{X^{*}(L)\oplus X^{*}(M)}{X^{*}(G)_{j}} \ar[r] & \frac{ X^{*}(M)}{X^{*}(G)_{\varphi}} \ar[r] & 0
%}$$
%
%The subscript $j$ here means that $X^{\text{*}}(G)$ is embedded into
%$X^{*}(L)\oplus X^{*}(M)$ through the embedding $j(\chi)=(-\chi_{|M},\chi_{|M})$.
%The subscript $\varphi$ means that $X^{\text{*}}(G)$ is embedded into
%$X^{*}(M)$ via the map $\chi\mapsto\left(\chi\circ\varphi-\chi\right)_{|M}$.
%The maps in the second line are respectively given by
%\[
%\chi\mapsto(-\chi\circ\varphi,\chi)\quad\mbox{and}\quad(\chi_{1},\chi_{2})\mapsto\chi_{1}+\chi_{2}\circ\varphi
%\]
%
%\begin{proposition}
%The group \textup{$\Pic^{E}(G)^{\rm triv}$} has no $p$-torsion.
%\end{proposition}
%
%\begin{proof}
%We have $\Pic^{E}(G)^{\rm triv}\cong\frac{X^{*}(M)}{X^{*}(G)_{\varphi}}$
%with the same notation as above. Let $\chi\in X^{*}(M)$ such that
%$p\chi=\theta+\theta\circ\varphi$ where $\theta$ is a character
%of $G$. Notice that the (natural) quotient $\frac{X^{*}(M)}{X^{*}(G)}$
%is torsion-free, since it embeds into $\Pic(G/Q)$ which is
%free. So the character $\chi$ extends to $G$. We want to prove that
%$\chi$ itself has the form $\alpha+\alpha\circ\varphi$, $\alpha\in X^{*}(G)$.
%To see this, consider the map
%\[
%f\colon X^{*}(G)\to X^{*}(G),\qquad\chi\mapsto-\chi+\chi\circ\varphi
%\]
%Write $f=-\mbox{id}+g$ where $g\colon \chi\mapsto\chi\circ\varphi$. Let
%$n\geq1$ such that all characters of $G$ are defined on $\mathbb{F}_{q^{n}}$.
%We see readily that $g^{n}(\chi)=q^{n}\chi$. We deduce that $f$
%is bijective modulo $p$, so its cokernel has no $p$ torsion.
%\end{proof}

%---------------------------------------------------------------------

\subsection{The open orbit and the stabilizer of $1\in G$}

Let $\eta = w_{0,J}w_0$ be the element of maximal length in $\leftexp{J}{W}$. Then $O^{\eta} = E\cdot (g_0\eta) = E\cdot 1$ is the unique open orbit. The stabilizer of $1 \in G$ in $E$ is a finite group scheme $S\subset E$ and $O^{\eta} \cong E/S$ is affine (in fact, every $E$-orbit in $G$ is affine by \cite{WdYa}~Theorem~2.2). Hence the complement $G \setminus O^{\eta}$ is a Cartier divisor whose irreducible components are fixed by the $E$-action because $E$ is connected. Hence~\eqref{EqEOrbits} induces bijections
\begin{equation}\label{EqEDivisors}
I \setminus J \bijective \set{w \in \leftexp{J}{W}}{\ell(w) = \ell(\eta)-1} \bijective \{\text{irreducible components of $G \setminus O^{\eta}$}\},
\end{equation}
where the first bijection is given by $s \sends s\eta$ and where the second bijection is $w \sends \overline{O^w}$. The remarks in the beginning of Subsection~\ref{GVarOpen} show that $Z^1(G)^E$ is freely generated by the right hand side. Hence we obtain an identification $\ZZ^{I \setminus J} \cong Z^1(G)^E$. Moreover, by \eqref{diag} and \eqref{DiagDenseOrbit} we have an exact sequence
\begin{equation}\label{EqDescribePicEG}
0 \lto Z^1(G)^E \lto \Pic^E(G) \lto X^*(S) \lto 1.
\end{equation}

We now study the stabilizer of $1$. First, we recall a basic result on the intersection of two parabolic
subgroups. If we write $g \in G$ we always work with $R$-valued points for a $k$-algebra $R$.

\begin{prop}\label{InterParab}
Let $P$ and $Q$ be two parabolics of a reductive group $G$ with unipotent radicals $U$ and $V$, respectively. Let $T$ be a maximal torus contained in $P$ and in $Q$ and let $L\subset P$ and $M\subset Q$ be the Levi subgroups with respect to $T$.
\begin{assertionlist}
\item\label{InterParab1}
The subgroups $P\cap Q$, $L\cap M$, $L\cap V$, $M\cap U$, $U\cap V$ are smooth and connected.
\item\label{InterParab2}
The group $\left(P\cap Q\right).U$ is a parabolic subgroup of $G$
contained in $P$, with Levi subgroups $L\cap M$.
\item\label{InterParab3}
Any element $x\in P\cap Q$ can be written uniquely as a product $x=abcd$,
with $a\in L\cap M$, $b\in L\cap V$, $c\in M\cap U$, $d\in U\cap V$.
\end{assertionlist}
\end{prop}

\begin{proof}
 The smoothness of $P\cap Q$ follows from \cite{SGA3}~Exp.~XXVI, Lemma 4.1.1. This implies the smoothness of the other subgroups. For the rest, see \cite{DM}~Proposition 2.1.
\end{proof}

The last statement means that $P\cap Q$ is the product of the varieties $L\cap M$, $L\cap V$, $M\cap U$, $U\cap V$. In particular, we obtain the following result.

\begin{cor}\label{contained}
We keep the same notations as in Proposition \ref{InterParab}.
If $U\cap V=\{1\}$ and $M\cap U=\{1\}$, then $P\cap Q\subset L$.
\end{cor}

If $x\in P$, define $\theta_{L}^{P}(x)\in L$ to be the unique element
of $L$ such that there exists $u\in U$ satisfying $x=\theta_{L}^{P}(x).u$.
We will use repeatedly the following corollary.

\begin{cor}\label{CorReduct}
We keep the same notations as in Proposition~\ref{InterParab}.
\begin{assertionlist}
\item For all $x\in P\cap Q$, one has $\theta_{L}^{P}(x)\in P\cap Q$.
\item For all $x\in P\cap Q$, one has $\theta_{L}^{P}(\theta_{M}^{Q}(x))=\theta_{M}^{Q}(\theta_{L}^{P}(x))$ (note that this makes sense because of the first assertion). 
\item Assume $G$ is defined over $\FF_q$ and let $\varphi:G\to G$ the $q$-th power Frobenius. Then $\varphi \left( \theta_{L}^{P}(x) \right)=  \theta_{\sigma L}^{\sigma P}(\varphi(x))$.
\item Assume $T\subset B\subset P\cap Q$ for some Borel $B$. Then $P\cap Q$ is a parabolic with Levi $L\cap M$ and for all $x\in P\cap Q$, one has $\theta_{L}^{P}(\theta_{M}^{Q}(x))=\theta_{M}^{Q}(\theta_{L}^{P}(x))=\theta_{L\cap M}^{P\cap Q}(x)$.
\end{assertionlist}
\end{cor}

\begin{proof}
Using the notation of Proposition~\ref{InterParab}~\eqref{InterParab3} we write $x=abcd \in P \cap Q$. Then $\theta_{L}^{P}(x)=ab$ and $\theta^Q_M(x) = ac$. This implies the first two assertions. The third assertion is obvious. The first part of the last assertion is Proposition \ref{InterParab}~(2). Finally, write $x=\theta_{L}^{P}(x)u$ with $u\in R_u(P)\subseteq R_u(P\cap Q)$. Now $\theta_{L}^{P}(x)=\theta_{M}^{Q} \left( \theta_{L}^{P}(x) \right) v$ with $v\in R_u(Q)\subseteq R_u(P\cap Q)$. The result follows.
\end{proof}

We return to the situation of Subsection~\ref{FrobZipDatum}.
%Now let $k$ again be an algebraically closed extension of a finite field $\FF_q$, let $G$ be a reductive group defined over $\mathbb{F}_{q}$ with a Borel pair $(B,T)$ defined over $\mathbb{F}_{q}$. Let $P$ be a parabolic
%subgroup containing $B^{-}$ and $Q := (P^{-})^{(q)}$ (defined over $k$). So $Q$ is a standard parabolic. Let $L\subset P$ and $M = L^{(q)} \subset Q$
%denote the Levi subgroups with respect to the maximal torus $T$.
%Let $E\subset P\times Q$ denote the zip group. % as usual and let $U\subset G$ be
%%the open $E-$orbit.
Let
\[
S = \set{(x,y) \in E}{x = y} \subset E
\]
be the scheme-theoretic stabilizer of $1\in G$. Note that $S$ is a finite group scheme that is usually not smooth. Let $S_{0} := S_{\rm red}$ be its underlying reduced subgroup scheme. Then $S_{0}$ is the finite constant group scheme over $k$ such that $S_{0}(k) = S(k)\subset E(k)$. We will often identify $S_{0}$ and $S_{0}(k)$. We view $S$ as a subgroup of $P\cap Q$ via the first projection. In the sequel we will use $X^*(E)^{S}$ and $X^*(S)$. The next few results show that we can replace $S$ by $S_0$ and give a description of $S_0$. We define:
\[
P_{0}:=\bigcap_{i\in\mathbb{Z}}P^{(q^i)}\quad\quad Q_{0}:=\bigcap_{i\in\mathbb{Z}}Q^{(q^i)}\quad\quad L_{0}:=\bigcap_{i\in\mathbb{Z}}L^{(q^i)}.
\]
It is clear that $P_{0}$ and $Q_{0}$ are opposite parabolic subgroups of $G$, defined over $\mathbb{F}_{q}$, with common Levi subgroup $L_{0}$. We have the following lemma:

\begin{lem}\label{ContainQ0}
One has $Q_{0}\cap P\subset L$.
\end{lem}

\begin{proof}
This follows from Corollary~\ref{contained}, because $L_{0}\cap R_{u}(P)=\{1\}$
and $R_{u}(Q_{0})\cap R_{u}(P)=\{1\}$ (since $R_{u}(Q_{0})\subset R_{u}(B)$
and $R_{u}(P)\subset R_{u}(B^{-})$).
\end{proof}


\begin{lem}\label{SandS0}
The group $S$ is contained in $Q_{0}\cap L$. The
group $S_{0}$ is contained in $L_{0}$. Further, one has $S_{0}=L_{0}(\mathbb{F}_{q})$
and $S\cap L_{0}=S_{0}$.
\end{lem}

\begin{proof}
Let $x\in S$. By definition, one has $\varphi(\theta_{L}^{P}(x))=\theta_{M}^{Q}(x)$.
Since $x\in Q$, one has $\theta_{L}^{P}(x)\in Q$ by Corollary \ref{CorReduct}.
It follows that $\theta_{M}^{Q}(x)\in\sigma(Q)$ and we deduce $x\in\sigma(Q)$
because $R_{u}(Q)\subset R_{u}(B)\subset\sigma(Q)$. Now we can apply
the same argument to $\sigma(Q)$ to show $x\in\sigma^{2}(Q)$. Continuing
this process, we get $x\in Q_{0}$. Since $S\subset P$, we also have
$S\subset L$ by Lemma~\ref{ContainQ0}.

Now, to prove $S_{0}\subset L_{0}$, we need only show that $S_{0}(k)\subset L_{0}(k)$,
because $S_{0}$ is smooth. Let $x\in S_{0}(k)$. By definition, one
has $\varphi(\theta_{L}^{P}(x))=\theta_{M}^{Q}(x)$. Since $x\in P$,
we have $\theta_{M}^{Q}(x)\in P$ by Corollary \ref{CorReduct}. We
deduce that $\theta_{L}^{P}(x)\in\sigma^{-1}(P)$ and then $x\in\sigma^{-1}(P)$.
Repeating the argument yields $x\in P_{0}$. We have showed that $S_{0}\subset L_{0}$.
Now for $x\in L_{0}$, $x$ lies in $S_{0}$ if and only if $\varphi(x)=x$,
so $S_{0}=L_{0}(\mathbb{F}_{q})$.

Since the Lang-Steinberg map $L_{0}\to L_{0}$, $x\mapsto x^{-1}\varphi(x)$
is �tale, it follows that the algebraic group $S\cap L_{0}=\left\{ x\in L_{0},\ \varphi(x)=x\right\} $
is smooth, equal to the constant group $L_{0}(\mathbb{F}_{q})$. So
we deduce $S\cap L_{0}=S_{0}$.
\end{proof}


In the next lemma, we consider the map $\theta_{L_{0}}^{Q_{0}}\colon Q_{0}\to L_{0}$.

\begin{lem}\label{ProjectS}
If $x\in S$, then $\theta_{L_{0}}^{Q_{0}}(x)\in S_{0}$.
\end{lem}

\begin{proof}
Let $x\in S$, so by definition one has $\varphi(\theta_{L}^{P}(x))=\theta_{M}^{Q}(x)$.
Now $x\in L\cap Q_{0}$ by Lemma \ref{SandS0}, so we have $\theta_{L}^{P}(x)=x$
and $\varphi(x)=\theta_{M}^{Q}(x)$. We deduce from this
\[
\varphi(\theta_{L_{0}}^{Q_{0}}(x))=\theta_{L_{0}}^{Q_{0}}(\varphi(x))=\theta_{L_{0}}^{Q_{0}}(\theta_{M}^{Q}(x)) = \theta_{M}^{Q}(\theta_{L_{0}}^{Q_{0}}(x)) = \theta_{L_{0}}^{Q_{0}}(x) \]
since $L_0\subset M$. We deduce $\theta_{L_{0}}^{Q_{0}}(x)\in L_{0}(\mathbb{F}_{q})=S_{0}$.
\end{proof}


\begin{remark}\label{IdentifyLevi}
The Levi subgroup $L_0$ is the group $H_w$ of \cite{PWZ1}~5.1 for $w = \eta \in \leftexp{J}{W}$ the longest element. Note that we chose the frame $(T,B,g_0)$ in such a way that $g_0\dot{\eta} = 1$. 
\end{remark}


\begin{prop}
We have $X^{*}(E)^{S}=X^{*}(E)^{S_{0}}$ and $X^{*}(S)=X^{*}(S_{0})$.\end{prop}
\begin{proof}
The first assertion follows from Lemma~\ref{ProjectS}, because for all
$x\in S$ and $\chi\in X^{*}(E)$, one has $\chi(x)=\chi(\theta_{L_{0}}^{Q_{0}}(x))$
so $\chi$ vanishes on $S$ if and only if it does on $S_{0}$.

For the second assertion, note that we already have proved $X^{*}(S)\subset X^{*}(S_{0})$.
We have an exact sequence
\[
1\to S^{0}\to S\to\pi_{0}(S)\to1
\]
And since we are working over a perfect field, this exact sequence is split by the group $S_{red}=S_{0}$. So $S_{0}\simeq\pi_{0}(S)$
and finally $X^{*}(S)=X^{*}(S_{0})$.
\end{proof}


Now, we can rewrite diagram (\ref{diag}) taking into account this new information :

\begin{equation}
\begin{gathered} \label{diagforG}
\xymatrix{ &  &  &  & \frac{\Ecal(U)}{X^{*}(G)}\ar@{^{(}->}[d]\\ &  &  & Z^{1}(G)^{E}\ar@{^{(}->}[d]^{\mathcal{L}}\ar[r]^{=} & Z^{1}(G)^{E}\ar[d]\\0\ar[r] & X^{*}(G) \ar[r]\ar@{^{(}->}[d] & X^{*}(E)\ar[r]\ar[d]^{=} & \Pic^{E}(G)\ar[r]\ar[d] & \Pic_{E}(G)\ar[r]\ar[d] & 0\\ 0\ar[r] & \Ecal(U) \ar[r]  & X^{*}(E) \ar[r] & X^{*}(S_0) \ar[d] \ar[r] & \frac{X^{*}(S_0)}{\im(X^{*}(E)\to X^{*}(S_0))} \ar[d]\ar[r] &  0 \\ &  &  & 0 & 0}
\end{gathered}
\end{equation}


\begin{cor}
The group $\Pic^{E}(G)$ has no $p$-torsion.
\end{cor}

\begin{proof}
This follows immediately from the previous diagram, knowing that $X^{*}(S_{0})$
has no $p$-torsion.
\end{proof}


We will denote by $\zeta$ the group homomorphism
\begin{equation}\label{EqDefineZeta}
\zeta\colon X^{*}(L_{0})\to X^{*}(L_{0}), \qquad \chi \sends \chi - \chi\circ\varphi.
\end{equation}


\begin{remark}\label{ZetaInjective}
The map $\zeta$ is injective, because if $\chi=\chi\circ\varphi$, then there exists $d\geq 1$ such that $\chi = \chi\circ\varphi^{d}=\varphi^{d}\circ\chi$, and this implies
$\chi=1$.
\end{remark}


\begin{cor}\label{zeta}
Assume that $\Pic(L_{0})=0$ (this is the case if $\Pic(G)=0$). Then we have an exact
sequence
\[
0\to X^{*}(L_{0})\overset{\zeta}{\to}X^{*}(L_{0})\to X^{*}(S_{0})\to0
\]
In particular, the order of the group $X^{*}(S_{0})$ is equal to
the absolute value of the determinant of $\zeta$.
\end{cor}

\begin{proof}
This follows from Proposition \ref{surject}, knowing that $S_{0}=L_{0}(\mathbb{F}_{q})$.\end{proof}


\begin{cor}\label{HasseSplit}
Assume $G$ is split over $\mathbb{F}_{q}$ and $\Pic(L_{0})=0$.
Then $$X^{*}(S_{0})\simeq\left(\frac{\mathbb{Z}}{(q-1)\mathbb{Z}}\right)^{d}$$
with $d := \rk(X^*(L_0))$.
\end{cor}

\begin{example}\label{ExampleGLn}
Let $G=GL_{n,\mathbb{F}_{q}}$, with the standard (upper) Borel pair
$(B,T)$. Let $r\geq s\geq0$ be two integers such that $r+s=n$.
Let $P$ be the parabolic of matrices of the form

\[
\left(\begin{array}{cc}
A & 0\\
C & B
\end{array}\right)\ \mbox{with}\ A\in GL_{r}(k),\ B\in GL_{s}(k),\ C\in M_{s,r}(k)
\]
and define $Q:=P^{-}$. Then $L_{0}=L\simeq GL_{r}\times GL_{s}$
and $S_{0}\simeq GL_{r}(\mathbb{F}_{q})\times GL_{s}(\mathbb{F}_{q})$.
Since $\mbox{Pic}(GL_{n})=0$, we deduce that
\[
X^{*}(S_{0})=\left(\frac{\mathbb{Z}}{(q-1)\mathbb{Z}}\right)^{2}
\]
In particular, $X^{*}(S_{0})$ is killed
by $q-1$.
\end{example}

\begin{example}\label{ExampleUnitary}
Let $G=U(n)$ be the unitary group associated to the $n\times n$-matrix
\[
J=\left(\begin{array}{ccc}
 &  & 1\\
 & \iddots\\
1
\end{array}\right)
\]
In other words, for every $\mathbb{F}_{q}$-algebra, one has :
\[
G(R)=\left\{ A\in GL_{n}(\mathbb{F}_{q^{2}}\otimes_{\FF_q} R),\ ^{t}\sigma(A)JA=J\right\} 
\]
We make the identification $G_{k}=GL_{n,k}$. The Galois action is
then given by $\sigma\cdot A=J^{t}\sigma(A)^{-1}J$ for all $A\in GL_{n}(k)$.
Let $(B,T)$ the standard (upper) Borel pair of $GL_{n,k}$. One sees
immediately that it is defined over $\mathbb{F}_{q}$. Let $r\geq s\geq0$
be two integers such that $r+s=n$ and let $P$ and $Q$ be the same
parabolics as in the previous example. For an $\mathbb{F}_{q}$-algebra $R$, the group $L_{0}(R)$ is easily
seen to be the set of elements in $G(R)$ of the form
\[
\left(\begin{array}{ccc}
A\\
 & B\\
 &  & C
\end{array}\right)\ \mbox{with}\ A\in GL_{s}(R\otimes_{\FF_q} \mathbb{F}_{q^2}),\ B\in GL_{r-s}(R\otimes_{\FF_q} \mathbb{F}_{q^2}),\ C\in GL_{s}(R\otimes_{\FF_q} \mathbb{F}_{q^2})
\]
with the conditions $\sigma^2(A)=A$, $C= \!^{t}\sigma(A)^{-1}$, and $B\in U(r-s)$.
In other words, one has 
\[
L_{0}\simeq U(r-s)\times\mbox{Res}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}}(GL_{s})
\]
And it follows that
\[
S_{0}=L_{0}(\mathbb{F}_{q})=U(r-s)(\mathbb{F}_{q})\times GL_{s}(\mathbb{F}_{q^{2}})
\]
For example if $n=3$, it can be shown that one has $S\simeq S_0\times \alpha_p$, so $S$ is in general not reduced. Since $\Pic(GL_{n})=0$, we can apply Corollary \ref{zeta} so
any group homomorphism $S_{0}\to k^{\times}$ comes from a character
of $L_{0}$. We deduce that
\[
X^{*}(S_{0})=\frac{\mathbb{Z}}{(q+1)\mathbb{Z}}\times\frac{\mathbb{Z}}{(q^{2}-1)\mathbb{Z}}
\]
In particular, $X^{*}(S_{0})$ is killed
by $q^{2}-1$.\end{example}

%---------------------------------------------------------------------

\subsection{Zip data attached to a cocharacter}\label{ZipDataChar}

In our applications to Shimura varieties, zip data will arise as follows. Let $G$ be a reductive group defined over $\FF_q$ and let $\chi\colon \GG_m \to G$ be a cocharacter of $G$ defined over $\FF_{q^r}$, $r\geq 1$. Then $\chi$ gives rise to an algebraic zip datum as follows. Let $P_{\pm} = P_{\pm}(\chi)$ be the pair of opposite parabolic subgroups of $G$ defined by $\chi$. The Lie algebra of $P_{+}$ (resp. $P_-$) is the sum of the weight spaces of weights $\leq 0$ (resp. $\geq 0$) in $\Lie (G)$ under the action of $\chi$. The parabolics $P_{\pm}$ have a common Levi subgroup $L$ which is the stabilizer of $\mu$.

We obtain an algebraic zip datum $\Zcal_{G,\chi} := (G,P_+,\sigma(P_-),\varphi)$, where $\varphi \colon L \to L^{(q)}$ is the relative Frobenius. In particular we obtain an attached zip group $E := E_{G,\chi}$. This zip datum is of the form studied in Subsection~\ref{FrobZipDatum}.

This setting is particularly convenient with respect to functoriality. Indeed, if $\alpha\colon G\to G'$ is a morphism of reductive groups defined over $\FF_q$, then a cocharacter $\mu\colon \GG_m \to G$ (defined over a finite extension of $\FF_q$) induces a cocharacter $\alpha \circ \mu$. Denote by $E$ and $E'$ the associated zip groups. Then $\alpha \times \alpha$ yields a homomorphism $E \to E'$ and we obtain an induced morphism of stacks
$$[E \backslash G ] \to [E' \backslash G']$$

%We set $P := P_+$ and $Q := P^{(\sigma)}_-$. Let $U$ (resp.~$V$) be the unipotent radiacal of $P$ (resp.~of $Q$).
%Then $\chi$ is $B$-dominant if and only if $B \subseteq P_+(\chi)$.

Zip data arising from small and minuscule cocharacters will be of great importance to us. Let us recall briefly the definitions. Let $G$ be a reductive group with a Borel pair $(T,B)$ and let $(X^*,\Phi,X_*,\Phi\vdual,\Delta)$ be the correspondig based root datum. Let $G^{\rm ad}$ be the adjoint group and let $G^{\rm ad} = \prod_{i=1}^rG^{(i)}$ be its decomposition into simple adjoint groups. The image of $(T,B)$ in $G^{(i)}$ is a Borel pair $(T^{(i)},B^{(i)})$. We obtain a corresponding decomposition of the set of roots $\Phi = \coprod_{i=1}^r\Phi^{(i)}$, similarly for the set of coroots and the root basis $\Delta$. Let us call a simple root $\alpha \in \Delta^{(i)}$ \emph{special} if it lies in the orbit of the affine root under the action of the automorphism group of the extended Dynkin diagram of  $(G^{(i)},B^{(i)},T^{(i)})$.

\begin{definition}\label{Minuscule}
A cocharacter $\chi \colon \GG_m \to G$ is called \emph{minuscule} if it satisfies the following equivalent properties (\cite{De_VarSh}~1.2).
\begin{equivlist}
\item
The representation $\ad \circ \chi$ of $\GG_m$ on $\Lie(G)$ has only weights $-1$, $0$, and $1$.
\item
It is conjugate to a cocharacter $\chi'$ of $T$ such that $\langle \chi',\alpha \rangle \in \{-1,0,1\}$ for all $\alpha \in \Phi$.
\item
Let $\chi^{(i)}$ be the unique $B^{(i)}$-dominant cocharacter of $T^{(i)}$ which is conjugate to the image of $\chi$ in $G^{(i)}$. Then there exists at most one simple root $\alpha \in \Delta^{(i)}$ such that $\langle \chi^{(i)},\alpha \rangle > 0$ and in this case $\langle \chi^{(i)},\alpha \rangle = 1$ and $\alpha$ is special.
\end{equivlist}
\end{definition}


\begin{definition}\label{defsmallcochar}
A cocharacter $\chi \colon \GG_m \to G$ is called \emph{small} if it satisfies Condition~(iii) of Definition~\ref{Minuscule} without the assumption that $\alpha$ is special.
\end{definition}

Let $\chi \in X_*$ be a dominant small cocharacter and let $P = P_{+}(\chi)$ be the associated parabolic subgroup. Then the image $P^{(i)}$ of $P$ in $G^{(i)}$ is either a maximal parabolic subgroup (if there exists $\alpha$ as in~(iii)) or equal to $G^{(i)}$ (if $\chi^{(i)}$ is central in $G^{(i)}$).
%Let $A \subseteq \Delta$ be the set of simple roots $\alpha$ occuring in (iii) and let $A \to \{1,\dots,r\}$, $\alpha \sends i(\alpha)$ be the injective map such that $\alpha \in \Delta_{i(\alpha)}$.


%========================================================================

\section{Positivity}

\subsection{Ample characters and ample functions}

\begin{definition}
Let $X$ be a smooth variety and $U\subset X$ an open subset such that $X-U$
is purely of codimension 1. A function $f\in \Ecal(U)$ will be called
\emph{$U$-ample} if it has an effective divisor, and if the support of the
divisor of $f$ is exactly the complement of $U$ in $G$. If $f\in \Ecal(U)_{\mathbb{Q}}$,
we say that $f$ is $U$-ample if some positive multiple of $f$ in
$\Ecal(U)$ is $U$-ample.
\end{definition}

When no confusion can occur about the open subset $U$, we will call
a $U$-ample function simply an ample function.

\begin{definition}\label{DefGAmple}
Let $G$ be a group and $H\subset G$ a closed subgroup. Let $\alpha \in X^{*}(H)$
be a character (see \ref{EqSectionLB}). We say that $\alpha$ is \emph{$G$-ample} or simply \emph{ample} if the line bundle $\xi(\alpha)$ on $G/H$ defined by $\alpha$ is ample. Again we define $\alpha \in X^{*}(H)_{\mathbb{Q}}$ to be $G$-ample if some positive multiple in $X^*(H)$ is $G$-ample.
\end{definition}

We define also an antiample character (resp. function) as the inverse
of an ample character (resp. function).

\begin{remark}\label{CharGAmple}
Let $G$ be a reductive group over an algebraically closed field $k$. Let $(B,T)$ be a Borel pair and let $(X^*,\Phi,X_*,\Phi\vdual,\Delta)$ be the associated root datum and $(W,I)$ be the Weyl group with its system of simple reflections. For $\alpha \in \Phi$ let $\alpha\vdual \in \Phi\vdual$ the dual root. For every $i \in I$ let $\alpha_i \in \Delta$ be the corresponding simple root and $\omega_i \in (X^*)_{\QQ}$ the corresponding fundamental weight.

Let $P_J$ be a parabolic subgroup containing $B$ of type $J \subseteq I$ and let $L$ be the unique Levi subgroup of $P$ containing $T$. Then restriction yields an injective map $X^*(P_J) \cong X^*(L) \mono X^* = X^*(T)$. Its image consists of those $\lambda \in X^*$ such that $\langle \alpha\vdual_j, \lambda\rangle = 0$ for all $j \in J$. It is well known (e.g.~\cite{Jantzen}~II, 4.4) that for a character $\beta \in X^*(P_J)$ the following assertions are equivalent.
\begin{eli}
\item
The character $\beta$ is $G$-ample.
\item
One has $\langle \alpha\vdual_i, \beta \rangle > 0$ for all $i \in I \setminus J$.
%\item
%For every $\lambda \in X^*(G)$ such that $\chi - \lambda = \sum_{i\in I}n_i\omega_i$ for some $n_i \in \QQ$ one has $n_i > 0$ for all $i \in I \setminus J$is a linear combination with $>0$ coefficients
%of the fundamental weights of the maximal parabolics containing $P$.
\end{eli}
In the sequel, $P$ will always denote a parabolic subgroup of type $J$ that contains the opposite Borel $B_-$ of $B$ with respect to $T$. Then $P_J = \leftexp{w_0}{P}$, where $w_0 \in W$ is the longest element. For $i \in I$, $-w_0(\alpha_i)$ is again in $\Delta$. Hence the image of $X^*(P)$ in $X^*$ consists of those $\lambda$ such that $\langle \alpha\vdual_j, \lambda\rangle = 0$ for all $j \in J^{\rm opp}$ and $\beta \in X^*(P)$ is $G$-ample if and only if $\langle \alpha\vdual_i, \beta\rangle < 0$ for all $i \in I \setminus J^{\rm opp}$.
\end{remark}


\begin{rem}\label{ampleprod}
Assume $G$ and $P$ decompose as products $G=G_{1}\times G_{2}$
and $P=P_{1}\times P_{2}$. Let $\alpha\in X^{*}(P)$ be a character
and let $\alpha=(\alpha_{1},\alpha_{2})$ be its decomposition. Then $\alpha$
is ample if and only if $\alpha_{1}$ and $\alpha_{2}$ are ample.
\end{rem}

\begin{rem}
Assume $G\subset G_{1}$ and $P=P_{1}\cap G$ for some parabolic $P_{1}$
in $G_{1}$. Then we get a closed immersion $G/P\to G_{1}/P_{1}$. It follows
that an ample character $\alpha\in X^{*}(P_{1})$ restricts to an ample
character of $P$.
\end{rem}

%---------------------------------------------------------------------

\subsection{The positivity conjecture}

Now we study again a Frobenius zip datum as in Subsection~\ref{FrobZipDatum}. To formulate the next conjecture, recall that if $U\subset G$ denotes the open $E$-orbit of $G$, the injection $\iota\colon \Ecal(U)\to X^{*}(E)$ has finite cokernel contained in $X^*(S_0)$, where $S_0$ is the underlying finite constant group scheme of the stabilizer of $1 \in G$ in the zip group $E$. In particular, it induces an isomorphism
\[
\iota\colon \Ecal(U)_{\mathbb{Q}} \liso X^{*}(E)_{\mathbb{Q}}.
\]
Furthermore, the first projection $p_1\colon E\to P$ induces an isomorphism $p_{1}^{*}\colon X^*(P)\simeq X^*(E)$.

\begin{conjecture}[Positivity Conjecture]\label{positivity}
Let $G$ be a reductive group defined over $\mathbb{F}_{q}$
which has a Borel pair $(B,T)$ defined over $\mathbb{F}_{q}$. Let $P$ be a parabolic containing $B^{-}$ and $Q:=\sigma(P^{-})$. Through the isomorphism
\begin{equation} \label{isomconj}
X^*(P)_{\mathbb{Q}} \overset{p_{1}^{*}}{\longrightarrow} X^*(E)_{\QQ} \overset{\iota^{-1}}{\longrightarrow} \Ecal(U)_{\QQ}
\end{equation}
an $G$-ample character $\alpha\in X^{*}(P)_{\mathbb{Q}}$ is mapped to an ample function in $\Ecal(U)_{\QQ}$.
\end{conjecture}


\begin{remark}\label{InterpretPositivity}
Since the map $\Ecal(U)\to Z^1(G)^E$ deduced from diagram (\ref{diagforG}) is $f\mapsto -\textrm{div}(f)$ (Remark \ref{minusdiv}), the conjecture asserts that an ample character in $X^*(P)_{\QQ}$ is mapped to an element of $Z^1(G)^{E}_{\mathbb{Q}}$ with $<0$ coefficients. Note also that this conjecture does not claim conversely that an ample function
corresponds to an ample character via the isomorphism \ref{isomconj}, which can be seen to be false in general, but true in the split case (see Remark \ref{whyconj}).
\end{remark}

We do not know whether the conjecture holds in general. For the applications to Shimura varieties we have in mind it suffices to prove the Positivity Conjecture for zip data defined by minuscule cocharacters (Subsection~\ref{ZipDataChar}). In particular it will be sufficient to show the following theorem.

\begin{theorem}\label{PositivityThm}
Suppose that the zip datum is defined by a small cocharacter (Definition~\ref{defsmallcochar}) or that $P$ and $Q$ are defined over $\FF_q$. Then Positivity Conjecture~\ref{positivity} holds.
\end{theorem}

We will prove this theorem in the remaining subsections of Section~3. We first deal with the case that $P$ and $Q$ are defined over $\FF_q$ in Subsection~\ref{PositiveRational}. In the next two subsections we reduce to the case that $G$ is of the scalar restriction of an adjoint simple group and make some remarks about embeddings of zip data. Finally, in Subsection~\ref{RestrictionAdjoint} we prove the Positivity Conjecture in a setting that particularly holds if $G$ is the scalar restriction of a simple adjoint group and if the zip datum is given by a small cocharacter.

%----------------------------------------------------------------------

\subsection{The case where $P$ and $Q$ are defined over $\FF_q$}\label{PositiveRational}

The goal of this section is the proof of the Positivity Conjecture in full generality (i.e., without assumption that $P$ is defined by a small cocharacter) but in the special case of a Frobenius zip datum as in Subsection~\ref{FrobZipDatum} where the parabolics $P$ and $Q$ are both defined over $\mathbb{F}_{q}$.
%We denote again by $\varphi$
%the $q$-th power Frobenius. More precisely, let $G$ be a reductive
%group over $\mathbb{F}_{q}$, $(B,T)$ a Borel pair defined over $\mathbb{F}_{q}$
%and $P$ a parabolic containing $B^{-}$, and which is defined over $\mathbb{F}_q$. This is always the case when $G$ is split.
Hence we have $Q = P^{-}$ and $L = M$ is a common Levi subgroup of $P$ and $Q$. As before we denote by $J \subseteq I$ the type of $P$ and by $K = J^{\opp}$ the type of $K$. We then have the following fact (\cite{Wd_Bruhat}~Proposition 3.1):
\begin{prop}
The open $P\times Q$-orbit in $G$ coincides with the open $E$-orbit
in $G$.
\end{prop}

In particular, one has $Z^{1}(G)^{P\times Q}=Z^{1}(G)^{E}$. Denoting
by $U$ the open $E$-orbit, one has a commutative diagram

$$\xymatrix@=30pt{
\Ecal(U)_{\QQ} \ar@{^{(}->}[r]^-{j_1} \ar[rd]^{\simeq}_-{j_2} & X^*(P\times Q)_{\QQ} \ar@<-1ex>[d] \\
&  X^*(E)_{\QQ} \ar@<-1ex>@{.>}[u]_j
}$$
which induces a natural section $j=j_1\circ j_2^{-1} : X^{*}(E)_{\mathbb{Q}}\to X^{*}(P\times Q)_{\mathbb{Q}}$
of the restriction map. To describe it concretely, consider the $\QQ$-linear map
\[
\zeta\colon X^{*}(L)\to X^{*}(L),\quad \chi \mapsto \chi- \chi \circ \varphi.
\]
It induces an automorphism of $X^{*}(L)_{\mathbb{Q}}$ (see the agrument in Remark~\ref{ZetaInjective}). We will always make identifications $X^{*}(L)_{\mathbb{Q}}=X^{*}(P)_{\mathbb{Q}}=X^{*}(Q)_{\mathbb{Q}}$ via the inclusions
$L\subset P$ and $L\subset Q$, and $X^{*}(E)_{\mathbb{Q}}=X^{*}(P)_{\mathbb{Q}}$ via the first projection $E\to P$.

\begin{lem}\label{whatmap}
There is a commutative diagram
$$\xymatrix@1{
X^{*}(E)_{\mathbb{Q}} \ar[r]^-{j} \ar[d]^{\simeq} & X^{*}(P\times Q)_{\mathbb{Q}} \ar[d]^{\simeq} \\
X^{*}(L)_{\mathbb{Q}} \ar[r] & X^{*}(L)_{\mathbb{Q}}\times X^{*}(L)_{\mathbb{Q}}
}$$
where the vertical maps are the above identifications and where the lower horizontal map is given by $\chi \mapsto (\zeta^{-1}(\chi),-\zeta^{-1}(\chi))$.
\end{lem}


\begin{proof}
Let $\chi\in X^{*}(L)_{\mathbb{Q}}$. The corresponding function $f\in \Ecal(U)_{\mathbb{Q}}$
satisfies $f(a^{-1}\varphi(a))=\chi(a)f(1)$ for all $a\in L$. We
want to determine the character $\chi'\in X^{*}(L\times L)_{\mathbb{Q}}$
such that $f(a^{-1}b)=\chi'(a,b)f(1)$ for all $(a,b)\in L\times L$.
Writing $\chi'=(\chi'_{1},\chi'_{2})\in X^{*}(L)_{\mathbb{Q}}\times X^{*}(L)_{\mathbb{Q}}$,
we get $f(a^{-1}b)=\chi_{1}'(a)\chi'_{2}(b)f(1)$. We see immediately
that $\chi_{2}'(a)=\chi'_{1}(a)^{-1}$ and $\chi(a)=\chi_{1}'(a)\chi_{2}'(\varphi(a))=\chi_{1}'(a)\chi_{1}'(\varphi(a))^{-1}=\zeta(\chi_{1}')(a)$.
In other words, $(\chi'_{1},\chi'_{2})=(\zeta^{-1}(\chi),-\zeta^{-1}(\chi))$.
\end{proof}


\begin{rem}\label{imageofEU}
It follows that the image of the map $\Ecal(U)_{\mathbb{Q}}\to X^{*}(P\times Q)_{\mathbb{Q}}=X^{*}(L)_{\mathbb{Q}}\times X^{*}(L)_{\mathbb{Q}}$
consists of elements of the form $(\alpha,-\alpha)$, $\alpha \in X^{*}(L)_{\mathbb{Q}}$.
\end{rem}


\begin{rem}\label{SplitZeta}
If $G$ is split, every character of $P$ is defined over $\mathbb{F}_{q}$,
so we deduce that $\zeta(\chi)=-(q-1)\chi$.
\end{rem}


\begin{lem}\label{FrobAmple}
Let $\chi\in X^{*}(P)_{\mathbb{Q}}$ be a $G$-ample character. Then $\chi\circ\varphi$ is $G$-ample.
\end{lem}

\begin{proof}
As $(B,T)$ are defined over $\FF_q$, the Frobenius induces an automorphism on the Coxeter system $(W,I)$ again denoted by $\varphi$. As $Q$ is defined over $\FF_q$, one has $\varphi(J^{\opp}) = J^{\opp}$. Hence the lemma follows from the characterisation of $G$-ample characters in Remark~\ref{CharGAmple}.
\end{proof}

\begin{lem}\label{antiample}
Let $\chi\in X^{*}(P)_{\mathbb{Q}}$ be an $G$-ample character. Then $\zeta^{-1}(\chi)$ is $G$-antiample.
\end{lem}

\begin{proof}
Write $\alpha=\zeta^{-1}(\chi)$, so $\chi=\alpha-\alpha\circ\varphi$.
Choose $d\geq1$ such that $\alpha$ is defined over $\mathbb{F}_{q^{d}}$.
We deduce from Lemma~\ref{FrobAmple} that
\[
\chi + \chi\circ\varphi + \dots + \chi\circ\varphi^{d} = \alpha-\alpha\circ\varphi^{d}=-(q^{d}-1)\alpha
\]
is $G$-ample, so $\alpha=\zeta^{-1}(\chi)$ is $G$-antiample.
\end{proof}


\begin{remark} \label{whyconj}
However, if $\chi$ is a $G$-ample character, then it is not in general true that $\zeta(\chi)$ is $G$-antiample. This is the main reason why in Conjecture \ref{positivity}, we cannot expect to have an equivalence, but only one implication.
\end{remark}

Denote by $p_2\colon X^*(P\times Q)\to X^*(Q)$ the projection onto the second factor. We have the following commutative diagram

\begin{equation}
\begin{gathered} \label{diageasycase}
\xymatrix@1{
& X^*(P)_{\mathbb{Q}} \ar[d]^{\simeq}&  \\
\Ecal(U)_{\mathbb{Q}}\ar[r]^-{j_2} \ar[dd]_{-\textrm{div}} \ar[rd]_{j_1} & X^*(E)_{\mathbb{Q}} \ar[d]^{j} & \\
& X^*(P\times Q)_{\mathbb{Q}} \ar[d] \ar[r]^-{p_2}& X^*(Q)_{\QQ} \ar[d] \\
Z^{1}_{\mathbb{Q}}(G)^{P\times Q} \ar[r] \ar@/_2pc/[rr]_{\delta} & \textrm{Pic}^{P\times Q}(G)_{\mathbb{Q}} \ar[r] &  \textrm{Pic}(G/Q)_{\mathbb{Q}} 
}
\end{gathered}
\end{equation}
In this diagram, the map $\textrm{Pic}^{P\times Q}(G)_{\mathbb{Q}}\to\textrm{Pic}(G/Q)_{\mathbb{Q}}$
is simply defined by identifying $\textrm{Pic}(G/Q)$ with
$\mbox{Pic}^{Q}(G)$, and forgetting the $P$-action. By composition this defines a map $\delta\colon Z_{\mathbb{Q}}^{1}(G)^{P\times Q}\to\Pic(G/Q)_{\mathbb{Q}}$.

\begin{lem} \label{isomCmodQ}
The map $\delta\colon Z_{\mathbb{Q}}^{1}(G)^{P\times Q}\to \Pic(G/Q)_{\mathbb{Q}}$ is an isomorphism. Further, if $C\in Z^{1}(G)^{P\times Q}$, then $\delta(C)$ is the line bundle attached to the Weil divisor $C/Q\subset G/Q$.
\end{lem}

\begin{proof}
Source and target of $\delta$ are $\mathbb{Q}$-vector spaces of the same dimension (equal to $\mbox{rk}(X^{*}(Q))-\mbox{rk}(X^{*}(G))$), so we need only show that the map is injective. Let $C\in Z_{\mathbb{Q}}^{1}(G)^{P\times Q}$ be mapped to zero in $\textrm{Pic}(G/Q)_{\mathbb{Q}}$. The image of $C$ in $\textrm{Pic}^{P\times Q}(G)_{\mathbb{Q}}$ has a preimage in $X^{*}(P\times Q)_{\mathbb{Q}}=X^{*}(L)_{\mathbb{Q}}\times X^{*}(L)_{\mathbb{Q}}$
of the form $(\alpha,0)$. But $C$ comes from a function $f\in \Ecal(U)_{\mathbb{Q}}$,
whose character in $X^{*}(P\times Q)_{\mathbb{Q}}=X^{*}(L)_{\mathbb{Q}}\times X^{*}(L)_{\mathbb{Q}}$
has the form $(\beta,-\beta)$, by Remark \ref{imageofEU}. So $\alpha=\beta=0$, and $C=0$.

Now consider the following commutative diagram :

$$\xymatrix{
Z^{1}(G)^{P\times Q} \ar[r]\ar[d] & \textrm{Pic}^{P\times Q}(G) \ar[d] & \\
Z^{1}(G)^{Q} \ar[r] & \textrm{Pic}^Q(G)\ar[r]^{\simeq} & \textrm{Pic}(G/Q)
}$$

The map $Z^{1}(G)^{P\times Q}\to Z^{1}(G)^{Q}$ is simply the inclusion
(notice that $Z^{1}(G)^{Q}$ has infinite rank). Now the map $Z^{1}(G)^{Q}\to\textrm{Pic}(G/Q)$ is given by sending $C$ to the line bundle attached to the Weil divisor $C/Q$, so we are done.
\end{proof}


\paragraph{The Borel case}

Assume for a moment that $P$ and $Q$ are Borel subgroups, so $P=B^-$ and $Q=B$. For a simple root $\alpha \in \Delta$, denote by $Z_{\alpha}$
the closure of the $B^{-}\times B$-orbit of $\dot{s}_{\alpha}$. The $Z_{\alpha}$ for $\alpha \in \Delta$ form a basis of $Z^{1}(G)^{B^{-}\times B}$. The right-$B$-invariant subvariety $Z_{\alpha}\subset G$
defines the Weil divisor $\delta(Z_\alpha) = \overline{Z}_{\alpha} := Z_{\alpha}/B$ (Lemma~\ref{isomCmodQ}). Let $\Lcal_{\alpha} \in \Pic(G/B)$ be the corresponding line bundle. Then one has the following (\cite{LMS_GB}~page 99):

\begin{prop}\label{BaseChangeBorel}
The natural map $X^{*}(B)\to \Pic(G/B)$ is given by
\[
\chi\mapsto\sum_{\alpha\in\Delta}\left\langle \alpha^{\vee},\chi\right\rangle \Lcal_{\alpha}.
\]
\end{prop}

We claim that this proves Conjecture~\ref{positivity} in the Borel case. Indeed, take a character $\chi \in X^*(P) = X^*(B^-) = X^*(T)$ and suppose that $\chi$ is $G$-ample or, equivalently, that $\langle \alpha^{\vee},\chi\rangle < 0$ for all $\alpha \in \Delta$ (Remark~\ref{CharGAmple}). Hence as an element of $X^*(Q) = X^*(B)$ the character $\chi$ is $G$-antiample. Consider the Diagram~\eqref{diageasycase}. By Lemma~\ref{whatmap} the image of $\chi$ in $X^*(Q)_{\QQ} = X^*(B)_{\QQ}$ under $p_2 \circ j$ is given by $\gamma := -\zeta^{-1}(\chi)$. Lemma~\ref{antiample} applied to $B$ shows that $\gamma$ is $G$-antiample and hence that $\chi$ is mapped to an antiample line bundle $\Lcal(\gamma) \in \Pic(G/B)_{\QQ}$. By Proposition~\ref{BaseChangeBorel} we have $\Lcal(\gamma) = \sum_{\alpha\in\Delta}\langle \alpha^{\vee}, \gamma\rangle \Lcal_{\alpha}$. But then $\delta^{-1}(L)= \sum_{\alpha}\langle \alpha^{\vee},\gamma\rangle Z_{\alpha}$ by Lemma~\ref{isomCmodQ} and since $\gamma$ is antiample, all the coefficients are $<0$.


\paragraph{The general case with $P$ and $Q$ defined over $\FF_q$}

\begin{prop}\label{conjdef}
In Conjecture~\ref{positivity}, assume further that $P$ is defined over $\mathbb{F}_{q}$. Then the conjecture holds.
\end{prop}

\begin{proof}
Again, we simply need to show that  $\delta^{-1}$ maps an ample line bundle in $\mbox{Pic}(G/Q)$ to a linear combination with $<0$ coefficients in $Z_{\mathbb{Q}}^{1}(G)^{P\times Q}$. We will deduce the result from the Borel case, using an elementary argument. Consider the following commutative
diagram :

$$\xymatrix{
Z^{1}(G)^{P\times Q}_{\mathbb{Q}} \ar[d] \ar[r]^{\simeq}  & \textrm{Pic}(G/Q)_{\mathbb{Q}} \ar[d] & X^*(Q)_{\mathbb{Q}} \ar[l] \ar[d]\\
Z^{1}(G)^{B^{-} \times B}_{\mathbb{Q}} \ar[r]^{\simeq}  & \textrm{Pic}(G/B)_{\mathbb{Q}} & X^*(B)_{\mathbb{Q}} \ar[l]
}$$

All the vertical maps are injective. The leftmost one is defined in
the following way. If $C\subset G$ is a $P\times Q$-orbit of codimension
one, it is mapped to the only $B^{-}\times B$-orbit of codimension
one contained in it. Now, let $\chi\in X^{*}(Q)$ be an ample character.
The restriction to $B$ of $\chi$ is definitely not ample (unless
$Q=B$), but it is a linear combination with $\leq 0$ coefficients
of the fundamental weights of the maximal parabolics containing $B$.
Using the explicit formula available in the Borel case, we deduce
that $\chi$ decomposes with $\leq 0$ coefficients in $Z^{1}(G)^{B^{-}\times B}$
and thus also in $Z^{1}(G)^{P\times Q}$. Now, the number
of simple roots $\beta$ such that $\left\langle \chi,\beta^{\vee}\right\rangle <0$
is equal to the cardinality of $I \setminus J$. This
proves that all the $P\times Q$-orbits of codimension one must appear
with $<0$ coefficients.
\end{proof}


\begin{example}\label{ExampleHB}
Let $G=\mbox{Res}_{\mathbb{F}_{q^{d}}/\mathbb{F}_{q}}(GL_{2,\mathbb{F}_{q^{d}}})$.
Let $(B_{0},T_{0})$ be the usual upper Borel pair in $GL_{2,\mathbb{F}_{q^{d}}}$,
and let $(B,T)$ be its Weyl restriction to $\mathbb{F}_{q}$. In
this example, we take $P=B_{-}$ and $Q=B$. The group $G_{k}$ is
isomorphic to a product of $d$ copies of $GL_{2,k}$. Let $\alpha:T_{0}\to k^{\times}$ be
the character defined by
$$\left(\begin{array}{cc}
a & 0\\
0 & c
\end{array}\right) \mapsto a.
$$
One checks easily that $\alpha$ is a $GL_2$-antiample character of $B_{0,-}$. Let $\alpha_{i}=\alpha\circ\mbox{pr}_{i}$
where $\mbox{pr}_{i}:G_{k}\to GL_{2,k}$ is the $i$-th projection.
The $(\alpha_{i})_{1\leq i\leq d}$ form a basis of $X^{*}(B)/X^{*}(G)$
and are the fundamental weights of the maximal parabolics containing
$B$. The map $\zeta$ is given in this basis by
\[
A=\left(\begin{array}{cccc}
1 & -q\\
 & 1 & \ddots\\
 &  & \ddots & -q\\
-q &  &  & 1
\end{array}\right)
\]
The inverse of $A$ is the matrix

\[
A^{-1}=-\frac{1}{q^d-1}\left(\begin{array}{ccccc}
1 & q & q^2 & \cdots & q^{d-1}\\
q^{d-1} & 1 & q & \cdots & q^{d-2}\\
\vdots & q^{d-1} & \ddots & \ddots & \vdots \\
q^2 & \vdots  & \ddots  & 1 & q \\
q & q^2 &\cdots   & q^{d-1} & 1
\end{array}\right)
\]
which has negative coefficients, as predicted by Lemma \ref{antiample}. The absolute value of the determinant of $A$
is $q^{d}-1$, which is the cardinality of $S_{0}=\mathbb{F}_{q^{d}}^{\times}$,
and this illustrates Proposition \ref{zeta}. Let $\left[C_{i}\right]\in Z_{\mathbb{Q}}^{1}(G)^{E}$
be the closure of the $E$-orbit of $(1,\dots,J,\dots,1)$ where $J=\left(\begin{array}{cc}
 & 1\\
1
\end{array}\right)$. Through the isomorphism
\[
\frac{X^{*}(P)_{\mathbb{Q}}}{X^{*}(G)_{\mathbb{Q}}}\simeq\mbox{Pic}^{E}(G)_{\mathbb{Q}}\simeq Z_{\mathbb{Q}}^{1}(G)^{E},
\]
the cycle $-\left[C_{i}\right]$ corresponds to the character $\zeta(\alpha_{i})=\alpha_{i}-p\alpha_{i-1}$ (where we define $\alpha_0=\alpha_n$).
This illustrates a result by Goren on Hasse invariants for Hilbert-Blumenthal Shimura varieties (see \cite{Goren_Hasse}).
\end{example}

%----------------------------------------------------------------------

\subsection{Reduction to the case of a scalar restriction of an adjoint group}

It is possible to reduce the proof of Theorem~\ref{PositivityThm} to the case when the group $G$ is semisimple adjoint. More precisely:

\begin{prop}
Keep the same notations as in Conjecture \ref{positivity}. Let $G'=G^{ad}$,
and define $T',B',P',Q'$ as the images of $T,B,P,Q$ by the natural
map $G\to G'$. Then $(G',P',Q',\varphi)$ is a zip datum. Further,
if Conjecture \ref{positivity} holds for $G'$, then it does
for $G$ as well.
\end{prop}

\begin{proof}
The first assertion is obvious. Denote by $E'$ the zip group of $(G',P',Q',\varphi)$. We get a map of stacks $\left[E\backslash G\right]\to\left[E'\backslash G'\right]$.
The $E$-orbits in $G$ are in one-to-one correspondance with the
$E'$-orbits in $G'$ \eqref{EqEOrbits}, so in particular the natural map $Z_{\mathbb{Q}}^{1}(G')^{E'}\to Z_{\mathbb{Q}}^{1}(G)^{E}$
is an isomorphism, and maps the canonical basis of $Z_{\mathbb{Q}}^{1}(G')^{E'}$
to the canonical basis of $Z_{\mathbb{Q}}^{1}(G)^{E}$. We have a commutative
diagram :

$$\xymatrix{
Z^{1}_{\mathbb{Q}}(G')^{E'} \ar[r] \ar[d] & \textrm{Pic}_{\mathbb{Q}}^{E'}(G') \ar[d]  & X^*(E')_{\mathbb{Q}} \ar[d] \ar[l] & X^*(P')_{\mathbb{Q}} \ar[l] \ar[d] \\
Z^{1}_{\mathbb{Q}}(G)^{E} \ar[r]  & \textrm{Pic}_{\mathbb{Q}}^{E}(G)  & \frac{X^*(E)_{\mathbb{Q}}}{X^*(G)_{\mathbb{Q}}} \ar[l] & \frac{X^*(P)_{\mathbb{Q}}}{X^*(G)_{\mathbb{Q}}} \ar[l]
}$$where all maps are isomorphisms. A $G$-ample character of $P$ is the
image of a $G'$-ample character of $P'$ (note that $G/P\simeq G'/P'$),
and the result follows.
\end{proof}

Now, assume that we have a direct product of two zip data. More precisely,
this means that $G=G_{1}\times G_{2}$ where $G_{1},G_{2}$ are reductive
groups over $\mathbb{F}_{q}$, and the groups $T,B,P,Q$ decompose
as $T=T_{1}\times T_{2}$, $B=B_{1}\times B_{2}$, $P=P_{1}\times P_{2}$,
$Q=Q_{1}\times Q_{2}$, such that $(G_{1},P_{1},Q_{1},\varphi)$ and
$(G_{2},P_{2},Q_{2},\varphi)$ are zip data.

\begin{prop}\label{PositivityProduct}
Assume that Conjecture \ref{positivity} holds for $G_{1}$ and
$G_{2}$. Then it holds for $G=G_{1}\times G_{2}$.
\end{prop}

\begin{proof}
This follows immediately from Remark \ref{ampleprod}.
\end{proof}

As every adjoint group over $\FF_q$ is isomorphic to a product of algebraic groups of the form $\Res_{\FF_{q^r}/\FF_q}(G_0)$, where $G_0$ is a simple adjoint group over some finite extension $\FF_{q^r}$ of $\FF_q$ (\cite{SGA3}~Exp.~XXIV, Prop.~5.5 and Prop.~5.9), this reduces the proof of Theorem~\ref{PositivityThm} to the case that $G$ is of this form.

%---------------------------------------------------------------------

\subsection{Perfect embeddings of zip data}

Let $G\subset G_{1}$ be two reductive groups defined over $\mathbb{F}_{q}$.
Let $(B,T)$ (resp. $(B_{1},T_{1})$) be a Borel pair defined over
$\mathbb{F}_{q}$ in $G$ (resp. $G_{1}$). Let $P$ (resp. $P_{1}$)
be a parabolic in $G$ (resp. $G_{1}$) containing $B^{-}$ (resp.
$B_{1}^{-}$). Define $Q:=\sigma(P^{-})$ and $Q_{1}:=\sigma(P_{1}^{-})$.
Let $L,M,L_{1},M_{1}$ the associated Levi subgroups of $P,Q,P_{1},Q_{1}$
respectively. Finally, let $U$ (resp. $U_{1}$) denote the open $E$-orbit
(resp. $E_{1}$-orbit) in $G$ (resp. $G_{1}$). We make the following
assumptions :
\begin{enumerate}
\item $B=B_{1}\cap G$, $T=T_{1}\cap G$, $P=P_{1}\cap G$, $Q=Q_{1}\cap G$,
$L=L_{1}\cap G$, $M=M_{1}\cap G$, $R_{u}(P)=R_{u}(P_{1})\cap G$,
$R_{u}(Q)=R_{u}(Q_{1})\cap G$.
\item $U = U_{1}\cap G$.
\end{enumerate}
We call this situation a perfect embedding of zip data. In this case,
the inclusion $P\times Q\to P_{1}\times Q_{1}$ induces an inclusion
$E\to E_{1}$, where $E$ and $E_{1}$ are the corresponding zip groups. Furthermore,
the inclusion $G\to G_{1}$ and the group actions of $E$ and $E_{1}$
are compatible through this map. We get a map of quotient stacks $\left[E\backslash G\right]\to\left[E_{1}\backslash G_{1}\right]$.

\begin{lemma}\label{embedded}
Assume that Conjecture \ref{positivity} holds for $G_{1}$. Assume further that the map $X^{*}(P_{1})\to X^{*}(P)$ induces a surjection from the set of $G$-ample characters of $P$ to the set of $G_1$-ample characters of $P_{1}$. Then Conjecture \ref{positivity} holds for $G$.
\end{lemma}

\begin{proof}
We have a commutative diagram :
$$\xymatrix{
X^{*}(P_1)_{\mathbb{Q}} \ar[r]^{\simeq} \ar[d] &  X^{*}(E_1)_{\mathbb{Q}} \ar[r]^{\simeq} \ar[d] & \Ecal(U_1)_{\mathbb{Q}} \ar[d] \\
 X^{*}(P)_{\mathbb{Q}} \ar[r]^{\simeq} &
 X^{*}(E)_{\mathbb{Q}} \ar[r]^{\simeq} & \Ecal(U)_{\mathbb{Q}}
}$$
It suffices to prove that an ample function $f_{1}\in \Ecal(U_{1})$ is
mapped to an ample function $f\in \Ecal(U)$. It can be seen readily that
$f$ is the restriction of $f_{1}$ to $U$. Since $\div(f_{1})$ is an
effective divisor, $f_1$ it extends to a regular function on $G_{1}$, so
$f$ extends to $G$. Thus $\mbox{div}(f)$ is effective. Now, $f$
vanishes exactly on $G-(U_{1}\cap G)=G-U$, so $f$ is ample.
\end{proof}

%-------------------------------------------------------------------

\subsection{The case of a Weil restriction}\label{RestrictionAdjoint}

We now prove the Positivity Conjecture in the following setting. Let $r\geq 1$ be an integer. Let $G_1$ be a connected reductive group over $\mathbb{F}_{q^r}$ and $G=\textrm{Res}_{\mathbb{F}_{q^r}/\mathbb{F}_q}(G_1)$. Let $k$ be an algebraic closure of $\FF_{q^r}$. We will also write $G_1$ and $G$ instead of $(G_1)_k$ and $G_k$. We denote by $\sigma\in\mbox{Gal}(k/\mathbb{F}_{q})$ the $q$-th power arithmetic Frobenius. Fix a Borel pair $(B_1,T_1)$ defined over $\mathbb{F}_{q^r}$ in $G_1$, and define $B=\textrm{Res}_{\mathbb{F}_{q^r}/\mathbb{F}_q}(B_1)$ and $T=\textrm{Res}_{\mathbb{F}_{q^r}/\mathbb{F}_q}(T_1)$. Then $(B,T)$ is a Borel pair of $G$ defined over $\FF_q$.
Over $k$, the group $G$ decomposes as a product
\begin{equation}
G=G_{1}\times\cdots\times G_{r}\label{prodG}
\end{equation}
where $G_{i}=\sigma^{i-1}(G_{1})$. The Frobenius sends $G_{i}$ onto $G_{i+1}$ (indices taken modulo $r$). We have similar decompositions $T = T_1 \times \dots \times T_r$ and  $B = B_1 \times \dots \times B_r$ over $k$, where $(B_i,T_i)$ is a Borel pair of $G_i$.

Let $P\subseteq G$ be a parabolic subgroup containing $B^{-}$ (defined over some finite extension of $\FF_q$) that decomposes as a product
\begin{equation}
P=P_{1}\times\cdots\times P_{r}\label{prodP}
\end{equation}
such that for each $i=1,\dots,r$, the parabolic $P_{i}\subset G_{i}$ (defined over some finite extension of $\FF_{q^r}$) is either maximal or equal to $G_{i}$. We have $B_i^- \subseteq P_i$. Let $\Delta\subseteq \left\{ 1,\dots,r\right\}$ be the subset where $P_{i}$ is maximal in $G_{i}$.

The Weyl group $(W,I)$ of $G$ with respect to $(T,B)$ decomposes naturally into a product (as a coxter group)
\[
(W,I) = (W_{1},I_1) \times \cdots \times (W_{r},I_r),
\]
where $(W_i,I_i)$ is the Weyl group of $G_i$ with respect to $(T_i,B_i)$. The Frobenius induces an automorphism of Coxeter groups of $(W,I)$ again denoted by $\sigma$ and we have $\sigma(W_i,I_i) = (W_{i+1},I_{i+1})$. In particular all factors are isomorphic. If $J$ denotes the type of $P$, we have $W_{J}=W_{J_{1}}\times\cdots\times W_{J_{r}}$ where $J_{i}\subset I_{i}$ is the type of the parabolic $P_{i}$.

Let $T\subset L\subset P$ denote the Levi subgroup of $P$ containing $T$ and let $P^-$ be the opposite parabolic subgroup with respect to $L$. Define the parabolic $Q\subset G$ by
\[
Q:=\sigma\left(P^{-}\right)= Q_1 \times \dots \times Q_r, \qquad Q_1 := \sigma(P_r^-), \quad Q_i := \sigma(P^-_{i-1}), 1 < i \leq r.
%\sigma(P_{r}^{-})\times\sigma(P_{1}^{-})\times\cdots\times\sigma(P_{r-1}^{-})
\]
As $B = \sigma(B)$, we have $B \subseteq Q$. The Levi subgroup $M$ of $Q$ containing $T$ is then $M = \sigma(L)$. We obtain a Frobenius zip datum (Subsection~\ref{FrobZipDatum}). Let $E$ denote its zip group. Recall that the $E$-orbits are parametrized by $^{J}W$ through the bijection $w \mapsto O^w := E\cdot (g_0 w)^{\cdot}$ where $g_0=w_0 w_{0,J}$, and we have the relation $\dim O^w = \ell(w) + \dim(P)$. Thus the $E$-orbits of codimension $1$ in $G$ are :
\begin{equation}
\label{Eorbs}
C_{j}:=E\cdot\left(1,\dots,(w_0w_{0,J} s_{j}w_{0,J} w_0)^{\cdot},\dots,1\right), \qquad j \in \Delta,
\end{equation}
where $s_{j} \in I_{j}$ is the simple reflection such that $I_j \setminus J_j = \{s_j\}$.

\begin{remark}\label{DescribERes}
An element of $E$
can be written in the form 
\[
\left(\left(x_{1},\dots,x_{r}\right),\left(y_1,y_2,\dots,y_{r}\right)\right)
\]
with $x_i \in P_i$, $y_i \in Q_{i}$ and $\varphi(\overline{x}_{i})=\overline{y}_{i+1}$ for all $1 \leq i \leq r$ (always taking modulo $r$).
\end{remark}
%For each $i\in\Delta$, let $\alpha_{i}\in X^{*}(P_{i})_{\mathbb{Q}}$
%denote the inverse of the fundamental weight of $P_{i}$.


\subsubsection{First step}

Define parabolic subgroups in $G_{1}$ by
\begin{align*}
P'_{1} &:= P_1 \cap \sigma^{-1}(P_2) \cap \dots \cap \sigma^{-(r-1)}(P_r),\\
Q'_{1} &:= Q_1 \cap \sigma(Q_r) \cap \sigma^2(Q_{r-1}) \cap \dots \cap \sigma^{r-1}(Q_2)\\
&= \sigma(P_r^-) \cap \sigma^2(P^-_{r-1}) \cap \dots \cap \sigma^r(P_1^-).
\end{align*}
Since the Borel subgroup $B$ is defined over $\mathbb{F}_{q}$, we see immediately that $P_1'$ and $Q_1'$ are parabolics in $G_{1}$ containing $B^-_{1}$ and $B_1$ respectively. Their respective Levi subgroups with respect to the torus
$T_{1}$ are
\[
L'_{1}:=\bigcap_{i=1}^{r}\sigma^{-(i-1)}(L_{i})\quad\mbox{and}\quad M'_{1}:=\bigcap_{i=1}^{r}\sigma^{r-i+1}(L_{i})=\sigma^{r}(L'_{1}).
\]
Since $\varphi^{r}$ maps $L'_{1}$ to $M'_{1}$, the quadruple $(G_{1},P'_{1},Q'_{1},\varphi^{r})$ is a Frobenius zip datum over $\FF_{q^r}$. The associated zip group is given by
\[
E'_{1}:=\left\{ (x,y)\in P'_{1}\times Q'_{1},\ \varphi^{r}(\overline{x})=\overline{y}\right\} ,
\]
where $\overline{x}=\theta_{L'_{1}}^{P'_{1}}(x)$ and $\overline{y}=\theta_{L'_{1}}^{Q'_{1}}(y)$.
It operates on $G_{1}$ in the usual way. For $(x,y)\in E'_{1}$ set
\begin{equation}\label{injEprime}
\begin{aligned}
u_x &:= (x,\varphi(\overline{x}),\varphi^{2}(\overline{x}),\dots,\varphi^{r-1}(\overline{x}))\in P \\
v_y &:= (y,\varphi(\overline{x}),\varphi^{2}(\overline{x}),\dots,\varphi^{r-1}(\overline{x}))\in Q.
\end{aligned}
\end{equation}
It is clear that $(u_x,v_y)\in E$. The map $E'_{1}\to E$, $(x,y)\mapsto(u_x,v_y)$
is an injective algebraic group homomorphism. The image of $E'_{1}$ is contained (in general strictly) in the stabilizer of $G_{1}$ in $E$ which has the following description (considering $G_1$ as the subvariety $G_1 \times \{1\} \dots \{1\}$ of $G$).

\begin{remark}\label{StabG1}
Let $a=\left(\left(x_{1},\dots,x_{r}\right),\left(y_1,y_2,\dots,y_{r}\right)\right)$ be an element of $E$. Then the following assertions are equivalent.
\begin{equivlist}
\item
$a(G_1) = G_1$.
\item
There exists $g_1 \in G_1$ such that $a\cdot g_1 \in G_1$.
\item
$x_i = y_i$ for all $1 < i \leq r$.
\end{equivlist}
\end{remark}


One has the following preliminary lemma.

\begin{lem}\label{technical}
Let $a=\left(\left(x_{1},\dots,x_{r}\right),\left(y_1,y_2,\dots,y_{r}\right)\right)$ be an element of $E(k)$ stabilizing $G_{1}$. Then for all $1 \leq j\leq r$, one has 
\begin{equation}\label{technical1}
x_{j}\in \Ptilde_j := \bigcap_{i=j,j+1,\dots,r}\sigma^{j-i}(P_{i})
\end{equation}
Further, for all $j = 2,3,\dots,r,r+1$, one has (taking indices modulo $r$)
\begin{equation}\label{technical2}
y_{j}\in \Qtilde_j := \bigcap_{i=0,\dots,j-2}\sigma^{i}(Q_{j+2-i})
\end{equation}
In particular, one has $x_1\in \Ptilde_1 = P'_j$ and $y_1 \in \Qtilde_1 = Q'_1$. Further, one has $(x_1,y_1)\in E'_1(k)$.
\end{lem}

%\begin{rem}
%Note that the lemma concerns only the $k$-points of scheme-theoretic
%stabilizer of $G_{1}$ in $E$, which may be non reduced. However,
%this slight restriction will not be an issue.
%\end{rem}

\begin{proof}
By Remark~\ref{StabG1} we have $x_{i} = y_{i}$ for all $1 < i \leq r$. By definition, one has
\[
\varphi\left(\theta_{L_{j}}^{P_{j}}(x_{j})\right) = \theta_{M_{j+1}}^{Q_{j+1}}(y_{j+1}). \tag{*}
\]
Now, let us fix an integer $1\leq j<r$ and let us prove the first asertion using decreasing induction on the integer $j$. For $j=r$ the assertion is $x_r \in P_r$ and there is nothing to prove. By induction we have 
\[
y_{j+1} = x_{j+1}\in \Ptilde_{j+1} = \bigcap_{i=j+1}^{r}\sigma^{-(i-j-1)}(P_{i})
\]
Now we apply Corollary \ref{CorReduct}~(ii) to the parabolics $\Ptilde_{j+1}$ and $Q_{j+1}$ in $G_{j+1}$ (both containing the torus $T_{j+1}$). We get
\[
\theta_{M_{j+1}}^{Q_{j+1}}(y_{j+1}) \in \Ptilde_{j+1}.
\]
From (*) we deduce $\theta_{L_{j}}^{P_{j}}(x_{j})\in\bigcap_{i=j}^{r}\sigma^{-(i-j)}(P_{i}) = \Ptilde_j$ (because we consider only $k$-valued points). Since $\Ptilde_j \subseteq P_j$ both contain the same Borel $B^-_j$, the unipotent radical of $P_{j}$ is contained in the unipotent radical of $\Ptilde_j$. Hence we find $x_{j}\in \Ptilde_j$ which proves \eqref{technical1}.

A similar argument shows \eqref{technical2}: The proof is by increasing induction on $j$ on the total order $2 < 3 < \dots < r < 1$. For $j = 2$ the assertion is $y_2 \in Q_2$. Now let $j > 2$. We have $x_{j-1} = y_{j-1} \in \Qtilde_{j-1} \cap P_{j-1}$ by induction hypothesis. Again we apply Corollary \ref{CorReduct}~(ii) to deduce $\theta^{P_{j-1}}_{L_{j-1}} \in \Qtilde_{j-1}$ and hence $\theta^{Q_j}_{M_j}(y_j) \in Q_j \cap \sigma(\Qtilde_{j-1}) = \Qtilde_j$ by (*). As the unipotent radical of $Q_j$ is contained in the unipotent radical of $Q_j$, we find $y_j \in \Qtilde_j$.

It remains to prove that $\varphi^{r}(\theta^{P'_1}_{L'_1}(x_1))=\theta^{Q'_1}_{M'_1}(y_1)$. For this, we use induction to prove:
\[
\varphi^j \left( \theta_{\sigma^{-(j-1)}L_{j}}^{\sigma^{-(j-1)}P_{j}} \ \theta_{\sigma^{-(j-2)} L_{j-1}}^{\sigma^{-(j-2)} P_{j-1}} \ \cdots \ \theta_{L_{1}}^{P_1}(x_1)\right) = \theta_{\sigma^{j-1} M_{2}}^{\sigma^{j-1} Q_2} \ \cdots \ \theta_{\sigma M_{j}}^{\sigma Q_{j}} \ \theta_{M_{j+1}}^{Q_{j+1}}(y_{j+1})\tag{**}
\]
For $j=1$ this is clear. Assume that (**) holds and apply the operator $\varphi \circ \theta^{P_{j+1}}_{L_{j+1}}$ to (**). We get:
\begin{align*}
\varphi^{j+1} \left( \theta_{\sigma^{-j}L_{j+1}}^{\sigma^{-j}P_{j+1}} \ \theta_{\sigma^{-(j-1)} L_{j}}^{\sigma^{-(j-1)} P_{j}} \ \cdots \ \theta_{L_{1}}^{P_1}(x_1)\right) & = \varphi \left( \theta^{P_{j+1}}_{L_{j+1}} \ \theta_{\sigma^{j-1} M_{2}}^{\sigma^{j-1} Q_2} \ \cdots \ \theta_{\sigma M_{j}}^{\sigma Q_{j}} \ \theta_{M_{j+1}}^{Q_{j+1}}(y_{j+1}) \right) \\
& = \varphi \left( \theta_{\sigma^{j-1} M_{2}}^{\sigma^{j-1} Q_2} \ \cdots \ \theta_{\sigma M_{j}}^{\sigma Q_{j}} \ \theta_{M_{j+1}}^{Q_{j+1}} \ \theta^{P_{j+1}}_{L_{j+1}}(y_{j+1}) \right) \\
& =  \theta_{\sigma^{j} M_{2}}^{\sigma^{j} Q_2} \ \cdots \ \theta_{\sigma^2 M_{j}}^{\sigma^2 Q_{j}} \ \theta_{\sigma M_{j+1}}^{\sigma Q_{j+1}} \ \varphi \left( \theta^{P_{j+1}}_{L_{j+1}}(x_{j+1}) \right) \\
& =  \theta_{\sigma^{j} M_{2}}^{\sigma^{j} Q_2} \ \cdots \ \theta_{\sigma^2 M_{j}}^{\sigma^2 Q_{j}} \ \theta_{\sigma M_{j+1}}^{\sigma Q_{j+1}} \ \theta^{Q_{j+2}}_{M_{j+2}}(y_{j+2})
\end{align*}
which is (**) for $j+1$. The above holds also for $j=r-1$ if we take indices modulo $r$. Now (**) for $j=r$ together with Corollary \ref{CorReduct}~(iv) gives exactly $\varphi^{r}(\theta^{P'_1}_{L'_1}(x_1))=\theta^{Q'_1}_{M'_1}(y_1)$ as claimed.
\end{proof}


\begin{lem}
\label{bijorbits}Let $X_{1}$ be the set of $E$-orbits in $G$ intersecting
$G_{1}$. Then the map $\sigma\mapsto\sigma\cap G_{1}$ defines a
bijection between $X_{1}$ and the set of $E'_{1}$-orbits in $G_{1}$.\end{lem}
\begin{proof}
First we need to prove that $\sigma\cap G_{1}$ is an $E'_{1}$-orbit.
Let $u,v\in\sigma\cap G_{1}$. We can find an element 
\[
a=\left(\left(x_{1},\dots,x_{r}\right),\left(y{}_{1},\dots,y_{r}\right)\right)\in E
\]
such that $a\cdot u=v$. But then $a$ is in the stabilizer of $G_{1}$ (Remark~\ref{StabG1}), so $(x_{1},y_{1})\in E'_{1}$ by Lemma~\ref{technical}. Thus $\sigma\cap G_{1}$ is contained in a $E'_{1}$-orbit.

Now, let $g\in\sigma\cap G_{1}$ and $(x,y)\in E'_{1}$. Define $\overline{x}:=\theta_{L'_{1}}^{P'_{1}}(x)$ and $\overline{y}:=\theta_{L'_{1}}^{Q'_{1}}(y)$. Consider the pair $(u_x,v_y)\in E$ defined as in \eqref{injEprime}. Then $xgy^{-1}=u_xgv_y^{-1}$, so $\sigma\cap G_{1}$ is exactly an $E'_{1}$-orbit. This also shows the injectivity of the map $\sigma\mapsto\sigma\cap G_{1}$. The surjectivity is clear.

%To prove the statement on codimensions, first notice that $G_{1}$
%is not contained in the open orbit $U$. Indeed, the $E'_{1}$-orbits
%are parametrized by $^{J'_{1}}W$, where $J'_{1}$ is the type of
%the parabolic $P'_{1}$ of $G_{1}$. Since $P_{1}'\neq G_{1}$, there
%are at least two $E'_{1}$-orbits. Now the result follows from the %following Lemma~\ref{codim1}. 
\end{proof}


\begin{lemma}\label{codim1}
Let $X$ be a regular integral noetherian scheme, let $Y \subset X$ be an integral subscheme of codimension 1, and let $Z \subseteq G$ be an irreducible subscheme of $X$ not contained in $Y$. Then every irreducible component of $Z \cap Y$ has codimension 1 in $Z$.
\end{lemma}

\begin{proof}
Since $X$ is regular, $Y$ is a Cartier divisor. Hence the open immersion $X \setminus Y \to X$ is an affine morphism. Therefore the open immersion $j\colon (X \setminus Y) \cap Z \to Z$ is affine. As $Z$ is not contained in $Y$ and $Z$ is irreducible, $j$ is dominant. Hence every irreducible component of $Z \setminus ((X \setminus Y) \cap Z) = Z \cap Y$ has codimension $1$ in $Z$ by \cite{WdYa}~Proposition~1.6.
\end{proof}




We may define also the groups $P'_{j},Q'_{j},E'_{j}$ for any $1\leq j\leq r$.
Note that we can permute the factors to form a new product 
\[
G_{j}\times G_{j+1}\times\dots\times G_{r}\times G_{1}\times\dots\times G_{j-1}
\]
and define $P_{j}',Q'_{j},E'_{j}$ as before with respect to this
new numbering. In other words, we define :
\[
P'_{j}=P_{j}\cap\sigma^{-1}(P_{j+1})\cap\dots\cap\sigma^{-(r-j)}(P_{r})\cap\sigma^{-(r-j+1)}(P_{1})\cap\dots\cap\sigma^{-(r-1)}(P_{j-1})
\]
\[
Q'_{j}=\sigma(P_{j}^{-})\cap\sigma^{r-1}(P_{j+1}^{-})\cap\dots\cap\sigma^{j}(P_{r}^{-})\cap\sigma^{j-1}(P_{1}^{-})\cap\dots\cap\sigma(P_{j-1}^{-})
\]
The tuple $(G_{j},P'_{j},Q'_{j},\varphi^{r})$ defines a group $E'_{j}$
as usual. Then there is a one-to-one correspondance between the $E$-orbits
in $G$ intersecting $G_{j}$ with the set of $E'_{j}$-orbits in
$G_{j}$. This bijection is defined by intersecting an $E$-orbit
with $G_{j}$. Let $(x,y)\in E'_{j}$ and write $\overline{x}:=\theta_{L'_{j}}^{P'_{j}}(x)$
and $\overline{y}:=\theta_{L'_{j}}^{Q'_{j}}(y)$. For all $1\leq j \leq r$, let $U_j$ denote the open $E'_j$-orbit in $G_j$. Then it is clear that $U_j=U\cap G_j$ (it is the $E'_j$-orbit of $1\in G_j$).  Define :
\[
u:=(\varphi^{r-j+1}(\overline{x}),\dots,\varphi^{r-1}(\overline{x}),x,\varphi(\overline{x}),\dots,\varphi^{r-j}(\overline{x}))\in P
\]
\[
v:=(\varphi^{r-j+1}(\overline{x}),\dots,\varphi^{r-1}(\overline{x}),y,\varphi(\overline{x}),\dots,\varphi^{r-j}(\overline{x}))\in Q
\]
It is clear that $(u,v)\in E$. This defines an embedding $\gamma_{j}:E'_{j}\to E$.
\begin{rem}
If $P_{1},\dots,P_{r}$ are all defined over $\mathbb{F}_{q^{r}}$,
then the groups $E'_{j}$ are simply Galois translates of each other.
In particular, the number of $E$-orbits intersecting $G_{j}$ is in
this case independant of $j$.
\end{rem}

\subsubsection{Second step}

We may define a map $\gamma_j : P'_j\to P$ compatible with the map $\gamma_j :E'_j\to E$ defined above. For $x\in P'_j$, define:
$$\gamma_j(x)=(\varphi^{r-j+1}(\overline{x}),\dots,\varphi^{r-1}(\overline{x}),x,\varphi(\overline{x}),\dots,\varphi^{r-j}(\overline{x}))$$ where $\overline{x}:=\theta_{L'_{j}}^{P'_{j}}(x)$. We will now construct a commutative diagram:

\begin{equation}
\begin{gathered}
\label{diagj}
\xymatrix{
X^*(P)_{\QQ} \ar[r]^{\simeq} \ar[d]^{\gamma_{j}^*} & X^*(E)_{\QQ} \ar[r]^{\simeq} \ar[d]^{\gamma_{j}^*} & \Ecal(U)_{\QQ} \ar[d]^{\iota_{j}^*} \ar[r]^{-\textrm{div}} & Z^1(G)^{E}_{\QQ} \ar[d]^{u_j} \\
X^*(P'_j)_{\QQ} \ar[r]^{\simeq}  & X^*(E'_j)_{\QQ} \ar[r]^{\simeq} & \Ecal(U_j)_{\QQ} \ar[r]^{-\textrm{div}} & Z^1(G_j)^{E'_j}_{\QQ} 
}
\end{gathered}
\end{equation}

The maps $\gamma_{j}^*$ is the composition with the embedding $\gamma_j$. If $f\in \Ecal(U)$, then $\iota^*_j(f)$ is the restriction of $f$ to $U_j$ via the natural embedding $\iota_j : G_j \to G$, $x\mapsto (1,\dots,x,\dots,1)$.

\begin{lemma}
The map $\iota_{j}^*$ extends uniquely to a map $u_j : Z^1(G)^{E}_{\QQ} \to Z^1(G_j)^{E'_j}_{\QQ}$.
\end{lemma}

\begin{proof}
The map $u_j$ is clearly unique. Now if $f\in \Ecal(U)$ is the restriction of a character $\chi \in X^*(G)$, then $f\circ \iota_j$ is the restriction of the character $\chi \circ \iota_j$ of $G_j$, so we may define $u_j(\div(f))=\div(f\circ \iota_j)$ for all $f\in \Ecal(U)_{\QQ}$.
\end{proof}

If $C$ is an irreducible component of $G-U$, then we can write $[C]=\div(f)$ for some $f\in \Ecal(U)_{\QQ}$. So $f$ extends to a non-vanishing function on $G-C$. The intersection $C\cap G_j$ is empty or has codimension one in $G_j$ (Lemma~\ref{codim1}). We conclude that the divisor of $f\circ \iota_j$ has support $C\cap G_j$ if this intersection is nonempty. In any case we may write 
\begin{equation} \label{defuj}
u_j([C])= a_{j}(C).[C\cap G_j], \quad a_{j}(C)\geq 0
\end{equation}
and $a_j(C)>0$ if and only if $C\cap G_j\neq \emptyset$.

\begin{lemma}
The diagram \ref{diagj} is commutative.
\end{lemma}

\begin{proof}
For $f\in \Ecal(U)$, the corresponding character $\chi \in X^*(E)$ is defined by the formula $f(e\cdot x)=\chi(e)^{-1}f(x)$ for all $e\in E$ and $x\in U$. Now take $x=\iota_j(y)$ and $e=\gamma_j(u)$ with $y\in U_j$ and $u\in E'_j$. Since $\gamma_{j}(u)\cdot \iota_j(y)=\iota_{j}(u\cdot y)$, we get
$f(\iota_{j}(u\cdot y))=\chi(\gamma_{j}(u))^{-1} f(\iota_{j} (y)$ and the result follows.
\end{proof}

\begin{lemma} \label{amplej}
Assume $\chi \in X^*(P)_{\QQ}$ is $G$-ample. Then $\gamma^{*}_{j}(\chi)\in X^*(P'_j)_{\QQ}$ is $G_j$-ample.
\end{lemma}

\begin{proof}
To simplify notations, we will assume $j=1$. Recall that $P'_{1}:=\bigcap_{i=1}^{r}\sigma^{-(i-1)}(P_{i})$, so the maximal parabolics of $G_1$ containing $P'_1$ are the $\sigma^{-(i-1)}(P_{i})$ for $i\in \Delta$ (some of them may be equal). For all $i\in \Delta$, let $\chi_i \in X^*(P_i)_{\QQ}$ be the fundamental weight of $P_i$. Then $\chi=\sum_{i\in \Delta} a_i \chi_i$ with $a_i<0$ (see Remark \ref{CharGAmple}). We deduce that for all $x\in P'_1$, one has:

$$\chi \circ \gamma_1 = \sum_{i\in \Delta} a_i (\chi_i \circ \varphi^{i-1} ) = \sum_{i\in \Delta} a_i q^{i-1} \chi_{i}^{\sigma^{-(i-1)}}$$
where $\chi_{i}^{\sigma^{-(i-1)}}$ is the fundamental weight of $\sigma^{-(i-1)}(P_{i})$. It follows that $\chi \circ \gamma_1$ is a linear combination with $<0$ coefficients of the fundamental weights of the maximal parabolics containing $P'_1$, so it is ample.
\end{proof}

\begin{lemma}
Let $D\in Z^1(G)^{E}_{\QQ}$. Assume that $u_j(D) > 0$ for all $1\leq j\leq r$. Then $D > 0$.
\end{lemma}

\begin{proof}
Write $D=\sum_{i\in \Delta} n_i [\overline{C}_i]$ where $C_i$ are the codimension one $E$-orbits (see equation \ref{Eorbs}) and $\overline{C}_i$ is its closure. Fix an integer $j\in \Delta$. Define $a_{i,j}:=a_j(\overline{C}_i)$ (see equation \ref{defuj}). Now we have
$$u_j(D)=\sum_{i\in \Delta} a_{i,j} n_i [\overline{C}_i \cap G_j]$$
Clearly $a_{j,j}>0$ because $C_j$ intersects $G_j$. To prove the claim, it suffices to show that in this sum, the only $i\in \Delta$ contributing to $[\overline{C}_j\cap G_j]$ is $i=j$. So assume that  $\overline{C}_j\cap G_j= \overline{C}_i\cap G_j$, $i\in \Delta$. Since the intersection $C_j\cap G_j$ is nonempty, it follows that $C_j \cap \overline{C}_i\neq \emptyset$, so $C_j\subset \overline{C}_i$ because $C_j$ is an $E-$orbit. Looking at dimensions, it follows that $C_j=C_i$, so $i=j$. From $u_j(D)>0$ we deduce $n_j>0$ and finally $D>0$.
\end{proof}

\begin{proposition}
Assume Conjecture \ref{positivity} holds for the zip datum $(G_{j},P'_{j},Q'_{j},\varphi^{r})$ for all $1\leq j \leq r$. Then it holds for $(G,P,Q,\varphi)$.
\end{proposition}

\begin{proof}
This is a simple application of the previous lemmas using the commutativity of diagram \ref{diagj}.
\end{proof}

%Let $f$ be a non-vanishing function on $U\subset G$ and $D=\sum n_{i}\left[ \overline{C}_{i} \right]$
%its divisor, where $C_{i}$ are the codimension one $E-$orbits
%in $G$ and $\overline{C}_{i}$ is the closure of $C_i$. Then $f|_{G_{j}}$ does not vanish on the open %$E'_{j}-$orbit in $G_{j}$. Thus, its divisor is $D_{j}=\sum n'_{i}\left[\overline{C}_{i}\cap G_{j}%\right]$
%where the sum is taken over the $\overline{C}_{i}$ which meet $G_{j}$
%(recall that the intersection $\overline{C_{i}}\cap G_{j}$ has codimension
%one in $G_{j}$, see Lemma \ref{codim1}). Note that the integer $n'_{i}$
%has the same sign as $n_{i}$, although we don't have $n_{i}=n'_{i}$
%in general (the differ by some intersection number).

%Let $j\in\Delta$ and let $\alpha_{j}\in X^{*}(P_{j})_{\mathbb{Q}}$
%be the inverse of the fundamental weight of $P_{j}$. Let $f\in E(U)_{\mathbb{Q}}$
%whose character is $\alpha_{j}$. Let $D=\sum n_{i}\left[C_{i}\right]$
%be the divisor of $f$. Let $1\leq m\leq r$ be an integer. As we
%mentionned above, the $m-$part of $D$ is the divisor of $f_{m}=f|_{G_{m}}$.
%The character $\alpha'_{m}$ of $f_{m}$ is the composition $E'_{m}\to E\overset{\alpha_{j}}{\to}k^%{\times}$.
%Viewed as a character of $P'_{m}$, $\alpha'_{m}=\alpha_{j}\circ\varphi^{d}$
%where $d$ is the integer
%\[
%d=\begin{cases}
%j-m & \mbox{if }j\geq m\\
%j-m+r & \mbox{if }1\leq j<m
%\end{cases}
%\]


\begin{cor}
\label{partial}Assume $P_{1},\dots,P_{r}$ are defined over $\mathbb{F}_{q^{r}}$.
Then Theorem \ref{positivity} holds.
\end{cor}

\begin{proof}
Assume that $P_{1},\dots,P_{r}$ are defined over $\mathbb{F}_{q^{r}}$.
Then $P'_{j}$ and $Q'_{j}$ are defined over $\mathbb{F}_{q^{r}}$,
so the result comes from Proposition \ref{conjdef}.
\end{proof}

\subsubsection{Third step}

In this third step we assume that $P_{1},\dots,P_{r}$ are defined over
some finite field $\mathbb{F}_{q^{rd}}$. We consider the group
\[
\widetilde{G}=\mbox{Res}_{\mathbb{F}_{q^{rd}}/\mathbb{F}_{q}}(G_{1}\times_{\mathbb{F}_{q^{r}}}\mathbb{F}_{q^{rd}})
\]
We embed $G=\mbox{Res}_{\mathbb{F}_{q^{r}}/\mathbb{F}_{q}}(G_{1})$
into $\widetilde{G}$, such that for any $\mathbb{F}_{q}$-algebra
$R$, the map $G(R)\to\widetilde{G}(R)$ is given by the natural map
$R\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^{r}}\to R\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^{rd}}$
induced by the inclusion $\mathbb{F}_{q^{r}}\subset\mathbb{F}_{q^{rd}}$.
Over the algebraic closure, it decomposes into a product
\[
\widetilde{G}_{k}=\left(G_{1}\times\dots\times G_{r}\right)\times\dots\times\left(G_{1}\times\dots\times G_{r}\right)
\]
where the product $G=G_{1}\times\dots\times G_{r}$ appears $d$ times.
Note that the groups $\widetilde{G}$, $G\times\dots\times G$ and $\mbox{Res}_{\mathbb{F}_{q^{d}}/\mathbb{F}_{q}}(G)$
are all different, although the become isomorphic over $k$. The Galois
action on $\widetilde{G}(k)$ is given by
\[
\sigma\cdot(x_{1},\dots,x_{rd})=(\sigma\cdot x_{rd},\sigma\cdot x_{1},\dots,\sigma\cdot x_{rd-1}).
\]
Define $\widetilde{P}=P\times\dots\times P$ and $\widetilde{Q}=Q\times\dots\times Q$
inside $\widetilde{G}_{k}=G_k \times\dots\times G_k$, and denote by $\widetilde{E}$ the associated zip group.
\begin{lem}
This defines a perfect embedding of zip data.\end{lem}
\begin{proof}
The only non trivial part is to show that $U=\widetilde{U}\cap G$ where $\widetilde{U}$ is the open orbit in $\widetilde{G}$. For this, it suffices to prove that the embedding $G\to \widetilde{G}$ induces an injection from the set of $E-$orbits in $G$ to the set of $\widetilde{E}-$orbits in $\widetilde{G}$. But these sets are parametrized respectively by ${}^J W$ and ${}^{\widetilde{J}} \widetilde{W} := {}^J W \times \dots \times {}^J W$, and the map ${}^J W \to {}^{\widetilde{J}} \widetilde{W}$ is the diagonal embedding, so we are done.
\end{proof}
Conjecture \ref{positivity} holds for $\widetilde{G}$ (see
Corollary \ref{partial}). It is clear that the map $X^{*}(\widetilde{P})\to X^{*}(P)$
defines a surjection on the ample characters of $\widetilde{P}$
and $P$. It follows from Lemma~\ref{embedded} that Conjecture
\ref{positivity} also holds for $G$. This terminates the proof.
%Finally, let us formulate our result :
%\begin{thm}
%Assume $G$ and $P$ are defined as in \eqref{prodG} and \eqref{prodP}.
%Then Conjecture \ref{positivity} holds.\end{thm}

%=====================================================================

\section{Hasse invariants for Shimura varieties of Hodge type}


\subsection{Shimura varieties of Hodge type}

Let $(\Gbf,\Xbf)$ be a Shimura datum. Hence $\Gbf$ is a reductive group over $\QQ$ and $\Xbf$ is a $G(\RR)$-conjugacy class of homomorphisms $h\colon \Res_{\CC/\RR}\GG_{m,\CC} \to \Gbf_{\RR}$ satisfying Deligne's conditions (\cite{De_VarSh}). We denote by $[\mu]$ the $\Gbf(\CC)$-conjugacy class of the component of $h_{\CC}\colon \prod_{\Gal(\CC/\RR)}\GG_{m,\CC} \to \Gbf_{\CC}$ corresponding to $\id \in \Gal(\CC/\RR)$. The elements of $[\mu]$ are minuscule by Deligne's axioms. The field of definition of $[\mu]$ is a finite extension $E$ of $\QQ$, the reflex field.

We assume that $(\Gbf,\Xbf)$ is of Hodge type, i.e., it can be embedded into a Shimura datum of the form $(\GSp(V),S^{\pm})$, where $V = (V,\psi)$ is a symplectic space over $\QQ$ and where $S^{\pm}$ is the double Siegel half space. We choose such an embedding $\iota$.

Let $p$ be a prime number such that $\Gbf$ has a reductive model $\Gcal$ over $\ZZ_{(p)}$ (equivalent, $\Gbf_{\QQ_p}$ has a reductive model over $\ZZ_p$ \cite{Ki_Integral}~(2.3.2) which we also denote by $\Gcal$). Hence $K_p := \Gcal(\ZZ_p)$ is a hyperspecial subgroup of $\Gbf(\QQ_p)$. We denote by $G$ the special fiber of $\Gcal$. Hence $G$ is a reductive group over $\FF_p$. 

Choose a place $v$ of the reflex field $E$ of $(\Gbf,\Xbf)$ over $p$. Let $K = K_pK^p \subseteq \Gbf(\AA_f)$ be a compact open subgroup. If $K^p$ is sufficiently small (which we assume form now on), Kisin (\cite{Ki_Integral}) and Vasiu (\cite{Vasiu_Integral}) have shown the existence of integral canonical models $\Scal_K(\Gbf,\Xbf)$ for the Shimura variety attached to $(\Gbf,\Xbf)$ and $K$ (with restrictions for $p = 2$) over $O_{E,v}$. Here we follow Kisin and hence assume that for $p = 2$ Condition~(2.3.4) of \cite{Ki_Integral} is satisfied. In particular, $\Gbf^{\rm ad}$ has no factor of Dynkin type $B$ for $p=2$. We denote by $S := S_K(\Gbf,\Xbf)$ the special fiber. It is a smooth quasi-projective scheme over $\kappa := \kappa(v)$ the residue field of the place $v$. 

There exists a $\ZZ_{(p)}$-lattice $\Lambda$ of $V$ such that the embedding $\iota$ in the Siegel Shimura datum is induced by an embedding $G_{\ZZ_{(p)}} \to \GL(\Lambda)$ (\cite{Ki_Integral}~Lemma~(2.3.1); see also the remarks in the proof of Lemma~4.6.2 in \cite{Pera_CompHodge}). By Zarhin's trick we may assume after changing $(V,\psi)$ and $\Lambda$ that $\psi$ induces a perfect $\ZZ_{(p)}$-pairing on $\Lambda$ (\cite{Ki_Points}~(1.3.3)). We obtain an embedding
\begin{equation}\label{EqEmbGroup}
\iota\colon \Gcal \mono \GSp(\Lambda)
\end{equation}
of reductive group schemes over $\ZZ_{(p)}$ whose generic fiber is an embedding of Shimura data. We call such an embedding a \emph{$p$-integral Hodge embedding}.

By \cite{Ki_Integral} 1.3.2, $\iota$ identifies $\Gcal$ with the scheme theoretic stabilizer of a finite set $s$ of tensors in $\Lambda^{\otimes}$. Here for a finite locally free module $M$ over a ring we write $M^{\otimes}$ for the direct sum of all $R$-modules that one obtgains from $M$ by applying the operations of taking duals, tensor products, symmetric powers and exterior powers finitely often. Then we can identify $\Lambda^{\otimes}$ with $(\Lambda^*)^{\otimes}$. Moreover we identify $\GL(\Lambda)$ with $\GL(\Lambda^*)$ via $g \sends g\vdual := \doubleexp{t}{g}{-1}$ and hence
\[
\Gcal = \set{g \in \GL(\Lambda^*)}{g\vdual(s) = s}.
\]

We set $\Ktilde_p := \GSp(\Lambda)(\ZZ_p)$. By \cite{Ki_Integral}~(2.1.2) there exists for $K^p$ sufficiently small an open compact subgroup $\Ktilde^p \subset \GSp(\AA^p_f)$ containing $K^p$ such that $\iota$ yields an embedding
\[
\eps^0\colon {\rm Sh}_K(\Gbf,\Xbf) \mono {\rm Sh}_{\Ktilde}(\GSp(V),S^{\pm}),
\]
where $\Ktilde := \Ktilde_p\Ktilde^p$. The left hand side can be identfied with a moduli spaces of polarized abelian varieties. More precisely, for $\Ktilde^p$ sufficiently small, let
\[
\tilde\Scal := \Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm})
\]
be the smooth quasi-projective $\ZZ_{(p)}$-scheme whose $T$-valued points ($T$ a $\ZZ_{(p)}$-scheme) the set of isomorphism classes $(\Acal,\lambda,\eta)$, where
\begin{definitionlist}
\item
$\Acal$ is an abelian scheme over $T$ up to prime to $p$ isogeny,
\item
$\lambda$ is an equivalence class of a prime to $p$ quasi-isogeny $\lambda\colon \Acal \to \Acal\vdual$ such that locally on $T$ some multiple is a polarization, and where two such quasi-isogenies are equivalent if they differ by a global section of the constant sheaf with value $\ZZ_{(p)}^{\times}$ on $T$,
\item
$\eta$ is a $\Ktilde^p$-level structure, i.e. a section over $T$ of $\underline{{\rm Isom}}(V_{\AA^p_f},\Vhat^p(\Acal))/\Ktilde^p$, where
\[
\Vhat^p(\Acal) := (\lim_{p \nmid n}\Acal[n])_{\QQ}.
\]
\end{definitionlist}
Then the generic fiber of $\Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm})$ is naturally identified with ${\rm Sh}_{\Ktilde}(\GSp(V),S^{\pm})$ and the integral canonical model $\Scal_K(\Gbf,\Xbf)$ is defined as the normalization of the closure of ${\rm Sh}_K(\Gbf,\Xbf)$ in $\Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm}) \otimes_{\ZZ_{(p)}} O_{E,v}$ (\cite{Ki_Points}~(1.3.4)). In particular, one obtains a finite morphism of $O_{E,v}$-schemes
\begin{equation}\label{SiegelMorph}
\eps\colon \Scal := \Scal_K(\Gbf,\Xbf) \to \Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm}) \otimes_{\ZZ_{(p)}} O_{E,v}
\end{equation}
extending $\eps^0$.

The conjugacy class of cocharacters $[\mu]$ defines a conjugacy class of cocharacters of $\Gcal$ (resp.~of $G$) defined over $O_{E_v}$ (resp.~over $\kappa$) which we again denote by $[\mu]$. As $\Gcal$ is quasi-split, there exists a representative in $[\mu]$ defined over $O_{E_v}$ (resp.~over $\kappa$). 

%---------------------------------------------------------------------

\subsection{The De Rham cohomology and the Hodge line bundle}

Let $\tilde\Acal \to \Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm})$ be the universal abelian scheme. We call its pullback $\Acal$ to $\Scal := \Scal_K(\Gbf,\Xbf)$ via $\eps$ \eqref{SiegelMorph} the universal abelian scheme over $\Scal_K(\Gbf,\Xbf)$%
%and write $\pi\colon \Acal \to \Scal_K(\Gbf,\Xbf)$ for its structure morphism
. We set
\begin{equation}\label{EqDefDeRham}
\Vcal_{\Scal} := H^1_{\rm DR}(\Acal/\Scal)
\end{equation}
This is a locally free $\Oscr_{\Scal}$-module canonically endowed with a finite set $s_{\rm dR}$ of sections of $\Vcal_{\Scal}^{\otimes}$ that are horizontal with respect to the Gau{\ss}-Manin connection.

Then $e^*\Omega^1_{\Acal/\Scal}$ (where $e$ is the zero section of $\Acal$) is a locally direct summand of $\Vcal_{\Scal}$, the \emph{Hodge filtration}.
We call the highest exterior power of the Hodge filtration
\[
\omega_{\Scal} := \det(e^*\Omega^1_{\Acal/\Scal})
\]
the \emph{Hodge line bundle on $\Scal$}. It depends on the chosen $p$-integral Hodge embedding $\iota$. In case we want to stress this dependency, we write $\omega_{\Scal}(\iota)$.

\begin{proposition}\label{HodgeAmple}
The line bundle $\omega_{\Scal}$ is ample. 
\end{proposition}

\begin{proof}
By \cite{MB}, $\omega_{\tilde\Scal}$ is ample. As $\eps$ is finite, $\eps^*\omega_{\tilde\Scal} = \omega_{\Scal}$ is ample.
\end{proof}

%---------------------------------------------------------------------

\subsection{Arithmetic compactifications}

Madapusi Pera has constructed in \cite{Pera_CompHodge} arithmetic toroidal compactifications $\Scal^{\Sigma} = \Scal^{\Sigma}_K(\Gbf,\Xbf)$ (depending on a complete admissible rational partial polyhedral cone decomposition $\Sigma$ for $(\Gbf,\Xbf,K)$) and the arithmetic minimal compactification $\Scal^{\min} = \Scal^{\min}_K(\Gbf,\Xbf)$. Here we use the following facts about these compactifications.
\begin{assertionlist}
\item
For every $\Sigma$ as above there exists a flat proper integral model over $O_{E,(v)}$ of the toroidal compactification of ${\rm Sh}_K(\Gbf,\Xbf)$ given by $\Sigma$ which contains $\Scal$ as an open dense subscheme (\cite{Pera_CompHodge}~4.6.13). It carries a canonical extension $\omega(\iota)^{\Sigma}$ of the Hodge lione bundle (\cite{Pera_CompHodge}~4.8.1).
\item
$\Scal^{\min}$ is a projective $O_{E,(v)}$-scheme containing $\Scal$ as an open dense subscheme (by \cite{Pera_CompHodge}~4.8.11). It is flat with geometric normal fibers over $O_{E,(v)}$ (in the PEL case this is \cite{Lan_Comp}~7.2.4.3 via the description of the local ring; for Shimura varieties of Hodge type the argument is the same).
\item
For every $\Sigma$ as above there is a natural proper surjective map $\int^{\Sigma}\colon \Scal^{\Sigma} \to \Scal^{\min}$ with geometrically connected fibers inducing the identity on $\Scal$ (\cite{Pera_CompHodge}~4.8.11~(3)) such that $(\int^{\Sigma})^*\omega(\iota)^{\min} \cong \omega(\iota)^{\min}$.
\item
For every $p$-integral Hodge embedding $\iota$ the Hodge line bundle $\omega_{\Scal}(\iota)$ extends to an ample line bundle $\omega_{\Scal}(\iota)^{\min}$ over $\Scal^{\min}$. Given different $p$-integral embeddings $\iota$ and $\iota'$, there exists $s,s' \geq 1$ such that $(\omega_{\Scal}(\iota)^{\min})^{\otimes s'} \cong (\omega_{\Scal}(\iota')^{\min})^{\otimes s}$ (\cite{Pera_CompHodge}~4.8.11~(2)).
\item
If $\PGL_{2,\QQ}$ does not occur as a simple factor of $\Gbf^{\rm ad}$, then for any $k \geq 1$ pullback of sections induces isomorphism
\begin{equation}\label{EqExtendSection}
\Gamma(\Scal^{\min},(\omega(\iota)^{\min})^{\otimes k}) \iso \Gamma(\Scal^{\Sigma},(\omega(\iota)^{\Sigma})^{\otimes k} \iso \Gamma(\Scal,\omega(\iota)^{\otimes k}).
\end{equation}
Indeed, the hypothesis implies that the codimension of $\Scal^{\min} \setminus \Scal$ is of codimension at least $2$ in $\Scal^{\min}$. Then the second bijection is the Koecher principle (\cite{Pera_CompHodge}~4.8.12). The composition is given by the pullback of the inclusion $\Scal \mono \Scal^{\min}$. As $\Scal^{\min}$ is normal, Hartogs' theorem implies that this composition is an isomorphism.

The same argument holds for the special fiber and the restriction of $\omega(\iota)^{\min}$ to the special fiber.
\end{assertionlist}



%---------------------------------------------------------------------


\subsection{Ekedahl-Oort stratification}\label{EOStrat}

The conjugacy class $[\mu^{-1}]$ is defined over $\kappa$. As $G$ is quasi-split, there exists a cocharacter $\chi$ of $G_{\kappa}$ whose conjugacy class is $[\mu^{-1}]$. Then $\chi$ yields an orbitally finite algebraic zip datum as follows. Let $P_{\pm} = P_{\pm}(\chi)$ be the attached pair of opposite parabolic subgroups of $G_{\kappa}$ with common Levi subgroup $L$ the centralizer of $\chi$. Let $U_{\pm}$ be the unipotent radical of $P_{\pm}$. We obtain an algebraic zip datum $\Zcal_{G,\chi} := (G,P_+,P^{\sigma}_-,\varphi)$, where $(\ )^{\sigma}$ denotes the pullback under absolute Frobenius $\sigma\colon x \sends x^p$ and where $\varphi\colon L \to L^{\sigma}$ is the relative Frobenius. In particular we obtain an attached zip group $E := E_{G,\chi}$. We set $P := P_+$ and $Q := P^{\sigma}_-$. Let $U$ (resp.~$V$) be the unipotent radiacal of $P$ (resp.~of $Q$). We define the algebraic quotient stack over $\kappa$
\begin{equation}\label{EqDefGZipStack}
\GZip^{\chi} := [E \bs G_{\kappa}].
\end{equation}
By \cite{PWZ2}~Proposition~3.11, for a $\kappa$-scheme $T$ the $T$-valued points of this quotient stack is the groupoid of tuples $\Iline = (I,I_+,I_-,\iota)$, where $I$ is a $G_{\kappa}$-torsor, where $I_+ \subseteq I$ a $P$-torsor, $I_- \subseteq I$ a $Q$-torsor, and where $\iota\colon I^{\sigma}_+/U^{\sigma} \iso I_-/V$ is an isomorphism of $L^{\sigma}$-torsors.

The following lemma shows that we can assume (after conjugating $\chi$ over $\kappa$) that there exists a Borel pair $(T,B)$ of $G$ (defined over $\FF_p$) such that $\chi$ is a $B_{\kappa}$-antidominant cocharacter of $T_{\kappa}$.

\begin{lemma}\label{RationalBorel}
Let $T$ be a maximal torus, $B$ be a Borel subgroup of $G$ containing $T$, and let $B^-$ be the opposite Borel subgroup of $B$ with respect to $T$ (all defined over $\FF_p$). Let $\chi\colon \GG_{m,\kappa} \to G_\kappa$ be a cocharacter. Then there exists $g \in G(\kappa)$ such that for $\chi' := {\rm int}(g) \circ \chi$ the following properties hold.
\begin{assertionlist}
\item
$B_{k} \subseteq P_-(\chi')$.
\item
$\chi'$ factors through $T_\kappa$.
\item
$L(\chi')$ is the unique Levi subgroup of $P_-(\chi')$ containing $T_\kappa$. In particular $B^- \subseteq P_+(\chi')$.
\end{assertionlist}
\end{lemma}

The proof of the lemma works for all quasi-split reductive groups over arbitrary base fields.

\begin{proof}
By \cite{SGA3}~Exp.~XXVI, Lemme~3.8 there exists a parabolic subgroup $P'$ of $G_k$ such that $B_k \subseteq P'$ and such that $P'$ has the same type as $P_-(\chi)$. By loc.~cit.\ Corollaire~5.5~(ii) there exists $g \in G(k)$ such that $P_-({\rm int}(g) \circ \chi) = \leftexp{g}{P_-(\chi)} = P'$. Hence we may assume that $B \subseteq P_-(\chi)$. Let $L'$ be the unique Levi subgroup of $P_-(\chi)$ containing $T$. By loc.~cit.\ Corollaire~1.8 there exists (a unique) $g \in U_-(\chi)(k)$ such that $\leftexp{g}{L(\chi)} = L'$. Hence we may assume that $L(\chi)$ is the Levi subgroup $P_-(\chi)$ containing $T_k$. But then $T_k$ is a maximal torus of $L(\chi)$ and hence contains the identity component of the center of $L(\chi)$. By definition of $L(\chi)$, the cocharacter $\chi$ factors through the center of $L(\chi)$ and hence through its identity component.
\end{proof}

From now on we choose $T$, $B$, and $\chi$ as in Lemma~\ref{RationalBorel}.

\bigskip

Zhang has constructed in \cite{Zhang_EO} a $G$-zip of type $\chi$ over $S_K := S_K(\Gbf,\Xbf)$ and he has shown in loc.~cit.\ that the corresponding classifying morphism $S_K \to \GZip^{\chi}$ is smooth. Here we use the (slightly different) Construction~5.13 of a $G$-zip $\Iline$ of type $\chi$ given by Wortmann in \cite{Wort_MuOrd}~\S5 and obtain a smooth morphism
\begin{equation}\label{EqDefZeta}
\zeta := \zeta_{G}\colon S_K \lto \GZip^{\chi}.
\end{equation}
The Ekedahl-Oort strata of $S_{K}$ are the fibers of $\zeta$.

Let us recall Wortmann's construction. Let $\bar\Acal \to S_K$ be the restriction of the universal abelian scheme to the special fiber. Define $\bar\Vcal := H^1_{\rm DR}(\bar\Acal/S_K)$, let $\Ccal \subseteq \bar\Vcal$ be the Hodge filtration, and let $\Dcal \subseteq \bar\Vcal$ be the conjugate filtration. The embedding $\iota$ \eqref{EqEmbGroup} yields an embedding
\begin{equation}\label{EqEmbSpecial}
G \mono \GL(\Lambda_{\FF_p}) \liso \GL(\Lambda^*_{\FF_p}),
\end{equation}
where $(\ )^*$ denotes the dual space and where the isomorphism is given by $g \sends g\vdual := \leftexp{t}{g}^{-1}$. Then $(\ )\vdual \circ \chi$ and $(\ )\vdual \circ \chi^{\sigma}$ define $\ZZ$-gradings
\[
\Lambda_{\kappa}^* = \bigoplus (\Lambda_{\kappa}^*)^n_{\chi}, \qquad \Lambda_{\kappa}^* = \bigoplus (\Lambda_{\kappa}^*)^n_{\chi^{\sigma}}
\]
with $(\Lambda_{\kappa}^*)^n_{\chi} = (\Lambda_{\kappa}^*)^n_{\chi^{\sigma}} = 0$ for all $n \ne 0,1$. We obtain a descending resp.~ascending filtration
\begin{gather*}
\Fil^0_{\chi} := \Lambda_{\kappa}^* \ \supset \ \Fil^1_{\chi} := (\Lambda_{\kappa}^*)^1_{\chi} \ \supset \ \Fil^2_{\chi} := 0, \\
\Fil_{-1}^{\chi^{\sigma}} := 0 \ \subset \ \Fil_0^{\chi^{\sigma}} := (\Lambda_{\kappa}^*)^0_{\chi^{\sigma}} \ \subset \ \Fil_1^{\chi^{\sigma}} := \Lambda_{\kappa}^*.
\end{gather*}
Then $P_+$ is the stabilizer of $\Fil^{\bullet}_{\chi}$ in $G_{\kappa}$, and $P_-^{\sigma}$ is the stabilizer of $\Fil_{\bullet}^{\chi^{\sigma}}$ in $G_\kappa$.

Let $\bar\Vcal_{\Scal} = H^1_{\rm DR}(\bar\Acal/S_K)$ be the restriction of $\Vcal_{\Scal}$ to the special fiber. Let $\bar{s}_{\rm dR} \subset \bar\Vcal_{\Scal}^{\otimes}$ be the reduction of the tensors $s_{\rm dR} \subset \Vcal_{\Scal}^{\otimes}$, and denote by $\sbar$ the base change of $s \subset (\Lambda^*)^{\otimes}$ to $(\Lambda_{\kappa}^*)^{\otimes}$. Define
\begin{align*}
I &:= \Isomscr_{S_K}((\Lambda^*_{\kappa},\sbar) \otimes \Oscr_{S_K}, (\bar\Vcal_{\Scal},\sbar_{\rm dR})), \\
I_+ &:= \Isomscr_{S_K}((\Lambda^*_{\kappa},\sbar,\Fil^{\bullet}_{\chi}) \otimes \Oscr_{S_K}, (\bar\Vcal_{\Scal},\sbar_{\rm dR}, \bar\Vcal_{\Scal} \supset \Ccal)), \\
I_- &:= \Isomscr_{S_K}((\Lambda^*_{\kappa},\sbar,\Fil_{\bullet}^{\chi^{\sigma}}) \otimes \Oscr_{S_K}, (\bar\Vcal_{\Scal},\sbar_{\rm dR}, \Dcal \subset \bar\Vcal_{\Scal})).
\end{align*}
Then $G_{\kappa}$ acts from the right on $I$ by $\beta\cdot g := \beta \circ g\vdual$, inducing right actions of $P_+$ and $P_-^{\sigma}$ on $I_+$ and $I_-$, respectively. The Cartier isomorphism on $\bar\Vcal_{\Scal}$ induces an isomorphism $\iota\colon I^{\sigma}_+/U^{\sigma}_+ \iso I_-/U^{\sigma}_-$ and $\Iline := (I,I_+,I_-,\iota)$ is a $G$-zip of type $\chi$ over $S_K$ (\cite{Wort_MuOrd}~5.14). We obtain the morphism $\zeta\colon S_K \lto \GZip^{\chi}$ \eqref{EqDefZeta}, which is smooth by \cite{Zhang_EO}~3.1.2.

%As we follow here Wortmann's construction, the type $\chi$ of the $G$-zip is the inverse of the type considered in \cite{ViWd} in the special case of Shimura varieties of PEL type. This is because the above construction uses De Rham cohomology and contravariant Dieudonn\'e theory (following Kisin) and in \cite{ViWd} De Rham homology and covariant Dieudonn\'e theory is used.
%

\begin{remark}\label{PropEO}
The following properties of the Ekedahl-Oort strata are known.
\begin{assertionlist}
\item
Each Ekedahl-Oort stratum is smooth (\cite{WdYa}).
\item
If the Ekedahl-Oort $S^w$ is non-empty, it has dimension $\ell(w)$ (\cite{Zhang_EO} Proposition 3.1.6).
\item
The closure of the Ekedahl-Oort stratum $S^w$ is $\bigcup_{w' \preceq w}S^{w'}$ for a certain refinement $\preceq$ of the Bruhat order; see \cite{PWZ1}~Definition~6.1 for the precise definition of $\preceq$ (\cite{Zhang_EO} Proposition~3.1.6).
\item
The inclusion $S^w \mono \overline{S^w}$ is affine, in particular every irreducible component of $\overline{S^w} \setminus S^w$ is of codimension $1$ in $\overline{S^w}$ (\cite{WdYa}).
\end{assertionlist}
\end{remark}

\begin{remark}\label{NonEmptyEO}
For Shimura varieties of PEL type it is shown in \cite{ViWd}~Theorem~10.1 that all Ekedahl-Oort strata are non-empty. In general this is expected, but it is not known.
%
%If $S$ is proper (see Remark~\ref{ShimuraCompact} below), then $\zeta$ is universally closed.
\end{remark}


\begin{example}\label{ExampleSymplecticZip}
Consider the case $\Gcal = \GSp(\Lambda)$. Recall that $\rk_{\ZZ_{(p)}}(\Lambda) = 2g$. We endow $\Lambda^*$ with the symplectic pairing $\psi^*$ corresponding to the symplectic pairing $\psi$ on $\Lambda$ (i.e., if $\psi$ is given by an isomorphism $\Lambda \iso \Lambda^*$, then $\psi^*$ is given by its inverse $\Lambda^* \iso \Lambda = \Lambda^{**}$). Then $\GL(\Lambda) \iso \GL(\Lambda^*)$, $g \sends g\vdual$ induces an isomorphism $\GSp(\Lambda) \iso \GSp(\Lambda^*)$.

We have $\kappa = \FF_p$ and $(\ )\vdual \circ \chi = (\ )\vdual \circ \chi^{\sigma}$ defines a decomposition $\Lambda_{\FF_p}^* = (\Lambda_{\FF_p}^*)_0 \oplus (\Lambda_{\FF_p}^*)_1$ in totally isotropic subspaces. Hence $P_+$ and $P_-^{\sigma} = P_-$ are opposite parabolic subgroups whose common Levi subgroup is the stabilizer of the grading of $\Lambda_{\FF_p}^*$.

By \cite{PWZ2}~8.4, a $\GSp(\Lambda^*_{\FF_p})$-zip of type $(\ )\vdual \circ \chi$ (resp.~a $\GSp(\Lambda_{\FF_p})$-zip of type $\chi$) over an $\FF_p$-scheme $S$ may be interpreted as a triple $(\Mcalline,\Lcalline,E)$ consisting of an $F$-zip $\Mcalline$ over $S$ of rank $2g$ of type $\tau\vdual$ with $\tau\vdual(i) = g$ for $i = 0,1$ (resp.~of type $\tau$ with $\tau(i) = g$ for $i = -1,0$), of an $F$-zip $\Lcalline$ of $S$ of rank $1$ and an admissible epimorphism $E\colon \bigwedge^2\Mcalline \epi \Lcalline$ such that corresponding morphism of $F$-zips $\Etilde\colon \Mcalline \to \Mcalline\vdual \otimes \Lcalline$ is an isomorphism. Here $\Mcalline\vdual$ denotes the dual $F$-zip (in the sense of \cite{PWZ2}~Definition~6.5) and the tensor product is in the tensor category of $F$-zips over $S$ (\cite{PWZ2}~Definition~6.4).

We apply functoriality of $G$-zips (\cite{Zhang_Func}~Theorem~0.2) to the isomorphism $\GSp(\Lambda) \iso \GSp(\Lambda^*)$. Then the correspondence in \cite{PWZ2}~8.4 shows that there is an equivalence between \mbox{$\GSp(\Lambda_{\FF_p})$-zips} of type $\chi$ and $\GSp(\Lambda^*_{\FF_p})$-zip of type $(\ )\vdual \circ \chi$ which is given by attaching to a $\GSp(\Lambda_{\FF_p})$-zip $(\Mcalline,\Lcalline,E)$ of type $\chi$ the $\GSp(\Lambda^*_{\FF_p})$-zip of type $(\ )\vdual \circ \chi$ given by $(\Mcalline\vdual, \Lcalline\vdual, E^{-1})$. Here $E^{-1}\colon \bigwedge^2(\Mcalline\vdual) \to \Lcalline\vdual$ is the symplectic pairing given by
\[
\Etilde^{-1} \otimes \id_{\Lcalline\vdual}\colon \Mcalline\vdual = \Mcalline\vdual \otimes \Lcalline \otimes \Lcalline\vdual \iso \Mcalline \otimes \Lcalline\vdual = (\Mcalline\vdual)\vdual \otimes \Lcalline\vdual.
\]

Let $\Vcal_{\Stilde}$ be the first De Rham cohomology of the universal abelian scheme over the special fiber $\Stilde := S_{\Ktilde}(\GSp(\Lambda),S^{\pm})$. The Hodge filtration, the conjugate filtration, and the Cartier isomorphism define a natural structure of an $F$-zip $\Vcalline_{\Stilde}$ on $\Vcal_{\Stilde}$ (\cite{MoWd}~\S7). The universal equivalence class $\lambda$ of prime to $p$ quasi-isogenies yields a class of symplectic pairings on $\Vcal_{\Stilde}$ up to multiplication with a locally constant function with values in $\FF_p^{\times}$. We choose a pairing $\gamma$ in this class. This yields by loc.~cit.~a morphism $\gamma\colon \bigwedge^2(\Vcalline_{\Stilde}) \to \One(1)$ of $F$-zips, where $\One(1)$ denotes the Tate-$F$-zip of weight $1$ (\cite{PWZ2}~Example~6.6). One obtains a $\GSp(\Lambda^*_{\FF_p})$-zip of type $(\ )\vdual \circ \chi$. Its dual $(\Vcalline\vdual_{\Stilde}, \One(-1), \gamma^{-1})$ as explained above is then the $\GSp(\Lambda_{\FF_p})$-zips of type $\chi$ defining the morphism 
\[
\zeta\colon \Stilde \lto \GSp(\Lambda_{\FF_p})-{\tt Zip}^{\chi}.
\]
\end{example}


\begin{proposition}\label{EOCodim1}
Assume that all Ekedahl-Oort strata are non-empty (e.g., if the Shimura variety is of PEL type, cf. Remark~\ref{NonEmptyEO}). Then the free rank of $\Pic(\GZip^{\chi})$ is the number of Ekedahl-Oort strata of codimension one.
\end{proposition}

\begin{proof}
The rank of $\Pic(\GZip^{\chi})$ is the number of irreducible
components of $G-U$, where $U$ is the open $E$-orbit in $G$ (Corollary~\ref{rankPic}).
\end{proof}

%\begin{thm} \label{equiv} The following are equivalent
%\begin{enumerate}
%\item The groups $P$ and $Q$ are maximal parabolics.
%\item The free rank of $\Pic(\mbox{G-Zip})$ is one.
%\item There exists a unique Ekedahl-Oort stratum of codimension one.
%\end{enumerate}
%\end{thm}
%
%\begin{proof}
%This is~\eqref{EqRankGP}.
%\end{proof}


\begin{remdef}\label{MuOrdLocus}
As $\zeta$ is open, the preimage of the generic point of $\GZip^{\chi}$ is open and dense in $S_K(\Gbf,\Xbf)$. Moreover, Wortmann has shown (\cite{Wort_MuOrd}~Theorem~6.10) that this generic Ekedahl-Oort stratum is the $\mu$-ordinary stratum. We denote it by $S^{\text{$\mu$-ord}} = S_K^{\text{$\mu$-ord}}(\Gbf,\Xbf)$.
\end{remdef}


%---------------------------------------------------------------------

%\subsection{The $\mu$-ordinary locus}



%---------------------------------------------------------------------

\subsection{Bruhat stratification}

We continue to assume $P$ and $Q$ contain both a Borel subgroup of $G_{\kappa}$ which is already defined over $\FF_p$. We call
\[
\Bscr_G^\chi := [P\bs G_{\kappa}/Q]
\]
the Bruhat stack attached to $(G,\chi)$. Let
\begin{equation}\label{EqZipBruhat}
\beta_G\colon \GZip^{\chi} \lto \Bscr_G^\chi
\end{equation}
be the canonical morphism.

\begin{example}\label{BruhatSymplectic}
We set $\bar\Lambda := \Lambda \otimes_{\ZZ_p} \FF_p$ endowed with the induced symplectic form and write by $2g = \dim_{\FF_p}(\bar\Lambda)$. Then $\tilde\chi := \iota \circ \chi$ is $\GSp(\bar\Lambda)(\kappa)$-conjugate to a cocharacter of $\GSp(\bar\Lambda)_{\kappa}$ which is defined over $\FF_p$. As explained in Example~\ref{ExampleSymplecticZip}, $\tilde\chi$ defines a cocharacter of $\GSp(\bar\Lambda)_{\kappa}$ which has only weights $-1$ and $0$ on $\bar\Lambda_{\kappa}$ and such that the weight decomposition $\bar\Lambda_{\kappa} = \bar\Lambda_{-1} \oplus \bar\Lambda_0$ is a decomposition into totally isotropic subspaces.

For every $\kappa$-scheme, $\Bscr(\GSp(\bar\Lambda),\tilde\chi)(S)$ can be identified with the groupoid of triples $(\Mcal,\Lcal,E,\Ccal,\Dcal)$, where $\Mcal$ is a finite locally free $\Oscr_S$-module of rank $2g$, $\Lcal$ is a finite locally free $\Oscr_S$-module of rank $1$, $E\colon \bigwedge^2(\Mcal) \to \Lcal$ is an alternating pairing suchnthat the corresponding homomorphism $\Etilde\colon \Mcal \iso \Mcal\vdual \otimes \Lcal$ is an isomorphism, and where $\Ccal, \Dcal \subseteq \Mcal$ are Lagrangian submodules (i.e., $\Etilde$ induces isomorphisms $\Ccal \iso \Ccal^{\bot} \otimes \Lcal$ and $\Dcal \iso \Dcal^{\bot} \otimes \Lcal$.

Using \cite{Wd_Bruhat}~Example~2.11 one sees that the map
\[
\beta_{\GSp(\bar\Lambda)}\colon \GSp(\Lambda_{\FF_p})-{\tt Zip}^{\tilde\chi} \lto \Bscr(\GSp(\bar\Lambda),\tilde\chi)
\]
is given by attaching to a $\GSp(\bar\Lambda)$-zip $(\Mcalline,\Lcalline,E)$ of type $\chi$ (in the sense of \ref{ExampleSymplecticZip}) the tuple $(\Mcal,\Lcal,E,\Ccal,\Dcal)$, where $\Mcal$ and $\Lcal$ is the underlying $\Oscr_S$-module of $\Mcalline$ and $\Lcalline$ and where $\Ccal = C^0(\Mcalline)$ and $\Dcal = D_{-1}(\Mcalline)$.

Hence by Example~\ref{ExampleSymplecticZip} the composition
\[
\beta_{\GSp(\bar\Lambda)} \circ \zeta_{\GSp(\Lambda)}\colon \Stilde := S_{\Ktilde}(\GSp(\Lambda),S^{\pm}) \to \Bscr_{\GSp(\bar\Lambda)}^{\iota \circ \chi}
\]
is given by the tuple $(\Vcal\vdual_{\Stilde},\One(-1),\gamma^{-1},\Ccal,\Dcal)$, where $\Ccal$ (resp.~$\Dcal$) is the orthogonal complement of the Hodge filtration (resp.~the conjugate filtration) in $\Vcal_{\Stilde}$.
%has the following interpretation. Let $\tilde\Vcal$ be the first De Rham cohomology of the universal $p$-principally polarized abelian scheme over $\tilde{S}_{\kappa} = \Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm}) \otimes_{\ZZ_{(p)}} \kappa$. The $p$-principal polarization induces a perfect alternating pairing $\lrangle$ on $\tilde\Vcal$. It is endowed with the Hodge filtration $\tilde\Ccal$ and the conjugate filtration $\tilde\Dcal$ which are Lagrangian submodules. Passing to dual $\Oscr_{\Stilde}$-modules we obtain a symplectic $\Oscr_{\Stilde}$-module $\tilde\Vcal\vdual$ endowed with Lagrangian submodules $\Ccal^{\bot}$ and $\Dcal^{\bot}$. The triple $(\tilde\Vcal\vdual, \Ccal^{\bot}, \Dcal^{\bot})$ corresponds to the morphism $\beta_{\GSp(\bar\Lambda)} \circ \zeta_{\GSp(\Lambda)}\colon \Stilde \to \Bscr(\GSp(\bar\Lambda),\tilde\chi)$.
\end{example}

We have a diagram of morphisms of algebraic stacks over $\kappa$
\begin{equation}\label{EqCommutativeZip}
\begin{gathered}\xymatrix{
S_K(\Gbf,\Xbf) \ar[rr]^{\zeta_G} \ar[d]_{\bar\eps} & & \GZip^{\chi} \ar[rr]^{\beta_G} \ar[d]_{\iota_{\tt Z}} & & \Bscr(G,\chi) \ar[d]^{\iota_{\Bscr}} \\
S_{\Ktilde}(\GSp(\Lambda),S^{\pm}) \ar[rr]^{\zeta_{\GSp(\Lambda)}} & & \GSp(\bar\Lambda)-{\tt Zip}^{\iota\circ\chi} \ar[rr]^{\beta_{\GSp(\bar\Lambda)}} & & \Bscr(\GSp(\bar\Lambda),\iota\circ\chi)
}\end{gathered}
\end{equation}
where the middle and the right vertical arrows are induced by functoriality by the embedding $\iota\colon G \to \GSp(\bar\Lambda)$.

\begin{lemma}\label{CommZip}
This diagram is commutative.
\end{lemma}

\begin{proof}
This is clear for the right square and is a special case of Theorem~0.2 in \cite{Zhang_Func} for the left square.
\end{proof}


%---------------------------------------------------------------------

\subsection{Hasse invariants for the $\mu$-ordinary locus}

Let $(\Mcal,\Lcal,E,\Ccal,\Dcal)$ be the universal tuple over $\Bscr_{\GSp(\bar\Lambda)}^{\iota\circ\chi}$ (Example~\ref{BruhatSymplectic}). Let
\[
\omega^{\dflat}_{\GSp(\bar\Lambda)} := \det(\Ccal) \in \Pic(\Bscr_{\GSp(\bar\Lambda)}^{\iota\circ\chi}).
\]

\begin{proposition}\label{IdentifyHodge}
The pullback of $\omega^{\dflat}_{\GSp(\bar\Lambda)}$ under $\delta := \beta_{\GSp(\bar\Lambda)} \circ \zeta_{\GSp(\Lambda)}$ is isomorphic to the Hodge line bundle on $S_{\Ktilde}(\GSp(\Lambda),S^{\pm})$. 
\end{proposition}

\begin{proof}
Example~\ref{BruhatSymplectic} shows that $\delta^*\Mcal \cong \Vcal\vdual_{\Stilde}$. Moreover, $E$ induces an isomorphism $\Mcal \iso \Mcal\vdual \otimes \Lcal$ and hence $\det(\Mcal\vdual) \cong \det(\Mcal) \otimes \Lcal^{\otimes-1}$. Moreover $\delta^*(\Ccal) = (\Vcal_{\Stilde}/\Hcal)\vdual$, where $\Hcal \subset \Vcal_{\Stilde}$ is the Hodge filtration. Hence the Hodge line bundle is given by
\[
\det(\Hcal) \cong \det(\Vcal_{\Stilde}) \otimes \det((\Vcal/\Hcal)\vdual) \cong \delta^*(\det(\Mcal) \otimes \Lcal^{\otimes-1} \otimes \det(\Ccal)).
\]
Now $\det(\Mcal)$ is the line bundle attached to the determinant character of $\GSp(\Lambda)$, and $\Lcal$ is the line bundle attached to the multiplier character of $\GSp(\Lambda)$. Hence $\det(\Mcal) \cong \Lcal^{\otimes g}$. Moreover $\delta^*(\Lcal)$ is the underlying line bundle of the Tate $F$-zip $\One(-1)$ and hence is trivial. Therefore $\delta^*(\det\Mcal)$ is also trivial. This proves the claim.
\end{proof}

We define a line bundle on $\Bscr^{\chi}_G$ by
\begin{equation}\label{EqHodgeDFlat}
\omega^{\dflat}_G := \iota^*_{\Bscr}(\omega^{\dflat}_{\GSp(\bar\Lambda)}).
\end{equation}
As the universal abelian scheme $\Acal$ over $\Scal_{K}(\Gbf,\Xbf)$ is defined as the pull back of the universal abelian scheme over $\Scal_{\Ktilde}(\GSp(\Lambda),S^{\pm})$, the same holds for the Hodge line bundle of $\Acal$. Hence the commutativity of \eqref{EqCommutativeZip} shows that
\begin{equation}\label{EqDFlatInverse}
(\beta_G \circ \zeta_G)^*(\omega^{\dflat}_G) = \omega_{\Scal}\rstr{S_K(\Gbf,\Xbf)},
\end{equation}
i.e., the pull back of $\omega^{\dflat}_G$ is the Hodge line bundle on $S_K(\Gbf,\Xbf)$. We define a line bundle on $\GZip^{\chi}$ by
\[
\omega^{\flat}_G := \beta_G^*(\omega^{\dflat}_G).
\]
Then $\zeta_G^*(\omega^{\flat}_G)$ is the Hodge line bundle on $S_K(\Gbf,\Xbf)$ by \eqref{EqDFlatInverse}.
By Proposition~\ref{GlobalSec}, we have
\begin{equation}\label{EqDimLessOne}
\dim\Gamma(\GZip^{\chi},(\omega^{\flat}_G)^{\otimes r}) \leq 1
\end{equation}
for all $r \in \ZZ$.

\begin{definition}\label{DefHasseNumber}
Let $S_0$ be the reduced stabilizer of $1$ (see Subsection~2.5). The exponent $N := N_{\Gbf,\Xbf}$ of its character group $X^*(S_0)$ is called the \emph{Hasse number of $(\Gbf,\Xbf)$}.
\end{definition}


\begin{theorem}\label{ExistHasse}
Let $N = N_{\Gbf,\Xbf}$ be the Hasse number. Then for every integer $d \geq 1$ we have
\[
\dim \Gamma(\GZip^{\chi},(\omega^{\flat}_G)^{\otimes {Nd}}) = 1.
\]
For every non-zero section $H \in \Gamma(\GZip^{\chi},(\omega^{\flat}_G)^{\otimes {Nd}})$ the non-vanishing locus of $\zeta_G^*(H) \in \Gamma(S_K(\Gbf,\Xbf),(\omega_{\Scal})^{\otimes ND})$ is the $\mu$-ordinary locus $S^{\mu}$ of $S_K(\Gbf,\Xbf)$.
\end{theorem}

We call any section of the form $\zeta_G^*(H)$ as above a \emph{Hasse invariant} (\emph{of parallel weight $Nd$}).

\begin{proof}
By~\eqref{EqDimLessOne} it suffices to consider the case $d = 1$. Let $\Ptilde = P_+(\iota\circ\chi)$ be the parabolic of $\Gtilde := \GSp(\bar\Lambda)$ defined by $\iota \circ \chi$. The pullback of $\det(\Ccal)$ to $\Gtilde/\Ptilde$ is anti-ample. It is defined by a character $\tilde\lambda \in X^*(\Ptilde)$ by Corollary~\ref{ExactGroupQuot} because $\Pic(\Gtilde) = 1$ (Corollary~\ref{PicRedTrivial}). Hence its restriction $\Rcal$ to $G/P$ is anti-ample and defined by a $G$-anti-ample character $\lambda \in X^*(P)$, the restriction of $\tilde\lambda$.
%As $B^- \subseteq P$, $\det(\Ccal)$ is defined by a character $\lambda$ such that $\langle \lambda, \beta\vdual \rangle$ for all simple coroots $\beta\vdual$ corresponding to elements of $I \setminus J$ (Proposition~3.3). Here $J$ is the type of $P$. Equivalently, $\lambda$ is a linear combination with positive coefficients of the fundamental weights of the maximal parabolics containing $P$.
The parabolic $P$ is defined by a minuscule cocharacter and in particular by a small cocharacter. Hence we can apply Theorem~\ref{PositivityThm} which shows that there exists a section $H$ of $(\omega^{\flat}_G)^{\otimes N}$ such that $\div(H)$ is an element of $Z^1(G)^E$ with coefficients $> 0$ (Remark~\ref{InterpretPositivity}). In particular, its support is the complement of the generic point in $\GZip^{\chi}$. As $\zeta_G$ is flat (even smooth) and dominant, the vanishing locus of $\zeta_G^*(H)$ is the complement of the $\mu$-ordinary stratum in $S_K(\Gbf,\Xbf)$.
\end{proof}


\begin{corollary}\label{MuOrdAffine}
Assume that $\Scal_K(\Gbf,\Xbf)$ is projective, then the $\mu$-ordinary stratum $S^{\mu}$ is affine.
\end{corollary}

\begin{proof}
This follows from Theorem~\ref{ExistHasse} because $\omega_{\Scal}$ is ample by Proposition~\ref{HodgeAmple}.
\end{proof}

\begin{remark}\label{ShimuraCompact}
Madapusi Pera has shown that $\Scal_K(\Gbf,\Xbf)$ is projective if and only if $\Gbf^{\rm ad}$ is an anisotropic group over $\QQ$ (\cite{Pera_CompHodge}~4.4.7).
\end{remark}

More generally, assume that $\Gbf^{\rm ad}$ has no factor isomorphic to $\PGL_{2,\QQ}$. Then every Hasse invariant $H$ of weight $k$ extends uniquely to a section in $\Gamma(\Scal^{\min} \otimes \kappa,(\omega_{\Scal}^{\min})^{\otimes k})$. We call its non-vanishing locus the \emph{$\mu$-ordinary locus of $\Scal^{\min}$} and denote it by $\Scal^{\min,\mu}$.

\begin{lemma}\label{MuOrdWellDefined}
The $\mu$-ordinary locus $\Scal^{\min,\mu}$ does not depend on the choice of the Hasse invariant.
\end{lemma}

\begin{proof}
Let $H$ and $H'$ are Hasse invariants of weight $k$ and $k'$, respectively. We may assume that $k' = dk$ for some $d \geq 1$. Then $H^{\otimes d}$ and $H'$ differ only by a non-zero scalar.
\end{proof}

As $\omega_{\Scal}^{\min}$ is ample and $\Scal^{\min}$ is projective, we deduce again:

\begin{corollary}\label{MuOrdMinAffine}
The $\mu$-ordinary locus $\Scal^{\min,\mu}$ of the minimal compactification is affine.
\end{corollary}


%---------------------------------------------------------------------

%\subsection{Non-emptiness of the ordinary locus}
%
%\begin{theorem}\label{NonEmptyOrdinary}
%The following assertions are equivalent.
%\begin{equivlist}
%\item
%$E_v = \QQ_p$.
%\item
%There exists a geometric point $\sbar$ of $S_K(\Gbf,\Xbf)$ such that the fiber of the universal abelian scheme in $\sbar$ is an ordinary abelian variety.
%\item
%For every geometric point $\sbar$ of the $\mu$-ordinary Newton stratum the fiber of the universal abelian scheme in $\sbar$ is an ordinary abelian variety.
%\end{equivlist}
%\end{theorem}
%
%
%\begin{lemma}\label{OrdLemma}
%Let $\Lambda$ be a finitely generated free abelian group. Let $\sigma \in \GL(\Lambda)$ such that there exists $r \geq 1$ with $\sigma^r = \id_{\Lambda}$. Let $v \in \Lambda$ such that $v = (1/r)\sum_{i=0}^{r-1}\sigma(v)$. Then $\sigma(v) = v$.
%\end{lemma}
%
%\begin{proof}
%We extend scalars from $\ZZ$ to $\CC$ and decompe $V = \Lambda_{\CC}$ into its direct sum $V = \bigoplus_{\zeta}V_{\zeta}$ of eigen spaces for $\sigma$. Here $\zeta$ runs through $e^{2\pi i/s}$ for $s$ a divisor of $r$ and $V_{\zeta}$ is the eigen spaces for $\sigma$ corresponding to $\zeta$ (of course $V_{\zeta} = 0$ for some $\zeta$ is possible). We decompose $v = \sum_{\zeta}v_{\zeta}$ with $v_{\zeta}$.
%
%Let $v_{\zeta} \ne 0$. We have $v_{\zeta} = (1/r)\sum_{i=0}^{r-1}\zeta^r(v_{\zeta})$ and hence $1 = (1/r)(1 + \zeta + \dots + \zeta^{r-1})$ which is only possible for $\zeta = 1$.
%\end{proof}

%---------------------------------------------------------------------


%
%
%Consider the decomposition of $G$ into $E$-orbits
%\[
%G=\bigsqcup_{\omega\in^{I}W}G^{\omega}
%\]
%We denote by $\eta\in^{I}W$ the element of maximal length. We define
%the colength of an element $\omega\in^{I}W$ to be $h(\omega)=\ell(\eta)-\ell(\omega)$.
%The length of an element in $^{I}W$ and the dimension of its corresponding
%$E$-orbit are related by the formula
%\[
%\mbox{dim}(G^{\omega})=\ell(\omega)+\mbox{dim}(P)
%\]
%Furthermore, the length $\ell(\omega)$ is equal to the dimension
%of the corresponding Ekedahl-Oort strata $\overline{M}^{\omega}$.
%The $E$-orbits of codimension one give $E$-stable Weyl divisors
%$D^{\omega}\colon =\left[\ \overline{G^{\omega}}\ \right]$. We also define 
%\[
%D_{H}=\sum_{h(\omega)=1}D^{\omega}
%\]
%We denote by $L^{\omega}$ and $L_{H}$ respectively the image of
%$D^{\omega}$ and $D_{H}$ by the injective map
%\[
%\left(A^{1}(G)\right)^{E}\to\Pic^{E}(G)\cong\Pic\left[E\backslash G\right]\cong\Pic(\mbox{G-Zip}^{\mu})
%\]
%By pulling back, we get line bundles $\mathscr{L}^{\omega}$ and $\mathscr{L}_{H}$
%on the Shimura variety $\overline{M}_{K}$.
%
%\begin{thm}
%Assume one of the conditions in theorem \ref{equiv} is satisfied.
%Then the line bundle $\mathscr{L}_{H}$ is ample.
%\end{thm}
%
%%---------------------------------------------------------------------

%\subsection{Applications}
%
%Let $T \subseteq S_K(\Gbf,\Xbf)$ be the complement of the $\mu$-ordinary locus considered as a closed reduced subscheme of $S_K(\Gbf,\Xbf)$.
%
%\begin{theorem}\label{ClassNonMuOrd}
%Assume that $E_v = \QQ_p$. Then $T = (p-1)c_1(\omega_{\Scal})$ in the Chow ring.
%\end{theorem}

%=====================================================================

\begin{thebibliography}{DDPW}
\setlength{\itemsep}{\itemsepamount}
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\bibitem[CTS]{CTS} J.-L. Colliot-Th\'el\`ene, J.-J.~Sansuc, {\em La $R$-\'equivalence sur les tores}, Ann.\ Sci.\ de l'ENS {\bf 10} (1977), 175--229.
\bibitem[De]{De_VarSh} P.~Deligne, {\em Vari\'et\'es de Shimura: Interpr\'etation modulaire, et techniques de construction de mod\`eles canoniques}, in A. Borel, W. Casselman (ed.): {\em Automorphic forms, representations, and $L$-functions}\/ 2, Proc. Symp. Pure Math. {\bf 33} (1977), 247--289.
\bibitem[DM]{DM} F.~Digne, J.~Michel, {\em Representations of finite groups of Lie type}, London Mathematical Society Student Texts {\bf 21}, Cambridge University Press (1991).
\bibitem[FoIv]{FoIv} R.~Fossum, B.~Iversen, {\em On Picard groups of algebraic fibre spaces}, Journal of Pure and Applied Algebra {\bf3} (1973), 269--280.
\bibitem[GoNi1]{GoNi_Hasse1} W.~Goldring, M.-H. Nicole, {\em A $\mu$-ordinary Hasse invariant}, preprint 2013, arXiv:1302.1614v2.
\bibitem[GoNi2]{GoNi_Hasse2} W.~Goldring, M.-H. Nicole, {\em The $\mu$-ordinary Hasse invariant of unitary Shimura varieties}, preprint 2013, arXiv:1305.6956.
\bibitem[Gor]{Goren_Hasse} E.~Goren, {\em Hasse invariants for Hilbert modular varieties}, Israel J.~Math. {\bf 122} (2001), 157--174.
\bibitem[Jan]{Jantzen} J.~C.~Janzten, {\em Representations of Algebraic Groups}, Academic Press (1987).
\bibitem[Ki1]{Ki_Integral} M.~Kisin, {\em Integral models for Shimura varieties of abelian type}, JAMS {\bf23} (2010), 967--1012.
\bibitem[Ki2]{Ki_Points} M.~Kisin, {\em Mod $p$ points on Shimura varieties of abelian type}, preprint 2013.
\bibitem[KKV]{KKV} F.~Knop, H.~Kraft, T.~Vust, {\em The Picard group of a $G$-variety}, in {\em Algebraic Transformation Groups and Invariant Theory}, DMV-Seminar {\bf13}, Birkh\"auser (1989).
\bibitem[Lan]{Lan_Comp} K.-W.~Lan, {\em Arithmetic compactification of PEL-type Shimura varieties}, Londom Math. Soc. Monographs {\bf36}, Princeton University Press (2013).
\bibitem[LMS]{LMS_GB} Lakshmibai, C.~Musili, C.S.~Seshadri, {\em Cohomology of line bundles on $G/B$}, Ann. Sci. ENS {\bf7} (1974), 89--137.
\bibitem[MB]{MB} L.~Moret-Bailly, {\em Pinceaux de vari\'et'es ab\'eliennes}, Ast\'erisque {\bf129}, SMF (1985).
\bibitem[MoWd]{MoWd} B.~Moonen, T.~Wedhorn, {\em Discrete invariants of varieties in positive characteristic}, IMRN {\bf2004:72} (2004), 3855--3903.
\bibitem[Per]{Pera_CompHodge} K.~Madapusi Pera, {\em Toroidal Compactifications of integral models of Shimura varieties of Hodge type}, preprint 2012, arXiv:1211.1731v4.
\bibitem[PWZ1]{PWZ1} R.~Pink, T.~Wedhorn, P.~Ziegler, {\em Algebraic zip data}, Doc.~Math.~{\bf16} (2011), 253--300.
\bibitem[PWZ2]{PWZ2} R.~Pink, T.~Wedhorn, P.~Ziegler, {\em F-zips with additional structure}, preprint 2012, to appear in Pacific Journal of Math., arXiv: 1208.3547.
\bibitem[San]{Sansuc} J.-J. Sansuc, {\em Groupe de Brauer et arithm\'etique des groupes
alg\'ebriques lin\'eaires sur un corps de nombres}, Journal f\"ur die reine und angewandte Mathematik {\bf 327} (1981), 12--80.
\bibitem[SGA3]{SGA3} M.~Demazure et al., \emph{Sch\'emas en groupes, I, II, III\/}, LNM {\bf 151}, {\bf 152}, {\bf 153}, Springer (1970).
%\bibitem[Spr]{Spr} T.~Springer, {\em Linear algebraic groups}, Reprint of the 1998 second edition, Modern Birkh\"auser Classics (2009).
\bibitem[St]{Steinberg_Endo} R.~Steinberg, \emph{Endomorphisms of linear algebraic groups}, Memoirs of the American Mathematical Society {\bf80}, AMS (1968).
\bibitem[Va]{Vasiu_Integral} A.~Vasiu, {\em Integral canonical models of Shimura varieties of preabelian type}, Asian J.~Math. {\bf3} (1999), 401--518.
\bibitem[ViWd]{ViWd} E.~Viehmann, T.~Wedhorn, \emph{Ekedahl-Oort and Newton strata for Shimura varieties of PEL type}, Math.\ Ann.\ {\bf356} (4) (2013), 1493--1550.
\bibitem[Wd1]{Wd_Ord} T.~Wedhorn, {\it Ordinariness in good reductions of Shimura varieties of PEL-type}, Ann.\ scient.\ \'Ec.\ Norm.\ Sup.\ (4), {\bf 32}, (1999), 575--618.
\bibitem[Wd2]{Wd_Bruhat} T.~Wedhorn, \emph{Bruhat strata and $F$-zips with additional structures}, preprint 2013, 31 pages, arXiv:1302.6715, to appear in M\"unster Journal of Mathematics.
\bibitem[WdYa]{WdYa} T.~Wedhorn, Y.~Yaroslav, {\em Purity of zip strata with applications to Ekedahl-Oort strata of Shimura varieties of Hodge type}, preprint 2014, arXiv:1404.0577.
\bibitem[Wor]{Wort_MuOrd} D.~Wortmann, {\em The $\mu$-ordinary locus for Shimura varieties of Hodge type}, preprint 2013, arXiv:1310.6444.
\bibitem[Zha1]{Zhang_EO} C.~Zhang, {\em Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type}, preprint 2013, arXiv:1312.4869.
\bibitem[Zha2]{Zhang_Func} C.~Zhang, {\em Remarks on Ekedahl-Oort stratifications}, preprint 2014, arXiv:1401.6632.
%\bibitem[Tev]{Tev} Tevelev, Projective duality and homogeneous spaces
\end{thebibliography}


\end{document}

1406.2178.extracted_bib
[BrKu]{Brion M.~Brion, S.~Kumar, {\em Frobenius splitting methods in geometry and representation theory, Progress in Mathematics {231, Birkh\"auser (2005).
1406.2178.extracted_bib
[CTS]{CTS J.-L. Colliot-Th\'el\`ene, J.-J.~Sansuc, {\em La $R$-\'equivalence sur les tores, Ann.\ Sci.\ de l'ENS {10 (1977), 175--229.
1406.2178.extracted_bib
[De]{De_VarSh P.~Deligne, {\em Vari\'et\'es de Shimura: Interpr\'etation modulaire, et techniques de construction de mod\`eles canoniques, in A. Borel, W. Casselman (ed.): {\em Automorphic forms, representations, and $L$-functions\/ 2, Proc. Symp. Pure Math. {33 (1977), 247--289.
1406.2178.extracted_bib
[DM]{DM F.~Digne, J.~Michel, {\em Representations of finite groups of Lie type, London Mathematical Society Student Texts {21, Cambridge University Press (1991).
1406.2178.extracted_bib
[FoIv]{FoIv R.~Fossum, B.~Iversen, {\em On Picard groups of algebraic fibre spaces, Journal of Pure and Applied Algebra {3 (1973), 269--280.
1406.2178.extracted_bib
[GoNi1]{GoNi_Hasse1 W.~Goldring, M.-H. Nicole, {\em A $\mu$-ordinary Hasse invariant, preprint 2013, arXiv:1302.1614v2.
1406.2178.extracted_bib
[GoNi2]{GoNi_Hasse2 W.~Goldring, M.-H. Nicole, {\em The $\mu$-ordinary Hasse invariant of unitary Shimura varieties, preprint 2013, arXiv:1305.6956.
1406.2178.extracted_bib
[Gor]{Goren_Hasse E.~Goren, {\em Hasse invariants for Hilbert modular varieties, Israel J.~Math. {122 (2001), 157--174.
1406.2178.extracted_bib
[Jan]{Jantzen J.~C.~Janzten, {\em Representations of Algebraic Groups, Academic Press (1987).
1406.2178.extracted_bib
[Ki1]{Ki_Integral M.~Kisin, {\em Integral models for Shimura varieties of abelian type, JAMS {23 (2010), 967--1012.
1406.2178.extracted_bib
[Ki2]{Ki_Points M.~Kisin, {\em Mod $p$ points on Shimura varieties of abelian type, preprint 2013.
1406.2178.extracted_bib
[KKV]{KKV F.~Knop, H.~Kraft, T.~Vust, {\em The Picard group of a $G$-variety, in {\em Algebraic Transformation Groups and Invariant Theory, DMV-Seminar {13, Birkh\"auser (1989).
1406.2178.extracted_bib
[Lan]{Lan_Comp K.-W.~Lan, {\em Arithmetic compactification of PEL-type Shimura varieties, Londom Math. Soc. Monographs {36, Princeton University Press (2013).
1406.2178.extracted_bib
[LMS]{LMS_GB Lakshmibai, C.~Musili, C.S.~Seshadri, {\em Cohomology of line bundles on $G/B$, Ann. Sci. ENS {7 (1974), 89--137.
1406.2178.extracted_bib
[MB]{MB L.~Moret-Bailly, {\em Pinceaux de vari\'et'es ab\'eliennes, Ast\'erisque {129, SMF (1985).
1406.2178.extracted_bib
[MoWd]{MoWd B.~Moonen, T.~Wedhorn, {\em Discrete invariants of varieties in positive characteristic, IMRN {2004:72 (2004), 3855--3903.
1406.2178.extracted_bib
[Per]{Pera_CompHodge K.~Madapusi Pera, {\em Toroidal Compactifications of integral models of Shimura varieties of Hodge type, preprint 2012, arXiv:1211.1731v4.
1406.2178.extracted_bib
[PWZ1]{PWZ1 R.~Pink, T.~Wedhorn, P.~Ziegler, {\em Algebraic zip data, Doc.~Math.~{16 (2011), 253--300.
1406.2178.extracted_bib
[PWZ2]{PWZ2 R.~Pink, T.~Wedhorn, P.~Ziegler, {\em F-zips with additional structure, preprint 2012, to appear in Pacific Journal of Math., arXiv: 1208.3547.
1406.2178.extracted_bib
[San]{Sansuc J.-J. Sansuc, {\em Groupe de Brauer et arithm\'etique des groupes alg\'ebriques lin\'eaires sur un corps de nombres, Journal f\"ur die reine und angewandte Mathematik {327 (1981), 12--80.
1406.2178.extracted_bib
[SGA3]{SGA3 M.~Demazure et al., Sch\'emas en groupes, I, II, III\/, LNM {151, {152, {153, Springer (1970). %
1406.2178.extracted_bib
[Spr]{Spr T.~Springer, {\em Linear algebraic groups, Reprint of the 1998 second edition, Modern Birkh\"auser Classics (2009).
1406.2178.extracted_bib
[St]{Steinberg_Endo R.~Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society {80, AMS (1968).
1406.2178.extracted_bib
[Va]{Vasiu_Integral A.~Vasiu, {\em Integral canonical models of Shimura varieties of preabelian type, Asian J.~Math. {3 (1999), 401--518.
1406.2178.extracted_bib
[ViWd]{ViWd E.~Viehmann, T.~Wedhorn, Ekedahl-Oort and Newton strata for Shimura varieties of PEL type, Math.\ Ann.\ {356 (4) (2013), 1493--1550.
1406.2178.extracted_bib
[Wd1]{Wd_Ord T.~Wedhorn, {Ordinariness in good reductions of Shimura varieties of PEL-type, Ann.\ scient.\ \'Ec.\ Norm.\ Sup.\ (4), {32, (1999), 575--618.
1406.2178.extracted_bib
[Wd2]{Wd_Bruhat T.~Wedhorn, Bruhat strata and $F$-zips with additional structures, preprint 2013, 31 pages, arXiv:1302.6715, to appear in M\"unster Journal of Mathematics.
1406.2178.extracted_bib
[WdYa]{WdYa T.~Wedhorn, Y.~Yaroslav, {\em Purity of zip strata with applications to Ekedahl-Oort strata of Shimura varieties of Hodge type, preprint 2014, arXiv:1404.0577.
1406.2178.extracted_bib
[Wor]{Wort_MuOrd D.~Wortmann, {\em The $\mu$-ordinary locus for Shimura varieties of Hodge type, preprint 2013, arXiv:1310.6444.
1406.2178.extracted_bib
[Zha1]{Zhang_EO C.~Zhang, {\em Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type, preprint 2013, arXiv:1312.4869.
1406.2178.extracted_bib
[Zha2]{Zhang_Func C.~Zhang, {\em Remarks on Ekedahl-Oort stratifications, preprint 2014, arXiv:1401.6632. %
1406.2178.extracted_bib
[Tev]{Tev Tevelev, Projective duality and homogeneous spaces