Quasi-Elliptic Cohomology and its Power Operations
Zhen Huan
Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant %a space...good enough words? loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.
1612.00930.tex
%-----------------------------------------------------------------------
% Beginning of article-template.tex
%-----------------------------------------------------------------------
%
%    This is a template file for proceedings articles prepared with AMS
%    author packages, for use with AMS-LaTeX.
%
%    Templates for various common text, math and figure elements are
%    given following the \end{document} line.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%    Remove any commented or uncommented macros you do not use.

%    Replace amsproc by the name of the author package.
\documentclass{amsproc}

%    If you need symbols beyond the basic set, uncomment this command.
\usepackage{amssymb}

%    If your article includes graphics, uncomment this command.
\usepackage{graphicx}

%    If the article includes commutative diagrams, ...
\usepackage[cmtip,all]{xy}

%    Include other referenced packages here.
%\usepackage{}

\usepackage[all]{xy}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
%\usepackage{mathtools}
%\usepackage{tikz-cd}
\usepackage{stmaryrd}
\usepackage{graphicx}%
\usepackage{hyperref}
\hypersetup{
    bookmarks=true,         % show bookmarks bar?
    unicode=false,          % non-Latin characters in Acrobat�s bookmarks
    pdftoolbar=true,        % show Acrobat�s toolbar?
    pdfmenubar=true,        % show Acrobat�s menu?
    pdffitwindow=false,     % window fit to page when opened
    pdfstartview={FitH},    % fits the width of the page to the window
    pdftitle={My title},    % title
    pdfauthor={Author},     % author
    pdfsubject={Subject},   % subject of the document
    pdfcreator={Creator},   % creator of the document
    pdfproducer={Producer}, % producer of the document
    pdfkeywords={keyword1, key2, key3}, % list of keywords
    pdfnewwindow=true,      % links in new PDF window
    colorlinks=false,       % false: boxed links; true: colored links
    linkcolor=red,          % color of internal links (change box color with linkbordercolor)
    citecolor=green,        % color of links to bibliography
    filecolor=magenta,      % color of file links
    urlcolor=cyan           % color of external links
}


\usepackage{amscd,keyval}

\makeatletter
\define@key{modCD}{cols}{\setlength{\minCDarrowwidth}{#1}}
\define@key{modCD}{rows}{\setlength{\modCD@rowsep}{#1}}
\newlength{\modCD@rowsep}

\renewenvironment{CD}[1][]
 {\modCD@rowsep=20\ex@ % default
  \setkeys{modCD}{#1}%
  \CDat
  \bgroup\relax\iffalse{\fi\let\ampersand@&\iffalse}\fi
  \CD@true\vcenter\bgroup\let\\\math@cr\restore@math@cr\default@tag
  \tabskip\z@skip\baselineskip20\ex@
  \lineskip3\ex@\lineskiplimit3\ex@\halign\bgroup
  &\hfill$\m@th##$\hfill\crcr}
 {\crcr\egroup\egroup\egroup}
\atdef@ V#1V#2V{\CD@check{V..V..V}{%
  \llap{$\m@th\vcenter{\hbox{$\scriptstyle#1$}}$}%
  \left\downarrow\vbox to.5\modCD@rowsep{}\right.\kern-\nulldelimiterspace
  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
\atdef@ A#1A#2A{\CD@check{A..A..A}{%
  \llap{$\m@th\vcenter{\hbox{$\scriptstyle#1$}}$}%
  \left\uparrow\vbox to.5\modCD@rowsep{}\right.\kern-\nulldelimiterspace
  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
\makeatother

%    Update the information and uncomment if AMS is not the copyright
%    holder.
%\copyrightinfo{2009}{American Mathematical Society}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\begin{document}

% \title[short text for running head]{full title}
\title{Quasi-Elliptic Cohomology and its Power Operations}

%    Only \author and \address are required; other information is
%    optional.  Remove any unused author tags.

%    author one information
% \author[short version for running head]{name for top of paper}


\author{Zhen Huan}

\address{Zhen Huan, Department of Mathematics,
University of Illinois at Champaign-Urbana,  1409 W. Green Street,
Urbana, IL 61801 } \curraddr{} \email{huan2@illinois.edu}
\thanks{The author was partially supported by NSF grant DMS-1406121.}

%    author two information
%\author{}
%\address{}
%\curraddr{} \email{}
%\thanks{}

%\subjclass[2000]{Primary }
%    The 2010 edition of the Mathematics Subject Classification is
%    now available.  If you are citing a classification from the
%    new scheme, use the following input coding instead.
\subjclass[2010]{Primary 55}

\date{September 20, 2016}

\begin{abstract}
Quasi-elliptic cohomology is a variant of
Tate K-theory. It is the orbifold K-theory of a space of constant %a space...good enough words?
loops. For global quotient orbifolds, it can be expressed in terms
of equivariant K-theories. In this paper we show how this theory
is equipped with power operations. We also prove that the Tate
K-theory of symmetric groups modulo a certain transfer ideal
classify the finite subgroups of the Tate curve.
\end{abstract}

\maketitle


\section{Introduction}


An elliptic cohomology theory is a generalized cohomology theory,
which is complex orientable, and whose associated formal group is
the formal completion of an elliptic curve. It is an old idea of
Witten, as shown in \cite{LProceeding}, that the elliptic
cohomology of  a space $X$ is related to the
$\mathbb{T}-$equivariant K-theory of the free loop space
$LX=\mathbb{C}^{\infty}(S^1, X)$ with the circle $\mathbb{T}$
acting on $LX$ by rotating loops.

It is surprisingly difficult to make this precise, especially if
one wishes to consider equivariant generalization of this
construction. In this case the loop space $LX$ is considered with
the natural rotation action and is a rich orbifold. In this paper
we show the relation between the loop space and Tate K-theory, via
a new theory, which we call quasi-elliptic cohomology.

Tate K-theory is the generalized elliptic cohomology associated to
the Tate curve. The Tate curve $Tate(q)$ is an  elliptic curve
over $Spec\mathbb{Z}((q))$, which is classified as the completion
of the algebraic stack of some nice generalized elliptic curves at
infinity. A good reference for $Tate(q)$ is Section 2.6 of
\cite{AHS}. We give a sketch of it in Section \ref{tatecurve}. The
relation between Tate K-theory and string theory is better
understood than most known elliptic cohomology theories. The
definition of $G-$equivariant Tate K-theory for finite groups $G$
is modelled on the loop space of a global quotient orbifold, which
is formulated explicitly in \cite{Gan07}.

The idea of quasi-elliptic cohomology is  motivated by Ganter's
construction of Tate K-theory. Instead of a theory over
$\mathbb{Z}((q))$, we consider a theory over
$\mathbb{Z}[q^{\pm}]$. It is not an elliptic cohomology  but from
it we can recover the Tate K-theory. This new theory can be
interpreted in a neat form by equivariant K-theories, which makes
many constructions and computations easier and more natural than
those on the Tate K-theories. Some formulations can be generalized
to other equivariant cohomology theories.


\subsection{Loop Space}

Quasi-elliptic cohomology is  modelled on a version of equivariant
loop space. For background on orbifolds and lie groupoids, we
refer the readers to \cite{LerStack} and \cite{Moe02}.

For any  compact Lie group $G$ and a
manifold $X$ with a smooth $G-$action, we can define the  Lie
groupoid $X/\!\!/G$ whose objects are the points $x\in X$, and
whose arrows $\alpha: x\longrightarrow y$ are those $\alpha\in G$
for which $\alpha\cdot x = y$.


The space of smooth unbased loops in the orbifold $X/\!\!/G$ in
principle carries a lot of structure: on the one hand, it includes
loops represented by smooth maps $\gamma:
\mathbb{R}\longrightarrow X$ such that $\gamma(t+1)=\gamma(t)g$
for some $g\in G$; other than the group action by the loop group
$LG:=\mathbb{C}^{\infty}(S^1, G)$, the loop space also has the
circle action by rotation. As indicated in \cite{LerStack}, maps
in the 2-category of orbifolds are bibundles.  Thus, the free loop
space of an orbifold $M$ is
$$Loop(M)=Orb(S^1/\!\!/\ast, M)=Bibun(S^1/\!\!/\ast, M),$$ the category of bibundles from the trivial groupoid
$S^1/\!\!/\ast$ to the Lie groupoid $M$. We will write
$$Loop_1(X/\!\!/G):=Bibun(S^1/\!\!/\ast, X/\!\!/G),$$ which
is discussed in Definition \ref{loopspacemorphism}.



Other than the action by the loop group $LG$, we also want to
consider the action on the loop space induced by rotation of the
circle. In Definition \ref{loopext3space}, we extend the groupoid
$Loop_1(X/\!\!/G)$ to a groupoid $Loop_1^{ext}(X/\!\!/G)$ by
adding the rotations. Especially we are interested in the ghost
loops $GhLoop(X/\!\!/G)$, which is defined to be the full
subgroupoid of $Loop^{ext}_1(X/\!\!/G)$ consisting of objects
$(\pi, f)$ with the image of $f$ contained in a single $G-$orbit.

This groupoid has several good properties. They are computed
locally in $X$. For instance, if $X=U\cup V$ where $U$ and $V$ are
$G-$invariant open subsets, then $$GhLoop(X/\!\!/G)\cong
GhLoop(U/\!\!/G)\cup_{GhLoop((U\cap V)/\!\!/G)}
GhLoop(V/\!\!/G).$$ In addition, if $H$ is a closed subgroup of
$G$ and $X$ is the quotient space $G/H$, $GhLoop(X/\!\!/G)$ is
equivalent to $GhLoop(\mbox{pt}/\!\!/H)$.

When $G$ is finite, $GhLoop(X/\!\!/G)$ is isomorphic to the full
subgroupoid $\Lambda(X/\!\!/G)$ of $Loop^{ext}_1(X/\!\!/G)$
consisting of constant loops. This groupoid $\Lambda(X/\!\!/G)$
can be regarded as an extended version of the inertia groupoid
$I(X/\!\!/G)$. Please see Definition \ref{inertiagroupoid} for
inertia groupoid.



In general, for any orbifold groupoid $M$, we can formulate the
loop space $Loop^{ext}_1(M)$, the space of constant loops
$GhLoop(M)$, and $\Lambda(M)$  in a similar way via bibundles.

%The groupoid $\Lambda(X/\!\!/G)$ is a full subgroupoid of $GhLoop(X/\!\!/G)$.


\subsection{Quasi-elliptic cohomology}

Given a compact group $G$ and a compact $G-$space $X$, recall the
equivariant K-theory $K_G(X)$ is the Grothendieck ring of the
isomorphic classes of $G-$vector bundles over $X$. In general, for
any  compact orbifold groupoid $\mathbb{G}$, the orbifold K-theory
$K_{orb}(\mathbb{G})$ is defined to be the Grothendieck ring of
isomorphism classes of $\mathbb{G}-$vector bundles on
$\mathbb{G}$. In particular, $K_{orb}(X/\!\!/G)$ is $K_G(X)$. A
reference for orbifold K-theory is \cite{ALRuan}.


Generally, quasi-elliptic cohomology can be defined for any
orbifold groupoid $M$ by considering the constant loops
$GhLoop(M)$ in $Bibun(S^1/\!\!/\ast, M)$. Quasi-elliptic
cohomology $QEll^*(M)$ is defined to be the orbifold K-theory of
the groupoid $\Lambda(M)$. This construction is equivalent to
Ganter's construction in \cite{Gan13}. In the global quotient case
$M=X/\!\!/G$ with $G$ a finite group, $QEll^*_{G}(X)$ can be
expressed in term of the equivariant K-theory of $X$ and its
subspaces as
\begin{equation}QEll^*_G(X):=K_{orb}(GhLoop(X/\!\!/G))\cong \prod_{\sigma\in
G_{conj}}K^*_{\Lambda_G(\sigma)}(X^{\sigma})=\bigg(\prod_{\sigma\in
G}K^*_{\Lambda_G(\sigma)}(X^{\sigma})\bigg)^G,\label{defintro}\end{equation}
where $G_{conj}$ is a set of representatives of $G-$conjugacy
classes in $G$. The group $\Lambda_G(\sigma):=
C_G(\sigma)\times\mathbb{R}/\langle(\sigma, -1)\rangle$ acts on
the fixed point space $X^{\sigma}$ by $[g, t]\cdot m=g\cdot m$.
%which can be constructed from equivariant K-theories.



For any orbifold groupoid  $M$,      $QEll_G(M)$ has the structure
of a $\mathbb{Z}[q^{\pm}]-$algebra. We have
\begin{equation}QEll^*(M)\otimes_{\mathbb{Z}[q^{\pm}]}\mathbb{Z}((q))=K^*_{Tate}(M).
\label{tateqellequiv}\end{equation} Thus, quasi-elliptic
cohomology  contains all the information we need to study the Tate
K-theory and the Tate K-theory can be constructed from it.


Quasi-elliptic cohomology can be set up in a way similar to the
equivariant K-theories. We formulate the K$\ddot{u}$nneth map,
restriction map, change of group isomorphism and transfer.

In general, if $H^*$ is an equivariant cohomology theory, then the
functor $$X/\!\!/G \mapsto H^*(GhLoop(X/\!\!/G))$$ gives a new
equivariant cohomology theory. Moreover, for each global
cohomology theory, i.e. equivariant cohomology theories with
change of group isomorphism, we can formulate a new global
cohomology theory via the ghost loops.



\subsection{Power operation}

One interesting point is that quasi-elliptic cohomology has power
operations, which was first observed by
Ganter, as shown in \cite{Gan07} and \cite{Gan13}. % correct?
In Section \ref{poweroperation} we construct the total power
operation of quasi-elliptic cohomology. It satisfies the axioms
that Ganter gave in Definition 4.3 in \cite{Gan06}. More
explicitly, as shown in Theorem \ref{main1p}, the family of maps
$$\mathbb{P}_n=\prod_{(\underline{g}, \sigma)\in
(G\wr\Sigma_n)^{tors}_{conj}}\mathbb{P}_{(\underline{g},\sigma)}:
QEll^*_G(X)\longrightarrow QEll^*_{G\wr\Sigma_n}(X^{\times n}),$$
satisfy

(i) $\mathbb{P}_1=$Id, $\mathbb{P}_0(x)=1$.


(ii) Let $x\in QEll^*_G(X)$, $(\underline{g}, \sigma)\in
G\wr\Sigma_n$ and $(\underline{h}, \tau)\in G\wr\Sigma_m$. The
external product of two power operations
$$\mathbb{P}_{(\underline{g}, \sigma)}(x)\boxtimes \mathbb{P}_{(\underline{h}, \tau)}(x)=res|^{\Lambda_{G\wr\Sigma_{m+n}}(\underline{g},\underline{h}; \sigma\tau)}
_{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}\Lambda_{G\wr\Sigma_m}(\underline{h},
\tau)} \mathbb{P}_{(\underline{g},\underline{h};
\sigma\tau)}(x).$$


(iii) The composition of two power operations is
$$\mathbb{P}_{((\underline{\underline{h}, \tau}); \sigma)}(\mathbb{P}_m(x))=res|^{\Lambda_{G\wr\Sigma_{mn}}(\underline{\underline{h}}, (\underline{\tau}, \sigma))}
_{\Lambda_{(G\wr\Sigma_m)\wr\Sigma_n}((\underline{\underline{h},
\tau}); \sigma)}\mathbb{P}_{(\underline{\underline{h}},
(\underline{\tau}, \sigma))}(x)$$ where
$(\underline{\underline{h}, \tau})\in (G\wr \Sigma_m)^{\times n}$,
and $\sigma\in\Sigma_n$. $(\underline{\tau}, \sigma)$ is in
$\Sigma_m\wr\Sigma_n$, thus, can be viewed as an element in
$\Sigma_{mn}$.


(iv) $\mathbb{P}$ preserves external product. For $\underline{(g,
h)}=((g_1, h_1), \cdots (g_n, h_n))\in (G\times H)^{\times n}$,
$\sigma\in \Sigma_n$,
$$\mathbb{P}_{(\underline{(g, h)},\sigma)}(x\boxtimes y)=res|^{\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)\times_{\mathbb{T}}\Lambda_{H\wr\Sigma_n}(\underline{h}, \sigma)}
_{\Lambda_{(G\times H)\wr\Sigma_n}(\underline{(g, h)},
\sigma)}\mathbb{P}_{(\underline{g}, \sigma)}(x)\boxtimes
\mathbb{P}_{(\underline{h}, \sigma)}(y).$$ \label{main1p}



The power operation $\{\mathbb{P}_n\}_{n\geq 0}$ mixes the power
operation in $K-$theory with the natural operations of dilating
and rotating loops. The key point of the construction of the power
operation is an intermediate groupoid $d_{(\underline{g},
\sigma)}(X)$ defined in Definition \ref{ADgroupoid}, for each
element $(\underline{g}, \sigma)$ in the wreath product
$G\wr\Sigma_n$. The groupoid $d_{(\underline{g}, \sigma)}(X)$ is
constructed from $\Lambda(X/\!\!/G)$ and is isomorphic to
$(X^{\times n})^{(\underline{g}, \sigma)}/\!\!/
\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$ via an isomorphism
$f_{(\underline{g},
\sigma)}$.  % in the definition of $QEll$
For more details of the construction, please see Section
\ref{s2complete}.


We illustrate what this power operation looks like by some simple
cases. Let $G$ be the trivial group and $X$ a space. Let $(-)_k$
denote the rescaling map defined in (\ref{lpok}).

When $n=2$, $\mathbb{P}_{(\underline{1}, (1)(1))}(x)=x\boxtimes x$
and $\mathbb{P}_{(\underline{1}, (12))}(x)=(x)_2$.


When $n=3$, $\mathbb{P}_{(\underline{1}, (1)(1)(1))}(x)=x\boxtimes
x\boxtimes x$, $\mathbb{P}_{(\underline{1},
(12)(1))}(x)=(x)_2\boxtimes x$, and $\mathbb{P}_{(\underline{1},
(123))}(x)=(x)_3$.



For more examples please see Example \ref{pointsymmpower}.

\bigskip

%For more details of the construction, please see Section \ref{s2complete}.

For any equivariant cohomology theory $\{H^*_G( -)\}_G$ with an
$H_{\infty}$-structure in Ganter's sense, we can formulate a power
operation for the equivariant cohomology theories
$$\mathbb{H}^*_G(-):=\prod_{\sigma \in G_{conj}}
H^*_{\Lambda_G(\sigma)}(-)^{\sigma}$$ in the same way.



In addition, we can formulate the total power operation for the
orbifold quasi-elliptic cohomology in the sense of Definition \ref{ng1}.  %the sense of Definition 3.8 in \cite{Gan13}.
The construction of the power operation is shown in Section
\ref{s2}.

Moreover, via the isomorphism in Theorem \ref{strickqell}, we can
define a ring homomorphism
\begin{align*}\overline{P}_N: &QEll_{G}(X) \buildrel{\mathbb{P}_N}\over\longrightarrow QEll_{G\wr\Sigma_N}(X^{\times N})
\buildrel{res}\over\longrightarrow QEll_{G\times
\Sigma_N}(X^{\times N})\\  &\buildrel{diag^*}\over\longrightarrow
 QEll_{G\times\Sigma_N}(X) \cong QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_{\Sigma_N}(\mbox{pt}) \\
  &\longrightarrow QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_{\Sigma_N}(\mbox{pt})/\mathcal{I}^{QEll}_{tr},\end{align*}
 as shown in Proposition
\ref{adamsqell}. %It is a new operation different from Adams operation.
Under the identification (\ref{tateqellequiv}), it extends
uniquely to the ring homomorphism
$$\overline{P^{string}}_N: K_{Tate}(X/\!\!/G)\longrightarrow
K_{Tate}(X/\!\!/G)\otimes_{\mathbb{Z}((q ))}
K_{Tate}(\mbox{pt}/\!\!/\Sigma_N)/I^{Tate}_{tr}$$ constructed in
Section 5.4, \cite{Gan07}.
%by tensoring the identity map  on $\mathbb{Z}((q))$,
%Taking the trace of $\overline{P^{string}}_n(x)$, it equals $nT_n(x)$ with $T_n$ the Hecke operator of Tate K-theory defined
%by the stringy power operation in \cite{Gan07}.


\subsection{Classification of the finite subgroups of the Tate curve}

Though the general formulas for the power operations in $QEll_G$
are complicated, to understand it, it is useful to consider
special cases. It is already illuminating to consider the case
that $X$ is a point and $G$ is the trivial group, the power
operation has a neat form, as shown in Example
\ref{pointsymmpower}. It has a natural interpretation in terms of
the Tate elliptic curve.



% With the families of power operations I formulate in Section \ref{s2complete},
In Section \ref{proofstrict} applying the power operation we prove
that the Tate K-theory of symmetric groups modulo the transfer
ideal classifies the finite subgroups of the Tate curve, which is
analogous to the principal result in Strickland \cite{Str98} that
the Morava $E-$theory of the symmetric group $\Sigma_n$ modulo a
certain transfer ideal classifies the power subgroups of rank $n$
of the formal group $\mathbb{G}_E$.
%With the equivariant power operations , I prove the parallel conclusion for quasi-elliptic cohomology.

The finite subgroups of the Tate curve are classified by
$$\prod_{d|N}\mathbb{Z}((q))[q']/\langle q^d-q'^{\frac{N}{d}}
\rangle.$$

First we prove the parallel conclusion for quasi-elliptic
cohomology. \begin{theorem} \label{strickqell}
\begin{equation}QEll^0(\mbox{pt}/\!\!/\Sigma_N)/\mathcal{I}^{QEll}_{tr}\cong
\prod_{d|N}\mathbb{Z}[q^{\pm}][q']/\langle q^d-q'^{\frac{N}{d}}
\rangle,\label{ellc}\end{equation} where $\mathcal{I}^{QEll}_{tr}$
is the transfer ideal defined in (\ref{transferidealqec}) and $q'$
is the image of $q$ under the power operation $\mathbb{P}_N$. % whose order matters.
%For example, $(1, 2)$ and $(2, 1)$ are considered as different pairs.
\end{theorem}

Then applying the relationship between $QEll^*$ and Tate K-theory
\begin{equation}QEll^*(X)\otimes_{\mathbb{Z}[q^{\pm}]}\mathbb{Z}((q))=K^*_{Tate}(X)\mbox{
for any topological groupoid  }X,
\label{tateqellequiv}\end{equation} we obtain the main theorem.
\begin{theorem}%\label{stricktheorem}
The Tate K-theory of symmetric groups modulo the transfer ideal
$I^{Tate}_{tr}$  defined in (\ref{transferidealtatek}) classifies
finite subgroups of the Tate curve. Explicitly,
\begin{equation}K^0_{Tate}(\mbox{pt}/\!\!/\Sigma_N)/I^{Tate}_{tr}\cong
\prod_{d|N}\mathbb{Z}((q))[q']/\langle q^d-q'^{\frac{N}{d}}
\rangle,\label{tatec}\end{equation} where $q'$ is the image of $q$
under the power operation $P^{Tate}$ constructed
in \cite{Gan13}. %The product goes over all the positive integers dividing $N$. % whose order matters.



\end{theorem}



\subsection{Acknowledgement}
I would like to thank Charles Rezk who is always a wonderful
advisor and very inspiring teacher. Most of this work was guided
by him and it is a great experience to work with him. I would also
like to thank Matthew Ando. We had many mathematical discussions,
which are also important for my work.



\section{Models for orbifold loops and ghost loops} \label{introduction}
Quasi-elliptic cohomology is modelled on the orbifold K-theory of
the constant loops in the free loop space. %in Definition \ref{ddd1}.
So to understand $QEll^*_G(X)$, it is essential to understand the
orbifold loop space.

Let $G$ be a finite group and $X$ be a $G-$space. Let $X/\!\!/G$
denote the translation groupoid.  In this section, we will
describe several models for the loop space of $X/\!\!/G$. In
Definition \ref{loopspacemorphism} we discuss the groupoid
$Loop_1(X/\!\!/G)$ consisting of bibundles from $S^1/\!\!/\ast$ to
$X/\!\!/G$, and introduce another model $Loop_2(X/\!\!/G)$  in
Definition \ref{loopspace3}.

The groupoid structure of $Loop_1(X/\!\!/G)$ generalizes $Map(S^1,
X)/\!\!/G$, which is a subgroupoid of it. Other than the
$G-$action, we also
consider the rotation by the circle group $\mathbb{T}$ on the objects %note r/lz
and form the  groupoids $Loop_1^{ext}(X/\!\!/G)$ and
$Loop_2^{ext}(X/\!\!/G)$. The groupoid $Loop_2^{ext}(X/\!\!/G)$
has a skeleton
$$\mathcal{L}(X/\!\!/G):=\coprod_{g}{_1}\mathcal{L}{_g}
X/\!\!/L^1_gG\rtimes\mathbb{T},$$ where each component
$\mathcal{L}{_g} X
=\mbox{Map}_{\mathbb{Z}/l\mathbb{Z}}(\mathbb{R}/l\mathbb{Z}, X)$
with $l$ the order of $g$ is equipped with an evident
$C_G(g)-$action. %And there is a $G-$action on the whole space $\mathcal{L} (X/\!\!/G)$, which gives us the loop groupoid $L(X/\!\!/G)$ discussed in \cite{LUloop}.
$\mathcal{L} (X/\!\!/G)$ has the same space of objects as the
groupoid $L(X/\!\!/G)$ discussed in \cite{LUloop}, but with the
circle action added. The circle group $\mathbb{T}$ acts on
$\mathbb{R}/l\mathbb{Z}$ by rotation, and so in principle on the
orbifold ${_1}\mathcal{L}_gX$.

The key groupoid $\Lambda(X/\!\!/G)$ in the construction of
quasi-elliptic cohomology is the full subgroupoid of $\mathcal{L}
(X/\!\!/G)$ consisting of the constant loops.  In order to unravel
the relevant notations in the construction of $QEll^*_G(X)$, we
study the orbifold loop space with its $\mathbb{T}-$action in
Section \ref{orbifoldloop} and Section \ref{ghost}.

%relation with Tate K-theory

Quasi-elliptic cohomology is closely related to equivariant Tate
K-theory, which is defined from the orbifold K-theory of the full
subgroupoid of
$$\coprod_{g\in G_{conj}}{_1}\mathcal{L}{_g} X/\!\!/C_G(g)$$ consisting of the constant loops.





In Section \ref{orbifoldspre} we introduce basic notions on Lie
groupoids and bibundle, and discuss the example $Loop_1(X/\!\!/G)$
relevant to the construction of quasi-elliptic cohomology. The
references are \cite{LerStack} and \cite{Moe02}. Moreover, we
enlarge this groupoid by adding the rotation.  In Section
\ref{orbifoldloop} we interpret the groupoid
$Loop^{ext}_1(X/\!\!/G)$ in an equivalent ways and introduce a
skeleton $\mathcal{L}(X/\!\!/G)$ of it. In Section \ref{ghost} we
show quasi-elliptic cohomology can be interpreted by ghost loops
constructed by Rezk in his unpublished manuscript \cite{Rez16}. In
Section \ref{lambdarepresentationlemma} we show the representation
ring of $\Lambda_G(g)$ and relevant groups. In Section
\ref{orbqec} we introduce the construction of quasi-elliptic
cohomology in term of orbifold K-theory first and then equivariant
K-theory. We show some properties of the theory in Section
\ref{propertiesqec}.





%And in Section \ref{qec}, I introduce basic constructions of
%quasi-elliptic cohomology and some properties of it. The main
%reference for it is \cite{Rez11}.


\subsection{Loop space} \label{loopspace}

%In Section \ref{orbqec} I
%construct orbifold quasi-elliptic cohomology. The orbifold
%$\Lambda(X/\!\!/G)$ defined in \ref{orbqec} is a full subgroupoid of
%orbifold loop space, which is shown in Section \ref{orbifoldloop}.




\subsubsection{Bibundles}\label{orbifoldspre}

A standard reference for groupoids and bibundles is
\cite{LerStack}. We recall the definitions here.
\begin{definition}[Lie groupoids]
A Lie groupoid $\mathbb{G}$ is a groupoid object,
$\mathbb{G}=(\mathbb{G}_1\rightrightarrows \mathbb{G}_0)$, in the
category of smooth manifolds in which the source map and the
target map $s, t: \mathbb{G}_1\longrightarrow\mathbb{G}_0$ are
surjective submersions (so that the domain
$\mathbb{G}_1\,\strut_s\!\times\!\strut_t\,  \mathbb{G}_1$ of the
composition $m$ is a smooth manifold).

Functors and natural transformations are defined as functors and
natural transformations inside the category of smooth manifolds.
Especially, a homormophism is a smooth functor.\end{definition}




\begin{example} [Translation Groupoids]
Let $X/\!\!/G$ denote the translation groupoid with as objects the
points $x\in X$, and as arrows $\alpha: x\longrightarrow y$ those
$\alpha\in G$ for which $\alpha\cdot x = y$. %ok to use same symbol as global quotient? right action? left action?
It is a topological groupoid. In the case when $X$ is a manifold
and  $G$ is a Lie group acting on it smoothly, $X/\!\!/G$ is a Lie
groupoid.


When the group $G$ is trivial, any smooth manifold $X$ can be
viewed as a Lie groupoid with only identity morphisms. And each
Lie group can be viewed as a Lie groupoid $\ast/\!\!/ G$.
\end{example}




For any manifold $X$, let $Man_X$ denote the category of manifolds
over $X$, that is, the category whose objects are manifolds $Y$
equipped with a smooth map $Y\longrightarrow X$, and whose
morphisms are smooth maps $Y\longrightarrow Y'$ making the
following triangle commute. $$\xymatrix{Y\ar[r]\ar[d] &Y'\ar[dl]\\
X &}$$
\begin{definition}
Let $\mathbb{G}$ and $\mathbb{H}$ be Lie groupoids. A (left
principal) bibundle from $\mathbb{H}$ to $\mathbb{G}$ is a smooth
manifold $P$ together with \\1. A map $\tau: P\longrightarrow
\mathbb{G}_0$, and a surjective submersion $\sigma:
P\longrightarrow \mathbb{H}_0$. \\2. Action maps in
$Man_{G_0\times H_0}$ \begin{align*}
\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\, P &\longrightarrow
P \\ P\,\strut_{\sigma}\!\times\!\strut_t\, \mathbb{H}_1
&\longrightarrow P\end{align*} which we denote on elements as
$(g,p)\mapsto g\cdot p$ and $(p, h)\mapsto p\cdot h$, such that
\\1. $g_1\cdot (g_2\cdot p)= (g_1g_2)\cdot p$ for all $(g_1, g_2,
p)\in \mathbb{G}_1\,\strut_s\!\times\!\strut_t\,
\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\,  P$;
\\2. $(p\cdot h_1)\cdot h_2= p\cdot (h_1 h_2)$ for all $(p, h_1, h_2)\in P\,\strut_{\sigma}\!\times\!\strut_t\,  \mathbb{H}_1 \,\strut_s\!\times\!\strut_t\,
\mathbb{H}_1$; \\ 3. $p\cdot u_H(\sigma(p))=p$ and
$u_G(\tau(p))\cdot p = p$ for all $p\in P$. \\ 4. $g\cdot (p\cdot
h)= (g\cdot p)\cdot h$ for all $(g, p, h)\in
\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\,
P\,\strut_{\sigma}\!\times\!\strut_t\, \mathbb{H}_1$.
\\5. The map \begin{align*}\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\,
P&\longrightarrow P\,\strut_{\sigma}\!\times\!\strut_{\sigma}\,  P
\\ (g, p)&\mapsto (g\cdot p, p)\end{align*} is an isomorphism.

\label{bibundle}

\end{definition}

\begin{definition}
A bibundle map is a map $P\longrightarrow P'$ over
$\mathbb{H}_0\times \mathbb{G}_0$ which commutes with the
$\mathbb{G}-$ and $\mathbb{H}-$actions, i.e. the following
diagrams commute.
$$\begin{CD}\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\,  P
@>>>
P  \\ @VVV @VVV \\
\mathbb{G}_1\,\strut_s\!\times\!\strut_{\tau}\, P' @>>> P'
\end{CD}  \quad \begin{CD} P \,\strut_{\sigma}\!\times\!\strut_{t}\, \mathbb{H}_1 @>>> P \\ @VVV @VVV\\
 P' \,\strut_{\sigma}\!\times\!\strut_{t}\,
\mathbb{H}_1 @>>>P'
\end{CD}$$

\label{bibundlemap}
\end{definition}

For each pair of Lie groupoids $\mathbb{H}$ and $\mathbb{G}$,  we
have a category $Bibun(\mathbb{H}, \mathbb{G})$ with as objects
bibundles from $\mathbb{H}$ to $\mathbb{G}$ and as morphisms the
bundle maps. %good enough?
The category of smooth functors from $\mathbb{H}$ to $\mathbb{G}$
is a subcategory of $Bibun(\mathbb{H}, \mathbb{G})$.

\begin{example}[$Bibun(S^1/\!\!/\ast, \ast/\!\!/G)$] A bibundle from $S^1/\!\!/\ast$
to $\ast/\!\!/G$ with $G$ a Lie group is equivalent to a principal
$G-$bundle over $S^1$. $Bibun(S^1/\!\!/\ast, \ast/\!\!/G)$ is a
topological groupoid with principal $G-$bundles over $S^1$ as
objects and bundle isomorphisms as morphisms.
%According to Definition \ref{bibundle}, the objects in $Bibun(S^1/\!\!/\ast, \ast/\!\!/G)$ are the diagrams
%$$\begin{CD}P:= \{S^1@<{\pi}<< P @>{f}>> \ast\}\end{CD}$$
\label{babyloop1}
 \end{example}

\begin{definition}[$Loop_1(X/\!\!/G)$]

Let $G$ be a Lie group acting smoothly on a manifold $X$. We use
$Loop_1(X/\!\!/G)$ to denote the category $Bibun(S^1/\!\!/\ast,
X/\!\!/G)$, which generalizes Example \ref{babyloop1}. By
Definition \ref{bibundle}, the objects are the diagrams
$$\begin{CD}P:= \{S^1@<{\pi}<< P @>{f}>> X\}\end{CD}$$
where $P\buildrel{\pi}\over\longrightarrow S^1$ is a smooth
principal $G-$bundle over $S^1$ and $f: P\longrightarrow X$ is a
$G-$equivariant map. We use the same symbol $P$ to denote both the
object and the smooth manifold when there is no confusion. A
morphism $P\longrightarrow P'$ is a $G-$bundle map $\alpha:
P\longrightarrow P'$ making the diagram below commutes.
$$\xymatrix{S^1
&P \ar[l]_{\pi}\ar[d]^{\alpha}\ar[r]^{f} & X \\
&P' \ar[lu]^{\pi'}\ar[ru]_{f'} &}$$ Thus, the morphisms in
$Loop_1(X/\!\!/G)$ from $P$ to $P'$ are bundle isomorphisms. And
$Loop_1(X/\!\!/G)$ is a topological groupoid. The morphism spaces
are infinite-dimensional if $G$ is not finite.





%Recall that the gauge group of a principal bundle is defined to be
%the group of its vertical automorphisms. The readers may refer \cite{MV} for more details on gauge groups.

\label{loopspacemorphism}\end{definition}


We notice that $Loop_1(X/\!\!/G)$ counts in only the $G-$action on
$X$ but not the rotation on $S^1$. So we add the rotation by
adding more morphisms and obtain the definition below, which is
the orbifold loop space of interest to us.

\begin{definition}[$Loop^{ext}_1(X/\!\!/G)$]\label{loopext3space}
Let $\mathbb{T}$ denote the circle group $\mathbb{R}/\mathbb{Z}$.
Let $Loop^{ext}_1(X/\!\!/G)$ denote the groupoid with the same
objects as $Loop_1(X/\!\!/G)$. A morphism
$$\begin{CD}\{S^1@<{\pi}<< P @>{f}>> X\}\end{CD}\longrightarrow \begin{CD}\{S^1@<{\pi'}<< P' @>{f'}>> X\}\end{CD}$$ consists of the pair $(t,
\alpha)$ where $t\in\mathbb{T}$ is a rotation of the circle and
$\alpha$ is a morphism between the objects in $Loop_1(X/\!\!/G)$
such that the diagrams below commute.
$$\xymatrix{S^1\ar[d]_{t}
&P \ar[l]_{\pi}\ar[d]_{\alpha}\ar[r]^{f} & X \\S^1 &P'
\ar[l]^{\pi'}\ar[ru]_{f'} &}$$

The groupoid $Loop_1(X/\!\!/G)$ is a subgroupoid of
$Loop^{ext}_1(X/\!\!/G)$.
\end{definition}


\subsubsection{Another model for orbifold loop space}\label{orbifoldloop}

There is another description of the orbifold loop space defined in
Definition \ref{loopext3space}.

\begin{definition}[$Loop_2(X/\!\!/G)$] \label{loopspace3}

Let $G$ be a topological group acting continuously on a space $X$.
Let $Loop_2(X/\!\!/G)$ denote the groupoid whose objects are
$(\sigma, \gamma)$ with $\sigma\in G$ and $\gamma:
\mathbb{R}\longrightarrow X$
a continuous map such that $\gamma(s+1)= \gamma(s)\cdot\sigma$, for any $s\in\mathbb{R}$. %$\mbox{Map}([0, 1], X)$
A morphism $\alpha: (\sigma, \gamma)\longrightarrow (\sigma',
\gamma')$ is a continuous map $\alpha: \mathbb{R}\longrightarrow
G$ satisfying $\gamma'(s)= \gamma(s)\alpha(s)$. Note that
$\alpha(s)\sigma'=\sigma\alpha(s+1)$, for any $s\in\mathbb{R}$.
\end{definition}

Moreover, we can extend the groupoid $Loop_2(X/\!\!/G)$ by adding
the rotations.


\begin{definition}[$Loop^{ext}_2(X/\!\!/G)$]\label{loopext3space2}


Let $Loop^{ext}_2(X/\!\!/G)$ denote the groupoid with the same
objects as $Loop_2(X/\!\!/G)$. A morphism $$(\sigma,
\gamma)\longrightarrow (\sigma', \gamma')$$ consists of the pair
$(\alpha, t)$ with $\alpha:\mathbb{R}\longrightarrow G$ a
continuous map and $t\in\mathbb{R}$ %a rotation on $S^1$
satisfying $\gamma'(s)=\gamma(s-t)\alpha(s-t)$. Note that
$(\alpha, t+1)$ and $(\alpha\sigma', t)$ are the same morphism.
Thus, each morphism can be represented by a pair $(\alpha, t)$
with $t\in[0, 1)$.

$Loop_2(X/\!\!/G)$ is a subgroupoid of $Loop^{ext}_2(X/\!\!/G)$.


\end{definition}



\begin{lemma}The groupoid $Loop^{ext}_1(X/\!\!/G)$ is isomorphic to a full subgroupoid of $Loop^{ext}_2(X/\!\!/G)$.\end{lemma}

\begin{proof}
Define a functor  $$F: Loop^{ext}_1(X/\!\!/G)\longrightarrow
Loop^{ext}_2(X/\!\!/G)$$ by sending an object
$$\begin{CD} S^1 @<{\pi}<< P @>{f}>> X\end{CD}$$ to $(\sigma,
\gamma)$ with $\gamma(s):= f([s, e])$ and
$\sigma=\gamma(0)^{-1}\gamma(1)$ and sending a morphism
$$\xymatrix{S^1\ar[d]_{t} &P \ar[l]_{\pi}\ar[d]^F\ar[r]^{f} & X
\\ S^1 &P' \ar[l]^{\pi'}\ar[ru]_{f'} &}$$ to $(\alpha, t): (\sigma,
\gamma)\longrightarrow (\sigma', \gamma')$ with $\alpha(s):=F([s,
e])^{-1}.$

$F$ is a fully faithful functor but not essentially surjective.

\end{proof}

Therefore, the groupoid $Loop^{ext}_2(X/\!\!/G)$ contains all the
information of $Loop^{ext}_1(X/\!\!/G)$. Next we will show a
skeleton of this larger groupoid when $G$ is finite. Before that,
we introduce some symbols.

Let $k\geq 0$ be an integer and $g\in G$. Let $L^k_{g}G$ denote
the twisted loop group
\begin{equation}
\{\gamma: \mathbb{R}\longrightarrow
G|\gamma(s+k)=g^{-1}\gamma(s)g\}.
\end{equation}
The multiplication of it is defined by
\begin{equation}(\delta\cdot\delta')(t)=\delta(t)\delta'(t)\mbox{,     for      any     }\delta, \delta'\in
L^k_{g}G,  \mbox{    and     }t\in\mathbb{R}.\end{equation} The
identity element $e$ is the constant map sending all the real
numbers to the identity element of $G$. We extend this group by
adding the rotations. Let $L^k_{g}G\rtimes\mathbb{T}$ denote the
group with elements $(\gamma, t)$, $\gamma\in L^k_{g}G$ and
$t\in\mathbb{T}$. The multiplication is define by
\begin{equation} (\gamma, t)\cdot (\gamma', t'):= (s\mapsto \gamma(s) \gamma'(s+t), t+t').\label{lkgtmulti}\end{equation}



The set of constant maps $\mathbb{R}\longrightarrow G$ in
$L^k_{g}G$ is a subgroup of it, i.e. the centralizer $C_G(g)$.
When $G$ is finite, $L^k_{g}G=C_G(g)$.

\bigskip

When $G$ is finite, the objects of $Loop_2(X/\!\!/G)$ can be
identified with the space
$$\coprod\limits_{g\in G}{_1}\mathcal{L}{_g} X$$ where
\begin{equation}{_k}\mathcal{L}{_g}
X:=\mbox{Map}_{\mathbb{Z}/l\mathbb{Z}}(\mathbb{R}/kl\mathbb{Z},
X),\label{chongL}\end{equation} and $l$ is the order of the
element $g$. The cyclic group $\mathbb{Z}/l\mathbb{Z}$ is
isomorphic to the subgroup $k\mathbb{Z}/kl\mathbb{Z}$ of
$\mathbb{R}/kl\mathbb{Z}$. The isomorphism
$\mathbb{Z}/l\mathbb{Z}\longrightarrow k\mathbb{Z}/kl\mathbb{Z}$
sends the generator $[1]$ corresponding to $1$ to the generator
$[k]$ of $k\mathbb{Z}/kl\mathbb{Z}$ corresponding to $k$.
$k\mathbb{Z}/kl\mathbb{Z}$ acts on $\mathbb{R}/kl\mathbb{Z}$ by
group multiplication. Thus, via the isomorphism,
$\mathbb{Z}/l\mathbb{Z}$ acts on $\mathbb{R}/kl\mathbb{Z}$.
$\mathbb{Z}/l\mathbb{Z}$ is also isomorphic to the cyclic group
$\langle g\rangle$ by identifying the generater $[1]$ with $g$. So
it acts on $X$ via the $G-$action on it.

${_1}\mathcal{L}{_g} X/\!\!/ L^1_gG$ is a full subgroupoid of
$Loop_2(X/\!\!/G)$.


Moreover, ${_1}\mathcal{L}{_g} X/\!\!/ L^1_gG\rtimes\mathbb{T}$ is
a full subgroupoid of $Loop^{ext}_2(X/\!\!/G)$ where  %  for each $k$,
$L^k_{g}G\rtimes\mathbb{T}$ acts on ${_k}\mathcal{L}{_g} X$ by
\begin{equation}\delta \cdot(\gamma, t):= (s\mapsto
\delta(s+t)\cdot\gamma(s+t)), \mbox{  for any }(\gamma, t)\in
L^k_gG\rtimes\mathbb{T},\mbox { and    }\delta \in
{_k}\mathcal{L}{_g} X.\label{chongaction}\end{equation}

The action by $g$ on ${_k}\mathcal{L}{_g} X$ coincides with that
by $k\in\mathbb{R}$. So we have the isomorphism
%we have a well-defined action on ${_k}\mathcal{L}{_g} X$ by the extended twisted loop group
\begin{equation}
L^k_gG\rtimes\mathbb{T}=L_g^k G\rtimes \mathbb{R}/\langle
(\overline{g}, -k)\rangle,\label{bengkui}
\end{equation} where $\overline{g}$ represents the constant loop
$\mathbb{T}\longrightarrow \{g\}\subseteq G$.

Definition \ref{loopext3space} implies the following result.

\begin{proposition}

Let $G$ be a finite group. The groupoid
$$\mathcal{L}(X/\!\!/G):=\coprod_{g}{_1}\mathcal{L}{_g}
X/\!\!/L^1_gG\rtimes\mathbb{T},$$ where the coproduct goes over
conjugacy classes in $\pi_0G$, is a skeleton of
$Loop^{ext}_2(X/\!\!/G)$.  \label{loop1equivske}

\end{proposition}



Before we go on, we show the  physical meaning of the twisted loop
group $L_{\sigma}^1 G$ with $\sigma\in G$. Recall that the gauge
group of a principal bundle is defined to be the group of its
vertical automorphisms. The readers may refer \cite{MV} for more
details on gauge groups. For a $G-$bundle $P\longrightarrow S^1$,
let $L_P G$ denote its gauge group.


We have the well-known facts below. %reference??

\begin{lemma}The principal $G-$bundles over $S^1$ are classified up to isomorphism by
homotopy classes $$[S^1, BG]\cong \pi_0G/\mbox{conj}.$$ Up to
isomorphism every principal $G-$bundle over $S^1$ is isomorphic to
one of the forms $P_{\sigma}\longrightarrow S^1$ with $\sigma\in
G$ and
$$P_{\sigma}:=\mathbb{R}\times G/ (s+1, g)\sim(s, \sigma g).$$ A complete collection of isomorphism classes is given by  a
choice of representatives for each conjugacy class of
$\pi_0G$.\label{ghostlem1}\end{lemma}

For the gauge group $L_{P_{\sigma}} G$ of the bundle $P_{\sigma}$,
we have the conclusion.

\begin{lemma}For the bundle
$P_{\sigma}\longrightarrow S^1$, $L_{P_{\sigma}}G$ is isomorphic
to the twisted loop group $L^1_{\sigma}G$.
\label{ghostlem2}\end{lemma}

\begin{proof}

%Let $\mathbb{T}\buildrel{\pi}\over\leftarrow
%P_{\sigma}\buildrel{\widetilde{\delta}}\over\rightarrow X$ be an
%object in the groupoid $L(G\curvearrowright X)$.

Each automorphism $f$ of the bundle $P_{\sigma}\longrightarrow
S^1$ has the form
\begin{equation}\begin{CD}P_{\sigma} @>{[s, g]\mapsto [s, \gamma_f(s)g]}>> P_{\sigma} \\ @VVV @VVV \\ S^1 @>{=}>> S^1
\end{CD}\end{equation} for some $\gamma_f: \mathbb{R}\longrightarrow
G$.  The morphism is well-defined if and only if
$\gamma_f(s+1)=\sigma^{-1}\gamma_f(s)\sigma$.


So we get a well-defined map $$F: L_{P_{\sigma}}G\longrightarrow
L^1_{\sigma}G\mbox{,    } f\mapsto\gamma_f.$$  It is a bijection.
Moreover, by the property of group action, $F$ sends the identity
map to the constant map $\mathbb{R}\longrightarrow G\mbox{,   }
s\mapsto e$, which is the trivial element in $L^1_{\sigma}G$, and
for two automorphisms $f_1$ and $f_2$ at the object, $F(f_1\circ
f_2)= \gamma_{f_1}\cdot \gamma_{f_2}$. So $L_{P_{\sigma}}G$ is
isomorphic to $ L^1_{\sigma}G$.





\end{proof}



\subsubsection{Ghost Loops} \label{ghost}

In this section  we introduce a subgroupoid $GhLoop(X/\!\!/G)$ of
$Loop^{ext}_1(X/\!\!/G)$, which can be computed locally.
%$\Lambda(X/\!\!/G)$ is a full subgroupoid of it. When $G$ is finite, $GhLoop(X/\!\!/G)$ is isomorphic to $\Lambda(X/\!\!/G)$.


\begin{definition}[Ghost Loops]The ghost loop is defined to be the full subgroupoid
$GhLoop(X/\!\!/G)$ of $Loop^{ext}_1(X/\!\!/G)$ consisting of
objects $S^1\leftarrow
P\buildrel{\widetilde{\delta}}\over\rightarrow X$ such that
$\widetilde{\delta}(P)\subseteq X$ contained in a single
$G-$orbit. \label{ghostloopdef} \end{definition}

For a given $\sigma\in G$, define the space
\begin{equation}
GhLoop_{\sigma}(X/\!\!/G):=\{\delta\in {_1}\mathcal{L}{_{\sigma}}
X |\delta(\mathbb{R})\subseteq G\delta(0) \}.
\end{equation}

We have a corollary of Proposition \ref{loop1equivske} below.
\begin{proposition}$GhLoop(X/\!\!/G)$ is equivalent to the
groupoid
$$\Lambda(X/\!\!/G):=\coprod_{[\sigma]}GhLoop_{\sigma}(X/\!\!/G)/\!\!/L^1_{\sigma}G\rtimes\mathbb{T}$$
where the coproduct goes over conjugacy classes in $\pi_0G$.
\label{skullofghost}
\end{proposition}

Unlike true loops, ghost loops have the property that they can be
computed locally. In other words we have the lemma below. The
proof is left to the readers.

\begin{lemma}If $X=U\cup V$ where $U$ and $V$ are $G-$invariant open subsets,
then $GhLoop(X/\!\!/G)$ is isomorphic to the fibred product of
groupoids $$GhLoop(U/\!\!/G)\cup_{GhLoop((U\cap
V)/\!\!/G)} GhLoop(V/\!\!/G).$$ \label{ghostmv} %obvious, should prove?

\end{lemma}

Thus, the ghost loop construction satisfies Mayer-Vietoris
property.

\begin{example}
If $G$ is a finite group, it has the discrete topology. In this
case, $LG$ consists of constant loops and, thus,  is isomorphic to
$ G$. And the space of objects of $GhLoop(X/\!\!/G)$ can be
identified with $X$. For $\sigma\in G$ and any integer $k$,
$L^k_{\sigma} G$ can be identified with $C_G(\sigma)$;
$L^k_{\sigma}G\rtimes\mathbb{T}\cong C_G(\sigma)\times
\mathbb{R}/\langle (\sigma, -k)\rangle$; and
$GhLoop_{\sigma}(X/\!\!/G)$ can be identified with $X^{\sigma}$.

\label{finiteghost}
\end{example}

\begin{proposition}Let $G$ be a compact Lie group and $H$ a closed subgroup of it. It acts on the
space of left cosets $G/H$ by left multiplication. Let $\mbox{pt}$
denote the single point space with the trivial $H-$action. Then we
have the equivalence  of topological groupoids between
$Loop^{ext}_1((G/H)/\!\!/G)$ and $Loop^{ext}_1(\mbox{pt}/\!\!/H)$.
Especially, there is an equivalence of topological groupoids
between $GhLoop((G/H)/\!\!/G)$ and $GhLoop(\mbox{pt}/\!\!/ H)$.
\label{globalghost}\end{proposition}

\begin{proof}
%According to Definition \ref{ghostloopdef}, $GhLoop((G/H)/\!\!/G)$ is the groupoid whose objects are $S^1\leftarrow
%P\buildrel{\widetilde{\delta}}\over\rightarrow G/H$ with $P\longrigtharrow S^1$ a principal $G-$bundle and
%$P\buildrel{\widetilde{\delta}}\over\longrightarrow G/H$ a $G-$equivariant map. A morphism $(\phi, t)$

First we define a functor $F:
Loop^{ext}_1((G/H)/\!\!/G)\longrightarrow
Loop^{ext}_1(\mbox{pt}/\!\!/ H)$ sending an object $S^1\leftarrow
P\buildrel{\widetilde{\delta}}\over\rightarrow G/H$  to
$S^1\leftarrow Q\rightarrow \{eH\}=\mbox{pt}$ where
$Q\longrightarrow eH$ is the constant map, and $Q\longrightarrow
S^1$ is the pull back bundle $$\xymatrix{Q \ar[r] \ar[d] &\{eH\}
\ar@{^{(}->}[d]
\\ P\ar[r] &G/H.}$$

It sends a morphism $$\xymatrix{P'\ar[r] \ar[d]&P \ar[r] \ar[d]
&G/H\\ S^1\ar[r] &S^1&}$$ to the morphism
$$\xymatrix{Q' \ar[r] \ar[d] &Q\ar[r] \ar[d] &\{eH\} \ar[d] \\ P'\ar[r]\ar[d] &P\ar[r]\ar[d] &G/H\\
S^1 \ar[r] &S^1 & } $$ where all the squares are pull-back.

In addition, we can define a functor $F': Loop^{ext}_1
(\mbox{pt}/\!\!/ H) \longrightarrow Loop^{ext}_1((G/H)/\!\!/G)$
sending an object $S^1\leftarrow
 Q\rightarrow \mbox{pt}$ to $S^1\leftarrow G\times_HQ\rightarrow
 G\times_H\mbox{pt}=G/H$ and sending a morphism $$\xymatrix{Q'\ar[r] \ar[d] & Q \ar[d] \\ S^1\ar[r]
 &S^1}$$ to $$\xymatrix{G\times_HQ'\ar[r] \ar[d] & G\times_HQ \ar[d] \ar[r] &G\times_H\mbox{pt}=G/H\\ S^1\ar[r]
 &S^1 &}$$

$F\circ F'$ and $F'\circ F$ are both identity maps. So the
topological groupoids $Loop^{ext}_1((G/H)/\!\!/G)$ and
$Loop^{ext}_1(\mbox{pt}/\!\!/ H)$ are equivalent.

\bigskip

We can prove the equivalence between $GhLoop((G/H)/\!\!/G)$ and
$GhLoop(\mbox{pt}/\!\!/ H)$ in the same way.
\end{proof}

%\begin{remark} In the case when $G$ is a finite group, $Loop_1^{ext}((G/H)/\!\!/G)$
%and $Loop_1^{ext}(\mbox{pt}/\!\!/ H)$ are equivalent, the proof of which is adapted directly from that of Proposition
%\ref{globalghost}. When $G$ is not finite, we have to consider the subgroupoid $GhLoop((G/H)/\!\!/G)$ and $GhLoop(\mbox{pt}/\!\!/ H)$ to get
%the equivalence. \end{remark}


\begin{remark}In general, if $H^*$ is an equivariant cohomology theory,
Proposition \ref{globalghost} implies the functor $$X/\!\!/G
\mapsto H^*(GhLoop(X/\!\!/G))$$ gives a new equivariant cohomology
theory. When $H^*$ has the change of group isomorphism, so does
$H^*(GhLoop(-))$.
\end{remark}




\section{Quasi-elliptic cohomology $QEll^*_G$}\label{conqec}

We assume $G$ is a finite group in the rest part of the paper. In
Section \ref{orbqec} we define $QEll^*_G$ and prove some of its
main properties. The main reference for Section \ref{conqec} is
\cite{Rez11}.
%in term of equivariant K-theory.
Before that we discuss in Section \ref{lambdarepresentationlemma}
the complex representation ring of
\begin{equation}\Lambda_G(g):=L^1_{g}G\rtimes\mathbb{T}\cong C_G(\sigma)\times
\mathbb{R}/\langle (g, -1)\rangle,\label{lambdadef}\end{equation}
which is a component of $QEll^*_G$ of a point. We assume
familiarity with \cite{SegalequiK} and \cite{KC}.
% The main reference for these sections is \cite{Rez11}.


\subsection{Preliminary: representation ring of
$\Lambda_G(g)$}\label{lambdarepresentationlemma}
 Let $q: \mathbb{T}\longrightarrow U(1)$
be the isomorphism $t\mapsto e^{2\pi it}$. The representation ring
$R\mathbb{T}$ of the circle group is $\mathbb{Z}[q^{\pm}]$.

We have an exact sequence
$$1\longrightarrow C_G(g)\longrightarrow
\Lambda_G(g)\buildrel{\pi}\over\longrightarrow\mathbb{T}\longrightarrow
0$$ where the first map is $g\mapsto [g, 0]$ and the second map is
\begin{equation}\pi([g, t])= e^{2\pi it}.\label{pizq}\end{equation}

There is a relation between the complex representation ring of
$C_G(g)$ and that of $\Lambda_G(g)$, which is shown as Lemma 1.2
in \cite{Rez11}.

\begin{lemma}

$\pi^*: R\mathbb{T}\longrightarrow R\Lambda_G(g)$ exhibits
$R\Lambda_G(g)$ as a free $R\mathbb{T}-$module.


In particular, there is an $R\mathbb{T}-$basis of $R\Lambda_G(g)$
given by irreducible representations $\{V_{\lambda}\}$, such that
restriction $V_{\lambda}\mapsto V_{\lambda}|_{C_G(g)}$ to $C_G(g)$
defines a bijection between $\{V_{\lambda}\}$ and the set
$\{\lambda\}$ of irreducible representations of
$C_G(g)$.\label{cl}
\end{lemma}
\begin{proof}

Let $l$ be the order of  $g$. Note that $\Lambda_G(g)$ is
isomorphic to
$$C_G(g)\times\mathbb{R}/l\mathbb{Z}/\langle(g, -1)\rangle.$$ Thus, it is the quotient of the product of two compact Lie groups.

Let $\lambda: C_G(g)\longrightarrow GL(n, \mathbb{C})$ be an
$n-$dimensional $C_G(g)-$representation with representation space
$V$ and $\eta: \mathbb{R}\longrightarrow GL(n, \mathbb{C})$ be a
representation of $\mathbb{R}$ such that $\lambda(g)$ acts on $V$
via scalar multiplication by $\eta(1)$. Define
 a $n-$dimensional
$\Lambda_G(g)-$representation $\lambda\odot_{\mathbb{C}} \eta$
with representation space $V$ by
\begin{equation}\lambda\odot_{\mathbb{C}} \eta([h, t]):
=\lambda(h)\eta(t).\end{equation}



Any irreducible $n-$dimensional representation of the quotient
group $\Lambda_G(g)=C_G(g)\times\mathbb{R}/\langle(g, -1)\rangle$
is an irreducible $n-$dimensional representation of the product
$C_G(g)\times\mathbb{R}$. And any finite dimensional irreducible
complex representation of the product of two compact Lie groups is
the tensor product of an irreducible representation of each
factor. So any irreducible representation of the quotient group
$\Lambda_G(g)$ is the tensor product of an irreducible
representation $\lambda$ of $C_G(g)$ with representation space $V$
and an irreducible representation $\eta$ of $\mathbb{R}$. Any
irreducible complex representation $\eta$ of $\mathbb{R}$ is one
dimensional. So the representation space of
$\lambda\odot_{\mathbb{C}} \eta$ is still $V$. $\eta(1)^l=I$. We
need $\eta(1)=\lambda (g)$. So $\eta(1)=e^{\frac{2\pi ik}{l}}$ for
some $k\in \mathbb{Z}$. So
$$\eta(t)= e^{\frac{2\pi i(k+lm)t}{l}}.$$ Any $m\in\mathbb{Z}$
gives a choice of  $\eta$ in this case. And $\eta$ is a
representation of $\mathbb{R}/l\mathbb{Z}\cong \mathbb{T}$.

Therefore, we have a bijective correspondence between


(1) isomorphism classes of irreducible
$\Lambda_G(g)-$representation $\rho$, and


(2) isomorphism classes of pairs $(\lambda, \eta)$ where $\lambda$
is an irreducible $C_G(g)-$representation and
$\eta:\mathbb{R}\longrightarrow \mathbb{C}^*$ is a character such
that $\lambda(g)=\eta(1)I$. $\lambda=\rho|_{C_G(g)}$.

Then as a corollary, $\pi^*: R\mathbb{T}\longrightarrow
R\Lambda_G(g)$ exhibits $R\Lambda_G(g)$ as a free
$R\mathbb{T}-$module.
\end{proof}

\begin{remark} We can make a canonical choice of $\mathbb{Z}[q^{\pm}]$-basis
for $R\Lambda_{G}(g)$. For each irreducible $G$-representation
$\rho: G\longrightarrow Aut(G)$, write $\rho(\sigma)=e^{2\pi
ic}id$ for $c\in[0,1)$, and set $\chi_{\rho}(t)=e^{2\pi ict}$.
Then the pair $(\rho, \chi_{\rho})$ corresponds to a unique
irreducible $\Lambda_{G}(g)$-representation.
\label{lambdabasis}\end{remark}

\begin{example}[$G=\mathbb{Z}/N\mathbb{Z}$]
Let $G=\mathbb{Z}/N\mathbb{Z}$ for $N\geq 1$, and let $\sigma\in
G$. Given an integer $k\in\mathbb{Z}$ which projects to
$\sigma\in\mathbb{Z}/N\mathbb{Z}$, let $x_k$ denote the
representation of $\Lambda_G(\sigma)$ defined by
\begin{equation}\begin{CD}\Lambda_{G}(\sigma)=(\mathbb{Z}\times\mathbb{R})/(\mathbb{Z}(N,0)+\mathbb{Z}(k,1))
@>{[a,t]\mapsto[(kt-a)/N]}>> \mathbb{R}/\mathbb{Z}=\mathbb{T}
@>{q}>> U(1).\end{CD}\label{xk}\end{equation} $R\Lambda_G(\sigma)$
is isomorphic to the ring $\mathbb{Z}[q^{\pm}, x_k]/(x^N_k-q^k)$.
\bigskip

For any finite abelian group
$G=\mathbb{Z}/N_1\mathbb{Z}\times\mathbb{Z}/N_2\mathbb{Z}\times\cdots\times\mathbb{Z}/N_m\mathbb{Z}$,
let $\sigma=(k_1, k_2, \cdots k_m)\in G.$ Then $\Lambda_G(\sigma)$
is isomorphic to the fibred product
$$\Lambda_{\mathbb{Z}/N_1\mathbb{Z}}(k_1)\times_{\mathbb{T}}\cdots\times_{\mathbb{T}}\Lambda_{\mathbb{Z}/N_m\mathbb{Z}}(k_m).$$
And
\begin{align*}R\Lambda_G(\sigma)&\cong
R\Lambda_{\mathbb{Z}/N_1\mathbb{Z}}(k_1)\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}R\Lambda_{\mathbb{Z}/N_m\mathbb{Z}}(k_m)\\
&\cong\mathbb{Z}[q^{\pm}, x_{k_1}, x_{k_2},\cdots
x_{k_m}]/(x^{N_1}_{k_1}-q^{k_1},x^{N_2}_{k_2}-q^{k_2}, \cdots
x^{N_m}_{k_m}-q^{k_m})\end{align*} where all the $x_{k_j}$'s are
defined as $x_k$ in (\ref{xk}).\label{ppex}
\end{example}



\begin{example}[$G=\Sigma_3$]\label{replambdasymm3}
$G=\Sigma_3$ has three conjugacy classes represented by $1$,
$(12)$, $(123)$ respectively.
\bigskip

$\Lambda_{\Sigma_3}(1)=\Sigma_3\times\mathbb{T}$, thus,
$R\Lambda_{\Sigma_3}(1)=R\Sigma_3\otimes R\mathbb{T}=\mathbb{Z}[X,
Y]/(XY-Y, X^2-1, Y^2-X-Y-1)\otimes\mathbb{Z}[q^{\pm}]$ where $X$
is the sign representation on $\Sigma_3$ and $Y$ is the standard
representation.
\bigskip

$C_{\Sigma_3}((12))=\langle(12)\rangle=\Sigma_2,$ thus,
$\Lambda_{\Sigma_3}((12))\cong\Lambda_{\Sigma_2}((12)).$ So we
have $$R\Lambda_{\Sigma_3}((12))\cong
R\Lambda_{\Sigma_2}((12))=\mathbb{Z}[q^{\pm},
x_1]/(x_1^2-q)\cong\mathbb{Z}[q^{\pm\frac{1}{2}}].$$

$C_{\Sigma_3}(123)=\langle(123)\rangle=\mathbb{Z}/3\mathbb{Z},$
thus,
$\Lambda_{\Sigma_3}((123))\cong\Lambda_{\mathbb{Z}/3\mathbb{Z}}(1).$
So we have $$R\Lambda_{\Sigma_3}((123))\cong\mathbb{Z}[q^{\pm},
x_1]/(x_1^3-q)\cong\mathbb{Z}[q^{\pm\frac{1}{3}}].$$\end{example}


\bigskip


Let $k$ be any integer. We describe the relation between
\begin{equation}\Lambda^k_G(g):=L^k_{g}G\rtimes\mathbb{T}\cong C_G(\sigma)\times
\mathbb{R}/\langle (g, -k)\rangle \label{lambdakdef}\end{equation}
and $\Lambda_G(g)$, as well as the relation between their
representation rings.

There is an exact sequence
$$\begin{CD}1 @>>> C_G(g) @>{g\mapsto [g, 0]}>> \Lambda^k_G(g) @>{\pi_k}>>\mathbb{R}/k\mathbb{Z} @>>> 0\end{CD}$$
where the second map $\pi_k: \Lambda^k_G(g)\longrightarrow
\mathbb{R}/k\mathbb{Z}$ is $\pi_k([g, t])= e^{2\pi i t}$.


Let $q^{\frac{1}{k}}: \mathbb{R}/k\mathbb{Z}\longrightarrow U(1)$
denote the composition
$$\begin{CD}\mathbb{R}/k\mathbb{Z} @>{t\mapsto \frac{t}{k}}>> \mathbb{R}/\mathbb{Z} @>{q}>> U(1).\end{CD}$$
The representation ring $R(\mathbb{R}/k\mathbb{Z})$ of
$\mathbb{R}/k\mathbb{Z}$ is  $\mathbb{Z}[q^{\pm\frac{1}{k}}]$. And
there is a canonical isomorphism of $\lambda-$rings
$$R\Lambda_G(g)\longrightarrow R\Lambda_G^k(g)$$ sending $q$ to
$q^{\frac{1}{k}}$.


Analogous to Lemma \ref{cl}, we have the conclusion about
$R\Lambda^k_G(g)$ below.
\begin{lemma}\label{lambdakrepresentation}
The map $\pi^*_k: R(\mathbb{R}/k\mathbb{Z})\longrightarrow
R\Lambda^k_G(g)$ exhibits it as a free
$\mathbb{Z}[q^{\pm\frac{1}{k}}]-$module. There is a
$\mathbb{Z}[q^{\pm\frac{1}{k}}]-$basis of $R\Lambda^k_G(g)$ given
by irreducible representations $\{\rho_k\}$ such that the
restrictions $\rho_k|_{C_G(g)}$ of them to $C_G(g)$ are precisely
the $\mathbb{Z}$-basis of $RC_G(g)$ given by irreducible
representations.

In other words, any irreducible $\Lambda^k_G(g)-$representation
has the form $\rho\odot_{\mathbb{C}} \chi$ where $\rho$ is an
irreducible representation of $C_G(g)$,
$\chi:\mathbb{R}/k\mathbb{Z}\longrightarrow GL(n, \mathbb{C})$
such that $\chi(k)=\rho(g)$, and
\begin{equation}\rho\odot_{\mathbb{C}} \chi([h, t]):
=\rho(h)\chi(t)\mbox{,     for any   }[h, t]\in
\Lambda^k_G(g).\end{equation}
\end{lemma}

$R\Lambda_G^k(g)$ is a $\mathbb{Z}[q^{\pm}]-$module via the
inclusion $\mathbb{Z}[q^{\pm}]\longrightarrow
\mathbb{Z}[q^{\pm\frac{1}{k}}]$.

\bigskip

There is a group isomorphism $\alpha_k:
\Lambda^k_G(g)\longrightarrow \Lambda_G(g)$ sending $[g, t]$ to
$[g, \frac{t}{k}]$.

Observe that there is a pullback square of groups
\begin{equation}\xymatrix{&\Lambda_G^k(g)\ar[r]^{\alpha_k}\ar[d]^{\pi_k}
&\Lambda_G(g)\ar[d]^{\pi}\\&\mathbb{R}/k\mathbb{Z}
\ar[r]^{t\mapsto
\frac{t}{k}}&\mathbb{R}/\mathbb{Z}}\label{alphaklambdagroup}\end{equation}


By Lemma \ref{lambdakrepresentation}, we can make a
$\mathbb{Z}[q^{\pm\frac{1}{k}}]$-basis
$\{\rho\odot_{\mathbb{C}}\chi_{\rho, k}\}$ for $R\Lambda^k_{G}(g)$
with each $\rho: G\longrightarrow Aut(G)$ an irreducible
$G$-representation and $\chi_{\rho, k}(t)= e^{2\pi i\frac{ct}{k}}$
with $c\in[0,1)$ such that $\rho(\sigma)=e^{2\pi ic}id$. This
collection $\{\rho\odot_{\mathbb{C}}\chi_{\rho, k}\}$ gives  a
$\mathbb{Z}[q^{\pm\frac{1}{k}}]-$basis of $R\Lambda^k_G(g)$.
%So we have a ring isomorphism $$R\Lambda_G(g)\otimes_{R\mathbb{T}}R(\mathbb{R}/k\mathbb{Z})\longrightarrow R\Lambda_G^k(g)$$ sending $\rho\odot_{\mathbb{C}}\chi_{\rho}$
%to $\rho\odot_{\mathbb{C}}\chi_{\rho, k}$.

So we have the commutative square of a pushout square in the
category of $\lambda-$rings. \begin{equation}\xymatrix{&
R\Lambda^k_G(g)&R\Lambda_G(g)\ar[l]\\
&R(\mathbb{R}/k\mathbb{Z})\ar[u]&R\mathbb{T}\ar[u]\ar[l]}\label{alphaklambdaring}\end{equation}
A good reference for $\lambda-$rings is the book \cite{Yau10}.





\bigskip




Moreover, we have the conclusion below about the relation between
the induced representations
$Ind|^{\Lambda_G(\sigma)}_{\Lambda_H(\sigma)}(-)$ and
$Ind|^{C_G(\sigma)}_{C_H(\sigma)}(-).$
\begin{lemma}
Let $H$ be a subgroup of $G$ and $\sigma$ an element of $H$. Let
$m$ denote $[C_G(\sigma):C_H(\sigma)]$. Let $V$ denote a
$\Lambda_H(\sigma)-$representation $\lambda\odot_{\mathbb{C}}\chi$
with $\lambda$ a $C_H(\sigma)-$representation, $\chi$ a
$\mathbb{R}-$representation and $\odot_{\mathbb{C}}$ is defined in
Lemma \ref{cl}. Then the induced representation
$$Ind^{\Lambda_G(\sigma)}_{\Lambda_H(\sigma)}
(\lambda\odot_{\mathbb{C}}\chi)$$ is isomorphic to the
$\Lambda_G(\sigma)-$representation
$$(Ind^{C_G(\sigma)}_{C_H(\sigma)}\lambda)\odot_{\mathbb{C}}\chi.$$
Their underlying vector spaces are both $V^{\oplus m}$.
\label{induequ}
\end{lemma}

The proof is straightforward and left to the readers.

\begin{remark}
By Lemma \ref{induequ}, the computation of the induced
representation $$Ind^{\Lambda_G(\sigma)}_{\Lambda_H(\sigma)}
(\lambda\odot_{\mathbb{C}}\chi)$$ can be reduced to the
computation of
$$(Ind^{C_G(\sigma)}_{C_H(\sigma)}\lambda)\odot_{\mathbb{C}}\chi.$$ Thus,  the conclusions about induced representations
for finite groups  can apply to the computation of
$Ind^{\Lambda_G(\sigma)}_{\Lambda_H(\sigma)}
(\lambda\odot_{\mathbb{C}}\chi)$.

\end{remark}

\begin{remark}
Let $G$ be a compact Lie group and $H$ a closed subgroup.  Let
$\sigma$ be a torsion element in $H$. %Let $m$ denote $[C_G(\sigma):C_H(\sigma)]$.
Let $V$ denote a $\Lambda_H(\sigma)-$representation
$\lambda\odot_{\mathbb{C}}\chi$ with $\lambda$ a
$C_H(\sigma)-$representation, $\chi$ a $\mathbb{R}-$representation
and $\odot_{\mathbb{C}}$ is defined in Lemma \ref{cl}.

Then it is straightforward to check \begin{equation}
res^{\Lambda_G(\sigma)}_{\Lambda_H(\sigma)}(\lambda\odot_{\mathbb{C}}\eta)=(res^{C_G(\sigma)}_{C_H(\sigma)}\lambda)\odot_{\mathbb{C}}\eta.\end{equation}

\end{remark}



\subsection{Quasi-elliptic cohomology}\label{orbqec} In this
section we introduce the definition of quasi-elliptic cohomology
$QEll^*_G$ in term of orbifold K-theory, and then express it via
equivariant K-theory. We assume familiarity with
\cite{SegalequiK}. The reader may read Chapter 3 in \cite{ALRuan}
and \cite{Moe02} for a reference of orbifold K-theory.

When $G$ is finite, quasi-elliptic cohomology is defined from the
ghost loops in Definition \ref{ghostloopdef}. By proposition
\ref{skullofghost} and Example \ref{finiteghost}, we can see the
groupoid $GhLoop(X/\!\!/G)$ is equivalent to the disjoint union of
some translation groupoids.  Before describing this equivalent
groupoid $\Lambda(X/\!\!/G)$ in detail, we recall what inertia
groupoid is. A reference for that is \cite{LU01}.


%We can construct a groupoid $\coprod\limits_{g\in G^{tors}_{conj}}X^g/\!\!/\Lambda_G(g)$ and define the quasi-elliptic cohomology in
%term of orbifold $K$-theory.



\begin{definition}Let $\mathbb{G}$ be a groupoid. The inertia
groupoid $I(\mathbb{G})$ of $\mathbb{G}$ is defined as follows.

An object $a$ is an arrow in $\mathbb{G}$ such that its source and
target are equal. %i.e. $$ob(I(\mathbb{G}))=\{a\in\}$$
A morphism $v$ joining two objects $a$ and $b$ is an arrow $v$ in
$\mathbb{G}$ such that $$v\circ a=b\circ v.$$ In other words, $b$
is the conjugate of $a$ by $v$, $b=v\circ a\circ v^{-1}$.

\label{inertiagroupoid}
\end{definition}

Let $G$ be a finite group and $X$ a $G-$space.

\begin{example} The
inertia groupoid $I(X/\!\!/G)$ of the translation groupoid
$X/\!\!/G$ is the groupoid with

\textbf{objects}: the space $\coprod\limits_{g\in G}X^{g}$

\textbf{morphisms}: the space $\coprod\limits_{g, g'\in
G}C_G(g,g')\times X^g$ where $C_G(g,g')=\{\sigma\in
G|g'\sigma=\sigma g\}\subseteq G.$

For $x\in X^g$ and $(\sigma, g)\in C_G(g,g')\times X^g$, $(\sigma,
g)(x)=\sigma x\in X^{g'}.$ \label{torsionquotient}
\end{example}

%Define $\Lambda_G(g,g')$ to be the quotient of $C_G(g,g')\times\mathbb{R}$ under the action of $\mathbb{Z}$,
%where the action of the generator of $\mathbb{Z}$ is given by
%$(\sigma, t)\mapsto (\sigma g, t-1)=(g' \sigma, t-1).$ It has a
%well-defined multiplication $[\sigma_1, t_1][\sigma_2,
%t_2]=[\sigma_1\sigma_2, t_1+t_2].$

\begin{definition}The groupoid $\Lambda(X/\!\!/G)$ has the same objects as
$I(X/\!\!/G)$ but richer morphisms
$$\coprod\limits_{g, g'\in G}\Lambda_G(g, g')\times X^g$$
where $\Lambda_G(g, g')$ is the quotient of $C_G(g, g')\times
\mathbb{R}$ under the equivalence $$(x, t)\sim (gx, t-1)=(xg',
t-1).$$ For an object $x\in X^g$ and a morphism $([\sigma, t],
g)\in \Lambda_G(g, g')\times X^g$, $([\sigma, t], g)(x)=\sigma
x\in X^{g'}.$ The composition of the morphisms is defined by
\begin{equation}[\sigma_1, t_1][\sigma_2,
t_2]=[\sigma_1\sigma_2, t_1+t_2].\end{equation}
\label{lambdaoidef}\end{definition}


We have a homomorphism of groupoids
$$\pi: \Lambda(X/\!\!/G)\longrightarrow\mathbb{T}$$ sending all the objects to the single object in $\mathbb{T}$, and
a morphism $([\sigma,t], g)$ to $e^{2\pi it}$ in $\mathbb{T}$.




\begin{definition}
The quasi-elliptic cohomology $QEll^*_G(X)$ is defined to be
$K^*_{orb}(GhLoop(X/\!\!/G))\cong K^*_{orb}(\Lambda(X/\!\!/G))$.
\label{qecdef}\end{definition} We can unravel the definition and
express it via equivariant K-theory.


Let $\sigma\in G$. The fixed point space $X^{\sigma}$ is a
$C_G(\sigma)-$space. We can define a $\Lambda_G(\sigma)-$action on
$X^{\sigma}$ by
$$[g, t]\cdot x:=g\cdot x.$$
Then we have
\begin{proposition}
\begin{equation}QEll^*_G(X)=\prod_{g\in
G_{conj}}K^*_{\Lambda_G(g)}(X^{g})=\bigg(\prod_{g\in
G}K^*_{\Lambda_G(g)}(X^{g})\bigg)^G,\end{equation}  where
$G_{conj}$ is a set of representatives of $G-$conjugacy classes in
$G$.
\end{proposition}
Thus, for each $g\in\Lambda_G(g)$, we can define the projection
$$\pi_g: QEll^*_G(X)\longrightarrow K^*_{\Lambda_G(g)}(X^{g}).$$
For the singe point space, we have
\begin{equation}QEll^0_G(\mbox{pt})\cong\prod_{g\in G_{conj}} R\Lambda_G(g). \label{qecpt}\end{equation}
%is exactly $QEll^*_G(X)$ defined in Section \ref{conqec}.
%In Section \ref{conqec} I describe $QEll_G(X)$ where $G$ is a compact Lie group and $X$ is a $G-$space.  In Section \ref{orbqec}
%we have seen the definition to orbifolds. The discussion there is similar to the construction of orbifold Tate K-theory in \cite{NT},




We have the ring homomorphism
$$\mathbb{Z}[q^{\pm}]=K^0_{\mathbb{T}}(\mbox{pt})\buildrel{\pi^*}\over\longrightarrow K^0_{\Lambda_G(g)}(\mbox{pt})\longrightarrow
K^0_{\Lambda_G(g)}(X)$$ where $\pi: \Lambda_G(g)\longrightarrow
\mathbb{T}$ is the projection defined in (\ref{pizq}) and the
second is via the collapsing map $X\longrightarrow \mbox{pt}$. So
$QEll_G^*(X)$ is naturally a
%$\mathbb{Z}[q^{\pm}]-$
$\mathbb{Z}[q^{\pm}]-$algebra. %why




\subsection{Properties}\label{propertiesqec} In this section
 we discuss some properties of $QEll^*_G$,
including the restriction map, the K\"{u}nneth map on it, its
tensor product and the change-of-group isomorphism.

Since each homomorphism $\phi: G\longrightarrow H$ induces a
well-defined homomorphism $\phi_{\Lambda}:
\Lambda_G(\tau)\longrightarrow\Lambda_H(\phi(\tau))$ for each
$\tau$ in $G$, we can get the proposition below directly.
\begin{proposition}For each homomorphism $\phi: G\longrightarrow H$, it induces a ring map
$$\phi^*: QEll^*_H(X)\longrightarrow QEll^*_G(\phi^*X)$$ characterized by the commutative diagrams

\begin{equation}\begin{CD}QEll^*_H(X) @>{\phi^*}>> QEll^*_G(\phi^*X) \\ @V{\pi_{\phi(\tau)}}VV  @V{\pi_{\tau}}VV  \\
K^*_{\Lambda_H(\phi(\tau))}(X^{\phi(\tau)}) @>{\phi^*_{\Lambda}}>>
 K^*_{\Lambda_G(\tau)}(X^{\phi(\tau)})\end{CD}\end{equation} for any $\tau \in G$. So $QEll^*_G$ is functorial in $G$.
\label{restrictionq}\end{proposition}

\bigskip
 And, we can define K\"{u}nneth map on $QEll^*_G$
induced from that on equivariant $K$-theory.

Let $G$ and $H$ be two finite groups. $X$ is a $G$-space and $Y$
is a $H$-space. Let $\sigma\in G$ and $\tau\in H$. Let
$\Lambda_G(\sigma)\times_{\mathbb{T}}\Lambda_H(\tau)$ denote the
fibered product of the morphisms
$$\Lambda_G(\sigma)\buildrel{\pi}\over\longrightarrow
\mathbb{T}\buildrel{\pi}\over\longleftarrow\Lambda_H(\tau).$$ It
is isomorphic to $\Lambda_{G\times H}(\sigma, \tau)$ under the
correspondence
$$([\alpha, t], [\beta, t])\mapsto [\alpha, \beta, t].$$

Consider the composition below
\begin{align*}T: &K_{\Lambda_G(\sigma)}(X^{\sigma})\otimes
K_{\Lambda_H(\tau)}(Y^{\tau})\longrightarrow
K_{\Lambda_{G}(\sigma)\times\Lambda_{H}(\tau)}(X^{\sigma}\times
Y^{\tau})\buildrel{res}\over\longrightarrow \\
&K_{\Lambda_{G}(\sigma)\times_{\mathbb{T}}\Lambda_{H}(\tau)}(X^{\sigma}\times
Y^{\tau}) \buildrel{\cong}\over\longrightarrow K_{\Lambda_{G\times
H}(\sigma, \tau)}((X\times Y)^{(\sigma, \tau)}).\end{align*} where
the first map is the K\"{u}nneth map of equivariant K-theory, the
second is the restriction map  and the third is the isomorphism
induced by the group isomorphism $\Lambda_{G\times H}(\sigma,
\tau)\cong\Lambda_G(\sigma)\times_{\mathbb{T}}\Lambda_H(\tau)$.

For any $g\in G$, let $1$ denote the trivial line bundle over
$X^g$ and let $q$ denote the line bundle $1\odot_{\mathbb{C}} q$
over $X^g$. The map $T$ above sends both $1\otimes q$ and
$q\otimes 1$ to $q$. So we get the well-defined map
\begin{equation}K^*_{\Lambda_G(\sigma)}(X^{\sigma})\otimes_{\mathbb{Z}[q^{\pm}]}K^*_{\Lambda_H(\tau)}(Y^{\tau})\longrightarrow
 K_{\Lambda_{G\times H}(\sigma, \tau)}((X\times
Y)^{(\sigma, \tau)}).\label{ku}\end{equation}

This map (\ref{ku}) gives the K\"{u}nneth map of quasi-elliptic
cohomology.

Thus, we have
\begin{equation}QEll^*_G(X)\otimes_{\mathbb{Z}[q^{\pm}]}QEll^*_H(Y)
\cong\prod_{\sigma\in G_{conj}\mbox{,   } \tau\in
H_{conj}}K^*_{\Lambda_G(\sigma)}(X^{\sigma})\otimes_{\mathbb{Z}[q^{\pm}]}K^*_{\Lambda_H(\tau)}(Y^{\tau}).\label{qectensor}\end{equation}
and the K\"{u}nneth map defined above produce a ring homomorphism
$$QEll^*_G(X)\otimes_{\mathbb{Z}[q^{\pm}]}QEll^*_H(Y)\longrightarrow
QEll^*_{G\times H}(X\times Y),$$ which is the direct product of
those maps defined in (\ref{ku}).

From this definition, by Lemma \ref{cl} we have
$$QEll^*_G(\mbox{pt})\otimes_{\mathbb{Z}[q^{\pm}]}QEll^*_H(\mbox{pt})=QEll^*_{G\times H}(\mbox{pt}).$$

%Let's consider the two extreme cases of $G$-actions on $X$.

\begin{proposition}

Let $X$ be a $G\times H-$space with trivial $H-$action and let
$\mbox{pt}$ be the single point space with trivial $H-$action.
 Then we have
$$QEll_{G\times H}(X)\cong QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_H(\mbox{pt}).$$

Especially, if $G$ acts trivially on $X$, we have
$$QEll_G(X)\cong   QEll(X)\otimes_{\mathbb{Z}[q^{\pm}]}
QEll_G(\mbox{pt}).$$ Here $QEll^*(X)$ is
$QEll^*_{\{e\}}(X)=K^*_{\mathbb{T}}(X)$.
\end{proposition}

\begin{proof}

\begin{align*}QEll_{G\times H}(X) &=\prod\limits_{\substack{g\in
G_{conj}
\\ h\in H_{conj}}}K_{\Lambda_{G\times H}(g, h)}(X^{(g,
h)})\cong \prod\limits_{\substack{g\in G_{conj} \\ h\in H_{conj}}
}K_{\Lambda_{G}(g)\times_{\mathbb{T}} \Lambda_{H}(h)}(X^{g})\\
&\cong \prod\limits_{\substack{g\in G_{conj} \\ h\in H_{conj}}}
K_{\Lambda_{G}(g)}(X^{g})\otimes_{\mathbb{Z}[q^{\pm}]}
K_{\Lambda_H(h)}(\mbox{pt})=
QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_H(\mbox{pt}).
\end{align*}


\end{proof}

%\begin{proof} Since $\mathbb{T}$ acts on $X$ trivially, $QEll^*(X)\cong
%K^*(X)\otimes R\mathbb{T}$. Since $G$ acts on $X$ trivially, we have $QEll^*(X^\sigma)\cong  QEll^*(X)$ for any $\sigma\in G$.
%Thus, \begin{align*}QEll^*_G(X)&\cong\prod_{\sigma\in G^{tors}_{conj}}K^*(X^{\sigma})\otimes R\Lambda_G(\sigma)
%\cong\prod_{\sigma\in G^{tors}_{conj}}QEll^*(X^{\sigma})\otimes_{\mathbb{Z}[q^{\pm}]}
%R\Lambda_G(\sigma)\\ &\cong\prod_{\sigma\in G^{tors}_{conj}}QEll^*(X)\otimes_{\mathbb{Z}[q^{\pm}]}
%R\Lambda_G(\sigma)\cong QEll^*(X)\otimes_{\mathbb{Z}[q^{\pm}]}
%QEll^*_G(\mbox{pt}).\end{align*}\end{proof}



\begin{proposition}
If $G$ acts freely on $X$,
$$QEll^*_G(X)\cong QEll^*_e(X/G).$$
\end{proposition}
\begin{proof}
Since $G$ acts freely on $X$, $$X^{\sigma}=\begin{cases}\emptyset,
&\text{if $\sigma\neq e$;}\\ X, &\text{if
$\sigma=e$.}\end{cases}$$ Thus,
$QEll^*_G(X)\cong\prod\limits_{\sigma\in
G_{conj}}K^*_{\Lambda_G(\sigma)/C_G(\sigma)}(X^{\sigma}/C_G(\sigma))\cong
K^*_{\mathbb{T}}(X/G).$

Since $\mathbb{T}$ acts trivially on $X$, we have
$K^*_{\mathbb{T}}(X/G)=QEll^*_e(X/G)$ by definition. And it is
isomorphic to $K^*(X/G)\otimes R\mathbb{T}$.\end{proof}
%\begin{remark}When $G$ acts freely on $X$, $\prod_{[\sigma]\in G^{tor}_{conj}}K^*(X^{\sigma}/C_G(\sigma))$ is in fact
%$K^*_{orb}(I^{tors}(X/\!\!/G))$ where $I^{tors}(X/\!\!/G)$ is the torsion inertia orbifold shown in Section \ref{def3}.\end{remark}

%Moreover, since $$(G\times_H X)^{\sigma}\cong\prod_{\tau}C_G(\sigma)\times_{C_H(\tau)}
%X^{\tau},$$   where $H$ is a closed subgroup of $G$ and the direct product goes over all the conjugacy classes $\tau\in
%H^{tors}_{conj}$ conjugate to $\sigma$ in $G$.


We also have the change-of-group isomorphism as  in equivariant
$K$-theory.

Let $H$ be a subgroup of $G$ and $X$ a $H$-space. Let $\phi:
H\longrightarrow G$ denote the inclusion homomorphism. The
change-of-group map $\rho^G_H: QEll^*_G(G\times_HX)\longrightarrow
QEll^*_H(X)$ is defined as the composite
\begin{equation}\rho^G_H:   QEll^*_G(G\times_HX)\buildrel{\phi^*}\over\longrightarrow
QEll^*_H(G\times_H X)\buildrel{i^*}\over\longrightarrow
QEll_H^*(X)\label{changeofgroup}
\end{equation}
where $\phi^*$ is the restriction map and $i: X\longrightarrow
G\times_HX$ is the $H-$equivariant map defined by $i(x)=[e, x].$

\begin{proposition} The change-of-group map
$$\rho^G_H: QEll^*_G(G\times_H X)\longrightarrow
QEll^*_H(X)$$ defined in (\ref{changeofgroup}) is an
isomorphism.\end{proposition}
\begin{proof}
For any $\tau\in H_{conj}$, there exists a unique
$\sigma_{\tau}\in G_{conj}$ such that
$\tau=g_{\tau}\sigma_{\tau}g_{\tau}^{-1}$ for some $g_{\tau}\in
G$.  Consider the maps \begin{equation}\begin{CD}
\Lambda_G(\tau)\times_{\Lambda_H(\tau)}X^{\tau}@>{[[a, t], x
]\mapsto [a, x]}>> (G\times_H X)^{\tau}@>{[u, x]\mapsto
[g_{\tau}^{-1}u, x]}>>
(G\times_HX)^{\sigma}.\end{CD}\end{equation} The first map is
$\Lambda_G(\tau)-$equivariant and the second is equivariant with
respect to the homomorphism $c_{g_{\tau}}:
\Lambda_{G}(\sigma)\longrightarrow \Lambda_G(\tau)$ sending $[u,
t]\mapsto [g_{\tau} u g_{\tau}^{-1}, t]$. Taking a coproduct over
all the elements $\tau\in H_{conj}$ that are conjugate to
$\sigma\in G_{conj}$ in $G$, we get an isomorphism
$$\gamma_{\sigma}: \coprod_{\tau}\Lambda_G(\tau)\times_{\Lambda_H(\tau)} X^{\tau}\longrightarrow
(G\times_HX)^{\sigma}$$ which is $\Lambda_G(\sigma)-$equivariant
with respect to $c_{g_{\tau}}$. Then we have the map
\begin{equation}\gamma:=\prod_{\sigma\in G_{conj}}\gamma_{\sigma}:
\prod_{\sigma\in G_{conj}}
K^*_{\Lambda_G(\sigma)}(G\times_HX)^{\sigma}\longrightarrow
\prod_{\sigma\in
G_{conj}}K^*_{\Lambda_G(\sigma)}(\coprod_{\tau}\Lambda_G(\tau)\times_{\Lambda_H(\tau)}
X^{\tau})
\end{equation}


It is straightforward to check the change-of-group map coincide
with the composite \begin{align*} QEll^*_{G}(G\times_H
X)\buildrel{\gamma}\over\longrightarrow \prod_{\sigma\in
G_{conj}}K^*_{\Lambda_G(\sigma)}(\coprod_{\tau}\Lambda_G(\tau)\times_{\Lambda_H(\tau)}
X^{\tau})\longrightarrow &\prod_{\tau\in
H_{conj}}K^*_{\Lambda_H(\tau)}(X^{\tau})\\&=QEll^*_{H}(X)\end{align*}
with  the second map  the change-of-group isomorphism in
equivariant $K-$theory.
\end{proof}

\begin{remark}

As shown in (\ref{qecpt}), by computing the representation rings
$R\Lambda_G(g)$, we get $QEll^*_G(-)$ for single point space. For
each subgroup $H$ of $G$, we can compute $QEll_G^*(G/H)$ using the
change-of-group isomorphism.  Then, using Mayer-Vietoris sequence,
we can compute $QEll^*_G(-)$ for any $G$-CW complex by patching
the $G$-cells together.

\end{remark}








\section{Power Operation}\label{poweroperation}

%In Chapter \ref{poweroperation} I define power operation on quasi-elliptic cohomology. % why the one I defined is the right one

In Section \ref{s2complete} we define power operations for
equivariant quasi-elliptic cohomology $QEll_G^*(-)$. And we show
in Theorem \ref{main1p} that they satisfy the axioms that Ganter
established in her paper \cite{Gan06} for equivariant power
operations.



The power operation of quasi-elliptic cohomology is of the form
\begin{align*}\mathbb{P}_n=&\prod_{(\underline{g}, \sigma)\in
(G\wr\Sigma_n)_{conj}}\mathbb{P}_{(\underline{g},\sigma)}:&
\\
&QEll^*_G(X)\longrightarrow QEll^*_{G\wr\Sigma_n}(X^{\times n})
=&\prod_{(\underline{g}, \sigma)\in
(G\wr\Sigma_n)_{conj}}K_{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)}((X^{\times n})^{(\underline{g}, \sigma)}),\end{align*}
where $\mathbb{P}_n$ maps a bundle over the groupoid
$$\Lambda(X/\!\!/G)$$ to a bundle over
$$\Lambda(X^{\times n}/\!\!/(G\wr\Sigma_n)),$$ and each
$\mathbb{P}_{(\underline{g},\sigma)}$ maps a bundle over
$$\Lambda(X/\!\!/G)$$ to a $\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)-$bundle over the space $(X^{\times
n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma).$

%$$(X^{\times n})^{(\underline{g}, \sigma)}\cong \prod_k\prod_{(i_1, \cdots i_k)}X^{g_{i_k}\cdots
%g_{i_1}},$$ where $(i_1, \cdots i_k)$ goes over all the $k-$cycles of $\sigma$.






\bigskip


%\subsection{Relevant Construction}\label{lov}

%In Section \ref{equivpower} I construct the power operation of quasi-elliptic cohomology. Before that I present some relevant
%constructions which are necessary for the construction of the power operation.


We construct each $\mathbb{P}_{(\underline{g}, \sigma)}$ as the
composition below.

\begin{align*}QEll^*_G(X)&\buildrel{U^*}\over\longrightarrow
K^*_{orb}(\Lambda^1_{(\underline{g}, \sigma)}(X))
\buildrel{(\mbox{ })_k^{\Lambda}}\over\longrightarrow
K^*_{orb}(\Lambda^{var}_{(\underline{g}, \sigma)}(X))\\
&\buildrel{\boxtimes}\over\longrightarrow
K^*_{orb}(d_{(\underline{g},
\sigma)}(X))\buildrel{f^*_{(\underline{g},
\sigma)}}\over\longrightarrow
K^*_{\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)}((X^{\times
n})^{(\underline{g}, \sigma)}),\end{align*}
 where
$k\in\mathbb{Z}$ and $(i_1, \cdots i_k)$ goes over all the
$k-$cycles of $\sigma$. We will explain the first three functors
in detail in Section \ref{s2complete}. In Section \ref{111} we
construct the isomorphism $f_{(\underline{g}, \sigma)}$ between
the groupoid $$\Lambda(X^{\times n}/\!\!/(G\wr\Sigma_n))$$ and the
groupoid $d((X/\!\!/G)\wr\Sigma_n)$ constructed in Definition
\ref{dxgsigman}.











\subsection{Loop Space of Symmetric Power}\label{111}

Before constructing the power operation
%$$\mathbb{P}=\prod_{(\underline{g}, \sigma)\in (G\wr\Sigma_n)^{tors}_{conj}}\mathbb{P}_{(\underline{g},\sigma)}:
%QEll^*_G(X)\longrightarrow QEll^*_{G\wr\Sigma_n}(X^{\times n}),$$
$$\mathbb{P}_n: K_{orb}(\Lambda(X/\!\!/G))=QEll_G(X)\longrightarrow QEll_{G\wr\Sigma_n}(X^{\times n})= K_{orb}(\Lambda(X^{\times n}/\!\!/(G\wr\Sigma_n))),$$
to make it explicit we introduce the groupoid
$d((X/\!\!/G)\wr\Sigma_n)$ in Section \ref{setupad}, which is
isomorphic to the groupoid $\Lambda(X^{\times
n}/\!\!/(G\wr\Sigma_n))$, as shown in Theorem \ref{iprm}. With it,
it is convenient for constructing the explicit formula of the
power operation.


%In Section \ref{111} I study the symmetric powers of the global quotient orbifold $X/\!\!/G$ and
%$$\mathcal{L}((X/\!\!/G)\wr\Sigma_n)\cong\coprod_{[\underline{g}, \sigma]\in (G\wr\Sigma_n)^{tors}_{conj}}\mathcal{L}_{(\underline{g}, \sigma)}(X^{\times
%n})/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma).$$  The main conclusion is Theorem \ref{iptm}.




\subsubsection{The groupoid $d((X/\!\!/G)\wr\Sigma_n)$}\label{setupad}

In this section we first recall wreath product and introduce the
symmetric powers of orbifolds in Definition
\ref{symmetricpowergroupoid}. In the case of a global quotient
orbifold $X/\!\!/G$, the $n-$th symmetric power is $X^{\times
n}/\!\!/(G\wr\Sigma_n)$.

We first construct a groupoid $d_{(\underline{g}, \sigma)}(X)$ for
each  $(\underline{g}, \sigma)\in G\wr\Sigma_n$.  Then by
combining all the groupoids $d_{(\underline{g}, \sigma)}(X)$
together we construct $d((X/\!\!/G)\wr\Sigma_n)$.

\bigskip

Let $G$ be a finite group and $X$ a right $G-$space. Recall
$X/\!\!/G$
is the translation groupoid. Let $\Sigma_n$ denote the $n-$th symmetric group. % where should I use groupoid and where should I use orbifold

The wreath product $G\wr\Sigma_n$ has elements
$$(\underline{g}, \sigma)\in G^n\times\Sigma_n.$$ Its
multiplication is defined by $$(g_1, \cdots g_n, \sigma)\cdot
(h_1, \cdots h_n, \tau)=(g_1h_{\sigma^{-1}(1)}, \cdots
g_nh_{\sigma^{-1}(n)}, \sigma\tau).$$ $G\wr\Sigma_n$ acts on
$X^{\times n}$ by
\begin{equation}\underline{x}\cdot(\underline{g},
\sigma)=(x_{\sigma(1)}g_{\sigma(1)}, \cdots
x_{\sigma(n)}g_{\sigma(n)})\mbox{,               }\mbox{    for
any }\underline{x}\in X^{\times n}, \mbox{    }(\underline{g},
\sigma)\in G\wr\Sigma_n.\end{equation}

\begin{remark}
Consider the space $$\underline{\underline{n}}:=\{1, 2, \cdots
n\}$$ with discrete topology. The symmetric group $\Sigma_n$ acts
on $\underline{\underline{n}}$ by permutation. For each
$(\underline{g}, \sigma)\in G\wr\Sigma_n$, we have the commutative
diagram
\begin{equation}\begin{CD}G\times \underline{\underline{n}} @>{(\underline{g}, \sigma)}>> G\times\underline{\underline{n}}\\
@VVV   @VVV \\ \underline{\underline{n}} @>{\sigma}>>
\underline{\underline{n}}   \end{CD}\end{equation} where the
vertical maps are the projection to the second factor and
$(\underline{g}, \sigma)$ acts on $\underline{\underline{n}}$ by
$$(x, k)\cdot (\underline{g}, \sigma):= (xg_k, \sigma^{-1}(k))\mbox{,     for each    }(x, k)\in G\wr\Sigma_n.$$

So $G\wr\Sigma_n$ gives a family of bundle isomorphisms
$$G\times \underline{\underline{n}}\longrightarrow G\times \underline{\underline{n}}$$  between the $G-$bundles.

\label{wrbuniso}

\end{remark}


Generally, for any orbifold groupoid %define this?
$$\mathbb{G}=(\mathbb{G}_0, \mathbb{G}_1, s, t, u, i)$$ we can
define the $n-$th symmetric power $\mathbb{G}\wr\Sigma_n$ of
$\mathbb{G}$.

\begin{definition}[Symmetric Power of $\mathbb{G}$]
The objects of the $n-$th symmetric power $\mathbb{G}\wr\Sigma_n$
are the points $$\underline{x}:=(x_1, \cdots x_n)$$ in
$\mathbb{G}_0^{\times n}$ and the arrows of it are
$(\underline{g}, \sigma)$ with each $g_i$ an arrow of $\mathbb{G}$
starting at $x_i$ and $\sigma\in\Sigma_n$.
$$\underline{x}\cdot(\underline{g}, \sigma)=(x_{\sigma(1)} g_{\sigma(1)}, x_{\sigma(2)}g_{\sigma(2)}, \cdots
x_{\sigma(n)}g_{\sigma(n)}).$$
%$\mathbb{G}^{\times n}\times \Sigma_n$.The source of each morphism $(\underline{g}, \sigma)$ is
%$(s(g_{\sigma(1)}), \cdots s(g_{\sigma(n)}))$ and the target of it is $(t(g_1), \cdots t(g_n))$.
The unit map sends the object $\underline{x}$ to $(u(x_1), \cdots
u(x_n), Id)$. The composition of arrows $(\underline{g}, \sigma)$
and $(\underline{h}, \tau)$ where $g_{\sigma(i)}=h_i$ is given by
$$(\underline{g}, \sigma)\circ(\underline{h}, \tau)=
(g_1h_{\sigma^{-1}(1)}, \cdots g_nh_{\sigma^{-1}(n)}).$$ The
inverse of $(\underline{g}, \sigma)$ is given by
$$(i(g_{\sigma(1)}), \cdots i(g_{\sigma(n)}), \sigma^{-1}).$$
\label{symmetricpowergroupoid}
\end{definition}




%\begin{example} The symmetric powers of Loop spaces $L(X/\!\!/G)$
%\label{symloop}\end{example}
The symmetric power $(X/\!\!/G)\wr\Sigma_n$ is isomorphic to the
translation groupoid $X^{\times n}/\!\!/(G\wr\Sigma_n)$. And note
that when $G$ is finite, $L^1_{g}G\rtimes\mathbb{T}$ is isomorphic
to $\Lambda_G(g)$.
%the orbifold loop space
So the groupoid $\Lambda((X/\!\!/G)\wr\Sigma_n)$ is isomorphic to
$\Lambda(X^{\times n}/\!\!/(G\wr\Sigma_n))$ which is defined in
Definition \ref{lambdaoidef}.


We also need the definition below.
\begin{definition}
Let $C_G(g, g')$ denote the set $$\{x\in G| gx=xg'\}.$$

Let $C_G^k(g)$ denote the quotient group of $C_G(g)\times
\mathbb{Z}/kl\mathbb{Z}$ by the normal subgroup generated by $(g,
-k)$. And let $C^k_G(g, g')$ denote the quotient of $C_G(g,
g')\times \mathbb{Z}/kl\mathbb{Z}$ under the equivalence $$(x,
t)\sim (gx, t-k)=(xg', t-k).$$  Let $\Lambda^k_G(g, g')$ denote
the quotient of $C_G(g, g')\times \mathbb{R}/kl\mathbb{Z}$ under
the equivalence $$(x, t)\sim (gx, t-k)=(xg',
t-k).$$\label{lambdak1}\end{definition}



Let $\sigma\in\Sigma_n$ correspond to the partition $n =
\sum\limits_kkN_k$, i.e. it has $N_k$ $k-$cycles. Assume that for
each cycle $(i_1, \cdots i_k)$ of $\sigma$,  $i_1< i_2\cdots <
i_k$.



For $(\underline{g}, \sigma)\in G\wr\Sigma_n$, we consider the
orbits of the bundle
$G\times\underline{\underline{n}}\longrightarrow\underline{\underline{n}}$
under the action by $(\underline{g}, \sigma)$. The orbits of
$\underline{\underline{n}}$ under the action by $\sigma$
corresponds to the cycles in the cycle decomposition of $\sigma$.
The bundle
$G\times\underline{\underline{n}}\longrightarrow\underline{\underline{n}}$
is the disjoint union of the $G-$bundles

$$\bigsqcup_{(i_1 \cdots i_k)} (G\times \{i_1, \cdots i_k\}\longrightarrow \{i_1, \cdots i_k\})$$ where $(i_1, \cdots i_k)$ goes over all the
cycles of $\sigma$. Each bundle $G\times \{i_1, \cdots
i_k\}\longrightarrow \{i_1, \cdots i_k\}$ is an orbit of
$G\times\underline{\underline{n}}\longrightarrow\underline{\underline{n}}$
under the action by $(\underline{g}, \sigma)$.
%$G\rtimes\langle (\underline{g}, \sigma)\rangle-$action.

Two $G-$subbundles $$G\times \{i_1, \cdots i_k\}\longrightarrow
\{i_1, \cdots i_k\}\mbox{    and     }G\times \{j_1, \cdots
j_m\}\longrightarrow \{j_1, \cdots j_m\}$$ are $(\underline{g},
\sigma)-$equivariant equivalent if and only if $k=m$ and
$C_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})$ is nonempty.
For each $k$-cycle $i=(i_1, \cdots i_k)$ of $\sigma$, let
$W^{\sigma}_i$ denote the set of all the $G-$subbundles $G\times
\{j_1, \cdots j_m\}\longrightarrow \{j_1, \cdots j_m\}$ that are
$(\underline{g},  \sigma)-$isomorphic to $G\times \{i_1, \cdots
i_k\}\longrightarrow \{i_1, \cdots i_k\}$. There is a bijection
between $W^{\sigma}_i$ and the set $$\{j=(j_1, \cdots j_k)\mbox{ }
| \mbox{  }(j_1, \cdots j_k)\mbox{ is a k-cycle of }\sigma\mbox{
and } C_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})\mbox{ is
nonempty}\}.$$ Let $M^{\sigma}_{i}$ denote the size of the set
$W^{\sigma}_i$.  Let $\alpha^i_1, \alpha^i_2, \cdots
\alpha^i_{M^{\sigma}_i}$ denote all the elements of the set
$W^{\sigma}_i$. Obviously, $i=(i_1, \cdots i_k)$ is in
$W^{\sigma}_i$. So we can assume $\alpha^i_1$ is $i$.





For any $k-$cycle $i$ and $m-$cycle $j$ of $\sigma$, if $k\neq m$,
$W^{\sigma}_i$ and $W^{\sigma}_j$ are disjoint. In the case that
$k=m$, if $C_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})$ is
nonempty,  $W^{\sigma}_i$ and $W^{\sigma}_j$ are the same set;
otherwise, $W^{\sigma}_i$ and $W^{\sigma}_j$ are disjoint. The set
of all the $k-$cycles of $\sigma$ can be divided into the disjoint
union of several $W^{\sigma}_i$s. We can pick a set of
representatives $\theta_k$ of $k-$cycles of $\sigma$ such that the
set of $k-$cycles of $\sigma$ equals the disjoint union
$$\coprod_{i\in\theta_k}W^{\sigma}_i.$$


\begin{definition}[The groupoid  $d_{(\underline{g}, \sigma)}(X)$]
The objects of $d_{(\underline{g}, \sigma)}(X)$ are the points of
the space
$$\prod_k\prod_{(i_1, \cdots
i_k)}X^{g_{i_k}\cdots g_{i_1}},$$ where the second product goes
over all the $k-$cycles of $\sigma$.  The set of morphisms
starting from an object
$$x=\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots \gamma_{i,
M^{\sigma}_i})\in\prod\limits_k\prod\limits_{(i_1, \cdots
i_k)\in\theta_k}X^{g_{i_k}\cdots g_{i_1}}$$ is the subspace of
\begin{align*}\{(\times_{k}\times_{i\in\theta_k}&([a^i_1, t^i_1], \cdots
[a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i), x)\mbox{
  }|\mbox{   }\tau_i\in\Sigma_{M^{\sigma}_i}, \\ &[a^i_j,
t^i_j]\in\Lambda^k_G(g_{\tau_i(j)_k}\cdots g_{\tau_i(j)_1},
g_{j_k}\cdots g_{j_1}) \mbox{  for each } j \in W^{\sigma}_i.\\
&\mbox{  All the   }t^i_j\mbox{s have the same image under the
quotient map
 }\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}.\}\end{align*}
where $\tau_i$ in each morphism permutes the cycles $\alpha^i_1,
\alpha^i_2, \cdots \alpha^i_{M^{\sigma}_i}$ in $W^{\sigma}_i$.

The target of the morphism
$(\times_{k}\times_{i\in\theta_k}([a^i_1, t^i_1], \cdots
[a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i),x )$ is
\begin{align*}&\big(\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots  \gamma_{i,
M^{\sigma}_i})\big) \cdot (\times_{k}\times_{i\in\theta_k}([a^i_1,
t^i_1], \cdots [a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i),x
)\\=&\times_k\times_{i\in\theta_k}(\gamma_{i, \tau_i(1)}\cdot
[a^i_{\tau_i(1)}, t^i_{\tau_i(1)}], \cdots \gamma_{i,
\tau_i(M^{\sigma}_i)}\cdot [a^i_{\tau_i(M^{\sigma}_i)},
t^i_{\tau_i(M^{\sigma}_i)}]),\end{align*} where each $(\gamma_{i,
\tau_i(j)}\cdot[a^i_{\tau_i(j)}, t^i_{\tau_i(j)}])(t)=(\gamma_{i,
\tau_i(j)}\cdot a^i_{\tau_i(j)})(t+t^i_{\tau_i(j)}),$ as shown in
(\ref{chongaction}).

The composition of two morphisms is defined by
\begin{align}&\bigg(\times_{k}\times_{i\in\theta_k}([a^i_1, t^i_1], \cdots
[a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i)\bigg)\circ
\bigg(\times_{k}\times_{i\in\theta_k}([b^i_1, p^i_1], \cdots
[b^i_{M^{\sigma}_i}, p^i_{M^{\sigma}_i}],
\varrho_i)\bigg)\\=&\times_{k}\times_{i\in\theta_k}([a^i_1b^i_{\tau_i^{-1}(1)},
t^i_1+p^i_{\tau_i^{-1}(1)}], \cdots
[a^i_{M^{\sigma}_i}b^i_{\tau_i^{-1}(M^{\sigma}_i)},
t^i_{M^{\sigma}_i}+p^i_{\tau_i^{-1}(M^{\sigma}_i)}],
\tau_i\varrho_i)\label{compositionagsigma} \end{align}
\label{ADgroupoid}
\end{definition}


%There is more general case than Theorem \ref{iptm}. %In light of Theorem \ref{iptm},
Combining all the groupoids $d_{(\underline{g}, \sigma)}(X)$, we
can define the groupoid $$d((X/\!\!/G)\wr\Sigma_n).$$ A skeleton
for $d((X/\!\!/G)\wr\Sigma_n)$ is the subgroupoid of the disjoint
union $$\coprod\limits_{(\underline{g},
\sigma)\in(G\wr\Sigma_n)_{conj}}d_{(\underline{g}, \sigma)}(X)$$
whose morphisms consists of those with all the rotation $t^i_j$s
having the same image under the quotient map
$\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}$.

\bigskip

\begin{definition}[$d((X/\!\!/G)\wr\Sigma_n)$]
%There is more general case than Theorem \ref{iptm}. %In light of Theorem \ref{iptm},
Let  $d((X/\!\!/G)\wr\Sigma_n)$ denote the groupoid with objects
the points of the space
$$\coprod\limits_{(\underline{g},
\sigma)\in(G\wr\Sigma_n)}\prod_k\prod_{(i_1, \cdots
i_k)}X^{g_{i_k}\cdots g_{i_1}}$$ where $(i_1, \cdots i_k)$ goes
over all the $k-$cycles of $\sigma$. The set of morphisms starting
from an object $$x\in\prod_k\prod_{(j_1, \cdots
j_k)}X^{g_{j_k}\cdots g_{j_1}}$$ to an object in the component
$\prod\limits_k\prod\limits_{(i_1, \cdots i_k)}X^{g'_{i_k}\cdots
g'_{i_1}}$ corresponding to $(\underline{g}', \sigma')$ is
\begin{align*}\{(\times_{k}\times_{j\in\theta_k}&([a^j_1, t^j_1], \cdots
[a^j_{M^{\sigma}_j}, t^j_{M^{\sigma}_j}], \tau_j), x)\mbox{
   }|\mbox{    }\tau_j\in\Sigma_{M^{\sigma}_j}, \\ &[a^j_{\rho},
t^j_{\rho}]\in\Lambda^k_G(g'_{\tau_j(\rho)_k}\cdots
g'_{\tau_j(\rho)_1}, g_{\rho_k}\cdots g_{\rho_1})\mbox{  for each
} \rho \in W^{\sigma}_j;
\\ &\mbox{  all the }t^j_{\rho}\mbox{s have the
same image under the quotient map   }
\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}.
\},\end{align*} where $\tau_j$ permutes the cycles $\alpha^j_1,
\alpha^j_2, \cdots \alpha^j_{M^{\sigma}_j}$ in $W^{\sigma}_j$.

For any $x=\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots
\gamma_{i, M^{\sigma}_i})\in\prod\limits_k\prod\limits_{(i_1,
\cdots i_k)\in\theta_k}X^{g_{i_k}\cdots g_{i_1}}$,
\begin{align*}&\big(\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots  \gamma_{i,
M^{\sigma}_i})\big) \cdot (\times_{k}\times_{i\in\theta_k}([a^i_1,
t^i_1], \cdots [a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i),
x)\\=&\times_k\times_{i\in\theta_k}(\gamma_{i,
\tau_i(1)}[a^i_{\tau_i(1)}, t^i_{\tau_i(1)}], \cdots \gamma_{i,
\tau_i(M^{\sigma}_i)}[a^i_{\tau_i(M^{\sigma}_i)},
t^i_{\tau_i(M^{\sigma}_i)}]),\end{align*} where each $(\gamma_{i,
\tau_i(j)}\cdot[a^i_{\tau_i(j)}, t^i_{\tau_i(j)}])(t)=(\gamma_{i,
\tau_i(j)}\cdot a^i_{\tau_i(j)})(t+t^i_{\tau_i(j)}).$

The formula of the composition of two morphisms is %check these formulas
\begin{align}&\big(\times_{k}\times_{i\in\theta_k}([a^i_1, t^i_1], \cdots
[a^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}], \tau_i)\big)\circ
\big(\times_{k}\times_{i\in\theta_k}([b^i_1, p^i_1], \cdots
[b^i_{M^{\sigma}_i}, p^i_{M^{\sigma}_i}],
\varrho_i)\big)\\=&\times_{k}\times_{i\in\theta_k}([a^i_1b^i_{\tau_i^{-1}(1)},
t^i_1+p^i_{\tau_i^{-1}(1)}], \cdots
[a^i_{M^{\sigma}_i}b^i_{\tau_i^{-1}(M^{\sigma}_i)},
t^i_{M^{\sigma}_i}+p^i_{\tau_i^{-1}(M^{\sigma}_i)}],
\tau_i\varrho_i)\label{compositionagsigma} \end{align}
\label{dxgsigman}
\end{definition} %% this definition is better after the isomorphism theorem. And note for different component, t_0 are the same under R/kZ\longrightarrow R/Z.



Proposition \ref{trans} shows the groupoid $d_{(\underline{g},
\sigma)}(X)$ is a translation groupoid. Thus, to study
$K_{orb}(d((X/\!\!/G)\wr\Sigma_n))$, we can start by
 studying the representation ring of a group.

Let $\Gamma_{i, p}$ denote the element
\begin{equation}g_{{\alpha^i_p}_k}\cdots g_{{\alpha^i_p}_1},\label{gammaip}\end{equation} where
$\alpha^i_p=((\alpha^i_p)_1, \cdots (\alpha^i_p)_k)$ is an element
in $W^{\sigma}_i$ and $p=1, 2\cdots M^{\sigma}_{i}$. Assume the
representative $i=(i_1, \cdots i_k)$ of $W^{\sigma}_i$ is
$\alpha^k_1$.

We consider the group
$$\prod_{k}\prod_{i\in\theta_k}\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i}.$$ For each $[b^i_j, t^i_j]\in
\Lambda^k_G(\Gamma_{i, 1}),$ the product $$[\sigma_{i, \tau(j),
1}, 0][b^i_j, t^i_j][\sigma_{i, 1, j}, 0]\in
\Lambda^k_G(\Gamma_{i, \tau(j)}, \Gamma_{i, j}).$$ It acts on any
$\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots \gamma_{i,
M^{\sigma}_i})\in\prod\limits_k\prod\limits_{(i_1, \cdots
i_k)}X^{g_{i_k}\cdots g_{i_1}}$ by
\begin{align*}&(\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots  \gamma_{i,
M^{\sigma}_i})) \cdot (\times_{k}\times_{i\in\theta_k}([b^i_1,
t^i_1], \cdots [b^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}],
\tau_i))\\:=&(\times_k\times_{i\in\theta_k}(\gamma_{i, 1}, \cdots
\gamma_{i, M^{\sigma}_i}))\mbox{       } \cdot\\
&(\times_{k}\times_{i\in\theta_k}([\sigma_{i, \tau(1), 1},
0][b^i_1, t^i_1][\sigma_{i, 1, 1}, 0], \cdots [\sigma_{i,
\tau(M^{\sigma}_i), 1}, 0][b^i_{M^{\sigma}_i},
t^i_{M^{\sigma}_i}][\sigma_{i, 1, M^{\sigma}_i}, 0],
\tau_i))\\=&\times_k\times_{i\in\theta_k}(\gamma_{i,
\tau_i(1)}([\sigma_{i, \tau(1), 1}, 0][b^i_1, t^i_1][\sigma_{i, 1,
1}, 0]),  \\ &\cdots \gamma_{i, \tau_i(M^{\sigma}_i)}([\sigma_{i,
\tau(M^{\sigma}_i), 1}, 0][b^i_{M^{\sigma}_i},
t^i_{M^{\sigma}_i}][\sigma_{i, 1, M^{\sigma}_i}, 0])).\end{align*}

Let
$$\Lambda_G^k(\Gamma_{i,
1})\wr_{\mathbb{T}}\Sigma_{M^{\sigma}_i}$$ denote the subgroup of
the wreath product $\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i}$ consisting of those elements
$$([b^i_1, t^i_1], \cdots
[b^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}],    \tau_i)$$ with all
the $t^i_j$s having the same image under the quotient map
$\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}$,
for each $j=1, \cdots M^{\sigma}_i$.

And let
$$\prod\limits_{k}\!{_{\mathbb{T}}}\prod\limits_{i\in\theta_k}\!{_{\mathbb{T}}}\Lambda_G^k(\Gamma_{i,
1})\wr_{\mathbb{T}}\Sigma_{M^{\sigma}_i}$$ denote the subgroup of
$\prod\limits_{k}\prod\limits_{i\in\theta_k}\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i}$   consisting of those elements
$$(\times_{k}\times_{i\in\theta_k}([b^i_1, t^i_1], \cdots
[b^i_{M^{\sigma}_i}, t^i_{M^{\sigma}_i}],    \tau_i))$$ with all
the $t^i_j$s having the same image under the quotient map
$\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}$,
for each $k$, each $k-$cycle $i$, and each $j=1, \cdots
M^{\sigma}_i$.


\begin{proposition}
Each $d_{(\underline{g}, \sigma)}(X)$ is isomorphic to the
translation groupoid $$\big(\prod_k\prod_{(i_1, \cdots
i_k)}X^{g_{i_k}\cdots
g_{i_1}}\big)/\!\!/\big(\prod\limits_{k}\!{_{\mathbb{T}}}\prod\limits_{i\in\theta_k}\!{_{\mathbb{T}}}\Lambda_G^k(\Gamma_{i,
1})\wr_{\mathbb{T}}\Sigma_{M^{\sigma}_i}\big)$$ %where $\Gamma_{i, p}$ is defined in (\ref{gammaip}).
\label{trans}\end{proposition} The proof is straightforward.





To study $K_{orb}(d_{(\underline{g}, \sigma)}(X))$, we need to
study the representation ring of the wreath product
$$\prod\limits_{k}\prod\limits_{i\in\theta_k}\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i}.$$ Theorem \ref{repwr} gives all the
irreducible representations of a wreath product. It is a
conclusion from \cite{JK81}.

\begin{theorem}Let $\{\rho_k\}^N_1$ be a complete family of irreducible representations of $G$ and let $V_k$ be the corresponding representation space for $\rho_k$. Let $(n)$ be a partition
of $n$. $(n)=(n_1, \cdots n_N).$ Let $D_{(n)}$ be the
representation $$\rho_1^{\otimes
n_1}\otimes\cdots\otimes\rho_N^{\otimes n_N}$$ of $G^{\times N}$
on $V_1^{\otimes n_1}\otimes\cdots\otimes V_N^{\otimes n_N}.$ Let
$\Sigma_{(n)}=\Sigma_{n_1}\times\cdots\times\Sigma_{n_N}.$

Let $(D_{(n)})^{\sim}$ be the extension of $D_{(n)}$ from
$G^{\times n}$ to $G\wr\Sigma_{(n)}$ defined by
\begin{align*}&(D_{(n)})^{\sim}((g_{1, 1},\cdots g_{1, n_1}, \cdots g_{N, 1}, \cdots g_{N, n_N}; \sigma))\\ &(v_{1,
1}\otimes\cdots\otimes v_{1, n_1}\otimes\cdots\otimes
v_{N,1}\otimes\cdots\otimes v_{N,
n_N})\\=&\bigotimes^{N}_{k=1}\rho_k(g_{k,
1})v_{k,\sigma_k^{-1}(1)}\otimes\cdots\otimes\rho_k(g_{k,
n_k})v_{k,\sigma_k^{-1}(n_k)},\end{align*} where
$\sigma=\sigma_1\times\cdots\times\sigma_N$ with each
$\sigma_k\in\Sigma_{n_k}$.

Let $D_{\tau}$ with $\tau\in R\Sigma_{(n)}$ be the representation
of $G\wr\Sigma_{(n)}$ defined by \begin{equation}D_{\tau}((g_{1,
n_1},\cdots g_{N,n_N};
\sigma)):=\tau(\sigma).\label{drepsymm}\end{equation}
%acting on $V_1^{\otimes n_1}\otimes\cdots\otimes V_N^{\otimes
%n_N}$ by
%$$D_{\tau}((g_1,\cdots g_n, \sigma))(v_1\otimes\cdots\otimes v_n)=g_{\tau(\sigma)(1)}v_{\tau(\sigma)(1)}\otimes\cdots\otimes g_{\tau(\sigma)(n)}v_{\tau(\sigma)(n)}.$$

Then,
\begin{align*}\{Ind|^{G\wr\Sigma_n}_{G\wr\Sigma_{(n)}}(D_{(n)})^{\sim}\otimes
D_{\tau}| &(n)=(n_1, \cdots n_N)\mbox{  goes over all the partitions};\\
&\tau\mbox{    goes over all the irreducible representations of }
\Sigma_{(n)}.\}\end{align*}goes over all the irreducible
representations  of $G\wr\Sigma_{n}$
nonrepeatedly.\label{repwr}\end{theorem}


Let
$$\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_n$$ denote the subgroup of
the wreath product $\Lambda_G(\sigma)\wr\Sigma_n$ consisting of
those elements
$$([b^i_1, t^i_1], \cdots
[b^i_{n}, t^i_{n}],    \tau_i)$$ with all the $t^i_j$s having the
same image under the quotient map
$\mathbb{R}/k{\mathbb{Z}}\longrightarrow\mathbb{R}/\mathbb{Z}$,
for each $j=1, \cdots n$.

And we can get all the irreducible representations of the fibred
wreath product $$\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_n$$ in a
similar way.

The proof of Theorem \ref{repfibwr} is analogous to that of
Theorem \ref{repwr} in \cite{JK81}, applying Clifford's theory in
\cite{Clifford1937} and \cite{CurtisReiner62}. Note that
$$\{\rho_1\otimes_{\mathbb{Z}[q^{\pm}]}\cdots
\otimes_{\mathbb{Z}[q^{\pm}]} \rho_n\mbox{    }|\mbox{   Each
}\rho_j\mbox{    is    an irreducible representation of     }
\Lambda_G(\sigma).\}$$ goes over all the irreducible
representations of the fibred product
$$\Lambda_G(\sigma)\times_{\mathbb{T}}\cdots \times_{\mathbb{T}}
\Lambda_G(\sigma).$$



\begin{theorem}Let $\{\rho_k\}^N_1$ be a basis of the $\mathbb{Z}[q^{\pm}]-$module $R\Lambda_G(\sigma)$ and let $V_k$ be the corresponding representation space for $\rho_k$.
Let $(n)$ be a partition of $n$. $(n)=(n_1, \cdots n_N).$ Let
$D^{\mathbb{T}}_{(n)}$ be the
$\Lambda_G(\sigma)^{\times_{\mathbb{T}} n}-$representation
$$\rho_1^{\otimes_{\mathbb{Z}[q^{\pm}]}
n_1}\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}\rho_N^{\otimes_{\mathbb{Z}[q^{\pm}]}
n_N}$$ on the space $V_1^{\otimes n_1}\otimes\cdots\otimes
V_N^{\otimes n_N}.$ Let
$\Sigma_{(n)}=\Sigma_{n_1}\times\cdots\times\Sigma_{n_N}.$

Let $(D^{\mathbb{T}}_{(n)})^{\sim}$ be the extension of $D_{(n)}$
from $\Lambda_G(\sigma)^{\times_{\mathbb{T}} n}$ to
$\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_{(n)}$ defined by
\begin{align*}&(D_{(n)}^{\mathbb{T}})^{\sim}(([g_{1, 1}, t],\cdots [g_{1, n_1},  t], \cdots [g_{N, 1},  t], \cdots [g_{N, n_N},   t]; \sigma))
\\ &(v_{1, 1}\otimes\cdots\otimes v_{1, n_1}\otimes\cdots\otimes
v_{N,1}\otimes\cdots\otimes v_{N,
n_N})\\=&{\bigotimes\nolimits_{\mathbb{Z}[q^{\pm}]}}\rho_k([g_{k,
1},
t])v_{k,\sigma_k^{-1}(1)}\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}\rho_k([g_{k,
n_k},   t])v_{k,\sigma_k^{-1}(n_k)},\end{align*} where $k$ is from
$1$ to $N$ and $\sigma=\sigma_1\times\cdots\times\sigma_N$ with
each $\sigma_k\in\Sigma_{n_k}$.

Let $D^{\mathbb{T}}_{\tau}$ with $\tau\in R\Sigma_{(n)}$ be the
representation of $\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_{(n)}$
defined by \begin{equation}D^{\mathbb{T}}_{\tau}(([g_{1, n_1},
t],\cdots [g_{N,n_N},   t];
\sigma)):=\tau(\sigma).\label{drepsymm}\end{equation}
%acting on $V_1^{\otimes n_1}\otimes\cdots\otimes V_N^{\otimes
%n_N}$ by
%$$D_{\tau}((g_1,\cdots g_n, \sigma))(v_1\otimes\cdots\otimes v_n)=g_{\tau(\sigma)(1)}v_{\tau(\sigma)(1)}\otimes\cdots\otimes g_{\tau(\sigma)(n)}v_{\tau(\sigma)(n)}.$$

Then,
\begin{align*}\{Ind|^{\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_n}_{\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_{(n)}}&(D^{\mathbb{T}}_{(n)})^{\sim}\otimes
D^{\mathbb{T}}_{\tau}\mbox{   }|\mbox{    } (n)=(n_1, \cdots n_N)\mbox{  goes over all the partitions};\\
&\tau\mbox{    goes over all the irreducible representations of }
\Sigma_{(n)}.\}\end{align*}goes over all the irreducible
representation nonrepeatedly of
$\Lambda_G(\sigma)\wr_{\mathbb{T}}\Sigma_{n}$.\label{repfibwr}\end{theorem}%check

From Theorem \ref{repwr}, the representation ring of each
$\Lambda_G^k(\Gamma_{i, 1})\wr\Sigma_{M^{\sigma}_i}$ is a
$\mathbb{Z}[q^{\pm\frac{1}{k}}]-$module. Thus, the representation
ring of each $\Lambda_G^k(\Gamma_{i, 1})\wr\Sigma_{M^{\sigma}_i}$
is a $\mathbb{Z}[q^{\pm}]-$module via the
%inclusion
map
$$\mathbb{Z}[q^{\pm}]\longrightarrow\mathbb{Z}[q^{\pm\frac{1}{k}}]\mbox{,
}q\mapsto q^{\pm\frac{1}{k}}.$$ And the representation ring
$$R(\prod_{k}\prod_{i\in\theta_k}\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i})\cong
\bigotimes_{k}\bigotimes_{i\in\theta_k}R(\Lambda_G^k(\Gamma_{i,
1})\wr\Sigma_{M^{\sigma}_i})$$ is a $\mathbb{Z}[q^{\pm}]-$module.

Moreover, $K_{orb}(d_{(\underline{g}, \sigma)}(X))$ is a
$\mathbb{Z}[q^{\pm}]-$module via the map
\begin{equation}R(\prod\limits_{k}\!{_{\mathbb{T}}}\prod\limits_{i\in\theta_k}\!{_{\mathbb{T}}}\Lambda_G^k(\Gamma_{i,
1})\wr_{\mathbb{T}}\Sigma_{M^{\sigma}_i})\cong
K_{orb}^0(d_{(\underline{g}, \sigma)}(\mbox{pt}))\longrightarrow
K_{orb}^0(d_{(\underline{g}, \sigma)}(X)),\end{equation} which is
induced by $X\longrightarrow\mbox{pt}$.




\subsubsection{The isomorphism $f_{(\underline{g}, \sigma)}$}\label{FunctorF}

In this section  we show in Theorem \ref{iprm} that the groupoid
$\Lambda(X^{\times n}/\!\!/(G\wr\Sigma_n))$ is isomorphic to the
groupoid $d((X/\!\!/G)\wr\Sigma_n)$ defined in Definition
\ref{dxgsigman}.


\bigskip

First we show each component $(X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$ is
isomorphic to the groupoid $d_{(\underline{g}, \sigma)}(X)$.


First we recall some properties of
$C_{G\wr\Sigma_n}((\underline{g}, \sigma), (\underline{g}',
\sigma'))$.

%For $(\underline{h}, \tau)\in C_{G\wr \Sigma_n} ((\underline{g},\sigma), (\underline{g}',\sigma'))$, there are
%similar equivalences to those in $C_{G\wr\Sigma_n}((\underline{g},\sigma))$.
$(\underline{h}, \tau)$ is in $C_{G\wr \Sigma_n}
((\underline{g},\sigma), (\underline{g}',\sigma'))$ if and only if
$\tau\sigma'=\sigma\tau$ and
$g_{\sigma(\tau(i))}h_{\tau(i)}=h_{\tau(\sigma'(i))}g'_{\sigma'(i)},$
$\forall i.$

We can reinterpret the two conditions above. Since $\tau\in
C_{\Sigma_n}(\sigma, \sigma')$, $\tau$ maps a $k$-cycle
$i=(i_1,\cdots i_k)$ of $\sigma'$ to a $k$-cycle $j=(j_1, \cdots
j_k)$ of $\sigma$. $\tau$ will still used to denote its map on the
cycles, such as $\tau(r)=s$. For each
$l\in\mathbb{Z}/k\mathbb{Z}$, let $\tau(i_l)=j_{l+m_i}$ where
$m_i$ depends only on $\tau$ and the cycle $i.$ Then, the second
conditions can be expressed as
\begin{equation}\forall l\in\mathbb{Z}/k\mathbb{Z},
g_{j_l}h_{j_{l-1}}=h_{j_l}g'_{i_{l-m_i}}.\label{rela2}\end{equation}

From this equivalence, we can induce that the element
$$h_{j_k}g'^{-1}_{i_{1-m_i}}\cdots
g'^{-1}_{i_{k-1}}g'^{-1}_{i_k}=g^{-1}_{j_1}\cdots g^{-1}_{j_{m_i}}
h_{j_{m_i}}$$ maps $g_{j_k}\cdots g_{j_1}$ to $g'_{i_k}\cdots
g'_{i_1}$ by conjugation. In other words,
\begin{equation}\beta^{\underline{h}, \tau}_{j,
i}:=h_{j_k}g'^{-1}_{i_{1-m_i}}\cdots
g'^{-1}_{i_{k-1}}g'^{-1}_{i_k}\label{betadef}\end{equation} is an
element in $C_G (g_{j_k}\cdots g_{j_1}, g'_{i_k}\cdots g'_{i_1})$.
Thus, $C_G (g_{j_k}\cdots g_{j_1}, g'_{i_k}\cdots g'_{i_1})$ is
nonempty.



We construct a functor $$f_{(\underline{g}, \sigma)}: (X^{\times
n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)
\longrightarrow d_{(\underline{g}, \sigma)}(X).$$ It sends a point
$$x=(x_1, \cdots x_n)\in
(X^{\times n})^{(\underline{g}, \sigma)}$$ to the product of
$\sum\limits_k N_k$ paths,
$$\prod_k\prod_{(i_1, \cdots
i_k)}x_{i_k},$$ which is an object of $d_{(\underline{g},
\sigma)}(X)$. Note that $x_{i_k}=x_{i_1}g_{i_1}=\cdots
=x_{i_{k-1}}g_{i_{k-1}}\cdots g_{i_1}.$

Let $[(\underline{h}, \tau), t]\in
\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$.

Let $\tau$ send the $k-$cycle $i=(i_1, \cdots i_k)$ of $\sigma$ to
a $k-$cycle $j=(j_1, \cdots j_k)$ of $\sigma$ and
$\tau(i_1)=j_{1+m_i}$. According to the definition above,
$$f_{(\underline{g}, \sigma)}( \gamma\cdot [(\underline{h}, \tau),
t_0])= \prod_k\prod_{(i_1, \cdots
i_k)}x_{j_{m_i}}h_{j_{m_i}}=\prod_k\prod_{(i_1, \cdots
i_k)}x_{j_k}\cdot\beta^{\underline{h}, \tau}_{j, i},$$ where
$\beta^{\underline{h}, \tau}_{j, i}$ is the symbol defined in
(\ref{betadef}).


$f_{(\underline{g}, \sigma)}$ maps the morphism $[(\underline{h},
\tau), t]$ to
$$\times_k\times_{i\in\theta_k}([\beta^{\underline{h}, \tau}_{\tau(1), 1}, m_1+t], \cdots [\beta^{\underline{h}, \tau}_{\tau(M^{\sigma}_i), M^{\sigma}_i}, m_{M^{\sigma}_i}+t],
\tau|_{W^{\sigma}_i})$$ where $\tau|_{W^{\sigma}_i}$ denotes the
permutation induced by $\tau$ on the set
$W^{\sigma}_i=\{\alpha^i_1, \alpha^i_2, \cdots
\alpha^i_{M^{\sigma}_i}\}$, $\tau^{-1}(j)$ is short for $\tau^{-1}
(\alpha^i_j)$ and $\tau(j_l)=\tau(j)_{l+m_j}$.

It sends the identity map $[(1, \cdots, 1, \mbox{Id}), 0]$ to the
identity map $$\times_k\times_{i\in\theta_k}([1, 0], \cdots [1,
0], \mbox{Id}).$$ And $f_{(\underline{g}, \sigma)}$ preserves
composition of morphisms. So it is a well-defined functor.


\begin{theorem} The two groupoids $(X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$ and
$d_{(\underline{g}, \sigma)}(X)$ are isomorphic. Thus,
%they are Morita equivalent and
this isomorphism induces a $\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)-$action on the space $$\prod_k\prod_{(i_1, \cdots
i_k)}X^{g_{i_k}\cdots g_{i_1}}.$$
 \label{iptm}
\end{theorem}
\begin{proof}

We construct $J_{(\underline{g}, \sigma)}:  d_{(\underline{g},
\sigma)}(X)\longrightarrow (X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$. For
an object $ \times_k\times_{i\in\theta_k}\nu_{i, k}$ in
$d_{(\underline{g}, \sigma)}(X)$, $J_{(\underline{g},
\sigma)}(\times_k\times_{i\in\theta_k}\nu_{i, k})=\{\nu_m\}_1^n$
 with
$$\nu_{i_k}=\nu_{i, k}|_{[0, 1]}$$ and
$$\nu_{i_s}(t):=\nu_{i, k}(s+t)g_{i_1}^{-1}\cdots g_{i_s}^{-1}.$$

Let
$$\prod_k\prod_{i\in\theta_k}((u^i_1,m'^i_1),(u^i_2,
m'^i_2),\cdots(u^i_{M^{\sigma}_{i}}, m'^i_{M^{\sigma}_{i}}),
\varrho^k_i)$$ be a morphism in $d_{(\underline{g}, \sigma)}(X)$.
Let $t$ be a representative of the image of $m'^i_1$ in
$\mathbb{R}/\mathbb{Z}$. Then, each $m^i_k:=m'^i_k-t$ is an
integer.

When we know how $\tau\in C_{\Sigma_n}(\sigma)$ permutes the
cycles of $\sigma$, whose information is contained in those
$\varrho^k_i\in\Sigma_{M^{\sigma}_{i}}$, and the numbers $m^i_1,
\cdots m^i_{M^{\sigma}_{i}},$  we can get a unique $\tau$.
Explicitly, for any number $j_r=1, 2 \cdots n$, if $j_r$ is in a
$k-$cycle $(j_1, \cdots j_k)$ of $\sigma$ and it is in the set
$W^{\sigma}_i$, then $\tau$ maps $j_r$ to
$\varrho^k_i(j)_{r+m^i_j}$, i.e. the $r+m^i_j$-th element in the
cycle $\varrho^k_i(j)$ of $\sigma$.

For any $a\in W^{\sigma}_i$, $\forall k$ and $i$, we want
$u^i_a=\beta^{\underline{h}, \tau}_{\tau(a), a}$ for some
$\underline{h}$. Thus,
\begin{equation}h_{\tau(a)_k}=u^i_a g_{a_k}\cdots
g_{a_{1-m^i_a}}.\end{equation} And by (\ref{rela2}) we can get all
the other $h_{\tau(a)_j}$.

It can be checked straightforward that $J_{(\underline{g},
\sigma)}$ is a well-defined functor. And it does not depend on the
choice of the representative $t$.

$J_{(\underline{g}, \sigma)}\circ f_{(\underline{g}, \sigma)}=
\mbox{Id}$; $f_{(\underline{g}, \sigma)}\circ J_{(\underline{g},
\sigma)}= \mbox{Id}$. So the conclusion is proved.
\end{proof}


\begin{theorem}The two groupoids $\Lambda((X/\!\!/G)\wr\Sigma_n)$ and $d((X/\!\!/G)\wr\Sigma_n)$ are isomorphic.\label{iprm}\end{theorem}


\begin{proof}If $(\underline{g}, \sigma)$ and $(\underline{g}', \sigma')$
are conjugate in $G\wr\Sigma_n$, each $$[(\underline{h}, \tau),
t_0]\in\Lambda_{G\wr\Sigma_n}((\underline{g}, \sigma),
(\underline{g}', \sigma'))$$ gives an isomorphism from $(X^{\times
n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$ to
$(X^{\times n})^{(\underline{g}',
\sigma')}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}', \sigma')$ by
$$x\mapsto x\cdot [(\underline{h}, \tau), t_0].$$ There is a
canonical isomorphism $f([(\underline{h}, \tau), t_0])$ from
$d_{(\underline{g}, \sigma)}(X)$ to $d_{(\underline{g}',
\sigma')}(X)$ so that the diagram below commutes in the category
of groupoids.
$$\xymatrix{&(X^{\times
n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\ar[r]^>>>>>>>{f_{(\underline{g},
\sigma)}}\ar[d]^{[(\underline{h}, \tau), t_0]} &d_{(\underline{g},
\sigma)}(X)\ar[d]^{f([(\underline{h}, \tau), t_0])}\\
&(X^{\times n})^{(\underline{g}',
\sigma')}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g}',
\sigma')\ar[r]_>>>>>>{f_{(\underline{g}', \sigma')}}
&d_{(\underline{g}', \sigma')}(X)}$$





Let $x=(x_1, \cdots x_n)\in (X^{\times n})^{(\underline{g},
\sigma)}$. $f_{(\underline{g}', \sigma')}(x\cdot[(\underline{h},
\tau), t_0])=
\prod\limits_k\prod\limits_{i\in\theta_k}x_{\tau(i_k)}h_{\tau(i_k)}.$
%And $f_{(\underline{g}, \sigma)}(x)=\prod\limits_k\prod\limits_{i\in\theta_k}x_{j_k}.$

Assume $\tau$ maps a $k$-cycle $i=(i_1,\cdots i_k)$ of $\sigma'$
to a $k$-cycle $j=(j_1, \cdots j_k)$ of $\sigma$. By the relation
(\ref{rela2}), $x_{\tau(i_r)}h_{\tau(i_r)}g'_{i_r}\cdots
g'_{i_1}=x_{j_{r+m_i}}g_{j_{r+m_i}}\cdots
g_{j_1}(g^{-1}_{j_1}\cdots g^{-1}_{j_{m_i}}h_{j_{m_i}}).$ And
$g^{-1}_{j_1}\cdots
g^{-1}_{j_{m_i}}h_{j_{m_i}}=h_{j_k}g'^{-1}_{i_{1-m_i}}\cdots
g'^{-1}_{i_k}.$




So $f([(\underline{h}, \tau), t_0])$ maps $\nu_j\in
X^{g_{j_k}\cdots g_{j_1}}$ corresponding to a $k-$cycle $(j_1,
\cdots j_k)$ of $\sigma$ to $\nu_j \cdot
h_{j_k}g'^{-1}_{i_{1-m_i}}\cdots g'^{-1}_{i_k}\in
X^{g'_{i_k}\cdots g'_{i_1}}.$ $f([(\underline{h}, \tau), t_0])$ is
an isomorphism between the two groupoids.

\end{proof}


\subsubsection{Properties of the functor $f_{(\underline{g}, \sigma)}$}

%\prod_k\prod_{[\alpha]}\prod_{(j_1, \cdots j_k)\in [\alpha]}\sideset{_k}{_{g_{i_k}\cdots g_{i_1}}}{\mathcal{L}} X

%$$(\prod_{(j_1, \cdots j_k)\in [(i_1, \cdots i_k)]}\sideset{_k}{_{g_{i_k}\cdotsg_{i_1}}}{\mathcal{L}} X)$$

In addition, we state in Proposition \ref{sim} some  properties
about the functor $f_{(\underline{g}, \sigma)}$.

%Let $\mathbb{T}$ denote the Lie group $\mathbb{R}/\mathbb{Z}$. For
%each $\Lambda_G(g)$, we can define a projection
%$\pi:\Lambda_G(g)\longrightarrow\mathbb{T}$ sending $[\alpha, t]$ to $[t]$.

\begin{proposition}
(i) If $\sigma=(1)\in\Sigma_1$, the morphism $f_{(g, (1))}$ is the
identity map on $X^g/\!\!/\Lambda_{G}(g)$.

(ii) Let $(\underline{g}, \sigma)\in G\wr\Sigma_n$ and
$(\underline{h}, \tau)\in G\wr\Sigma_m$. The groupoids
$$(X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}(X^{\times m})^{(\underline{h},
\tau)}/\!\!/\Lambda_{G\wr\Sigma_m}(\underline{h}, \tau)$$ and
$$(X^{\times (n+m)})^{(\underline{g}, \underline{h}
\sigma\tau)}/\!\!/\Lambda_{G\wr\Sigma_{n+m}}(\underline{g},
\underline{h}, \sigma\tau)
$$ are isomorphic.


(iii)$f_{(\underline{g}, \sigma)}$ preserves cartesian product of
loops. The following diagram of groupoids commutes.

$$\xymatrix{(X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{\!G\wr\Sigma_n\!}(\underline{g},
\sigma)\!\!\times_{\mathbb{T}}\!\!(X^{\times m})^{(\underline{h},
\tau)}/\!\!/\Lambda_{\!G\wr\Sigma_m\!}(\underline{h},
\tau)\ar[r]^>>>>>>{\cong}%^{(\gamma_1, \gamma_2)\mapsto (\gamma_1, \gamma_2)}
\ar[d]_{f_{(\underline{g}, \sigma)}\times f_{(\underline{h},
\tau)}} &(X^{\times (n+m)})^{(\underline{g}, \underline{h},
\sigma\tau)}/\!\!/\Lambda_{\!G\wr\Sigma_{n+m}\!}\!(\underline{g},\!
\underline{h}, \!\sigma\tau)\!\ar[d]^{
f_{(\underline{g}, \underline{h}, \sigma\tau)}}\\
d_{(\underline{g}, \sigma)}(X)\times_{\mathbb{T}}
d_{(\underline{h}, \tau)}(X)\ar[r]^>>>>>>>>>>>>>>>>>>>>>>{\cong}
&d_{(\underline{g}, \underline{h}, \sigma\tau)}(X)}$$ \label{sim}
\end{proposition}

\begin{proof}




(i) is indicated in the proof of Theorem \ref{iptm}.





(ii)

We can define a functor $\Phi:(X^{\times n})^{(\underline{g},
\sigma)}/\!\!/\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}(X^{\times m})^{(\underline{h},
\tau)}/\!\!/\Lambda_{G\wr\Sigma_m}(\underline{h},
\tau)\longrightarrow (X^{\times (n+m)})^{(\underline{g},
\underline{h}
\sigma\tau)}/\!\!/\Lambda_{G\wr\Sigma_{n+m}}(\underline{g},
\underline{h}, \sigma\tau)$. It sends an object $(x_1, x_2)$ to
$(x_1, x_2)$. And it sends a morphism $([\alpha,t],[\beta, t])$ to
$[\alpha, \beta, t].$  It is straightforward to check $\Phi$ is an
isomorphism between the groupoids.




(iii) The proof is left to the readers.

\end{proof}





\subsection{Total Power Operation of
$QEll^*_G$}\label{s2complete}

In this section   we construct the total power operations
 for quasi-elliptic cohomology and give
its explicit formula in (\ref{ee}). In Theorem \ref{main1p} we
check they satisfy the axioms that Ganter concluded in Definition
4.3, \cite{Gan06} for equivariant power operation.

%Before that in Section \ref{lov} I present some relevant
%constructions which are necessary for the construction of the power operation.




%\subsection{The Construction of Power Operation}\label{equivpower}

%In this section I construct the power operation for quasi-elliptic
%cohomology in (\ref{poweroperationqec}) and give its explicit formula in (\ref{ee}).

%Let $\mathcal{V}$ be an orbifold $\mathbb{S}^1$-equivariant vector bundle over the subspace of constant loops of $\mathcal{L}X/\!\!/G$.


The power operation of quasi-elliptic cohomology is of the form
\begin{align*}\mathbb{P}_n=&\prod_{(\underline{g}, \sigma)\in
(G\wr\Sigma_n)_{conj}}\mathbb{P}_{(\underline{g},\sigma)}:&
\\
&QEll^*_G(X)\longrightarrow QEll^*_{G\wr\Sigma_n}(X^{\times n})
=&\prod_{(\underline{g}, \sigma)\in
(G\wr\Sigma_n)_{conj}}K_{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)}((X^{\times n})^{(\underline{g}, \sigma)}),\end{align*}
where each $\mathbb{P}_{(\underline{g}, \sigma)}$ is the
composition below.

\begin{align*}QEll^*_G(X)&\buildrel{U^*}\over\longrightarrow
K^*_{orb}(\Lambda^1_{(\underline{g}, \sigma)}(X))
\buildrel{(\mbox{ })_k^{\Lambda}}\over\longrightarrow
K^*_{orb}(\Lambda^{var}_{(\underline{g}, \sigma)}(X))\\
&\buildrel{\boxtimes}\over\longrightarrow
K^*_{orb}(d_{(\underline{g},
\sigma)}(X))\buildrel{f^*_{(\underline{g},
\sigma)}}\over\longrightarrow
K^*_{\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)}((X^{\times
n})^{(\underline{g}, \sigma)}),\end{align*}
 where
$k\in\mathbb{Z}$ and $(i_1, \cdots i_k)$ goes over all the
$k-$cycles of $\sigma$. We will explain in detail what each map in
this formula is: $U: \Lambda^1_{(\underline{g},
\sigma)}(X)\longrightarrow \Lambda(X/\!\!/G)$ is the groupoid
homomorphism defined in (\ref{functorU}). The second map is the
pullback $(\mbox{ })^{\Lambda}_k$ defined in
(\ref{rescalelambda}). The third map $\boxtimes$ is the external
product in (\ref{lambdaproductd}). The fourth one is the pullback
by $f_{(\underline{g}, \sigma)}$, which is defined in Section
\ref{FunctorF}.

The power operation has the explicit formula  %defined in (\ref{poweroperationqec})
\begin{equation}\mathbb{P}_{(\underline{g},
\sigma)}(\mathcal{V})=f^*_{(\underline{g},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(\mathcal{V}_{g_{i_k}\cdots g_{i_1}})_k),\end{equation} as
indicated in (\ref{ee}).
\bigskip



\textbf{The Functor $U$}


 For each element $(\underline{g},
\sigma)\in G\wr\Sigma_n$, $r\in\mathbb{Z}$,
let $\Lambda^r_{(\underline{g}, \sigma)}(X)$ denote the groupoid %of $\Lambda^r(X/\!\!/G)$
with objects
$$\coprod_k\coprod_{(i_1, \cdots i_k)}X^{g_{i_k}\cdots g_{i_1}}$$
where $(i_1, \cdots i_k)$ goes over all the $k-$cycles of
$\sigma$, and with morphisms
$$\coprod_k\coprod_{(i_1, \cdots i_k), (j_1, \cdots
j_k)}\Lambda^r_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots
g_{j_1})\times X^{g_{i_k}\cdots g_{i_1}},$$ where $(i_1, \cdots
i_k)$ and $(j_1, \cdots j_k)$ go over all the $k-$cycles of
$\sigma$ respectively. It may not be a subgroupoid of
$\Lambda^r(X/\!\!/G)$ because there may be cycles $(i_1, \cdots
i_k)$ and $(j_1, \cdots j_m)$
such that $$g_{i_k}\cdots g_{i_1}=g_{j_m}\cdots g_{j_1}.$$ %or conjugate

Let \begin{equation}U:\Lambda^1_{(\underline{g},
\sigma)}(X)\longrightarrow \Lambda(X/\!\!/
G)\label{functorU}\end{equation} denote the functor sending $x$ in
the component $X^{g_{i_k}\cdots g_{i_1}}$ to the $x$ in the
component $X^{g_{i_k}\cdots g_{i_1}}$ of $\Lambda(X/\!\!/ G)$, and
send each morphism $$([h, t], x)\mbox{   in
}\Lambda_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})\times
X^{g_{i_k}\cdots g_{i_1}}$$ to $$([h, t], x)\mbox{   in
  }\Lambda_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})\times
X^{g_{i_k}\cdots g_{i_1}}.$$ In the case that  $g_{i_k}\cdots
g_{i_1}$ and $g_{j_k}\cdots g_{j_1}$ are equal, $([h, t], x)$ is
an arrow inside a single connected component.

\bigskip

\textbf{The Functor $(\mbox{                  })_k$}%\label{functorrescalek}

\begin{definition}The groupoid $\Lambda^k(X/\!\!/G)$ has the same objects as
$\Lambda(X/\!\!/G)$ but different morphisms
$$\coprod\limits_{g, g'\in G}\Lambda^k_G(g, g')\times X^g$$
where $\Lambda^k_G(g, g')$ is defined in Definition
\ref{lambdak1}. For an object $x\in X^g$ and a morphism $([\sigma,
t], g)\in \Lambda^k_G(g, g')\times X^g$, $([\sigma, t],
g)(x)=\sigma x\in X^{g'}.$ The composition of the morphisms is
defined by
\begin{equation}[\sigma_1, t_1][\sigma_2,
t_2]=[\sigma_1\sigma_2, t_1+t_2].\end{equation}
\label{lambdaoidefk}\end{definition}




For each integer $k$, there is a functor of groupoids $(\mbox{
})_k:  \Lambda^k(X/\!\!/G)\longrightarrow \Lambda(X/\!\!/G)$ %${_k}\mathcal{L} (X/\!\!/G)$
sending an object $x$ to $x$ and  a morphism $([h, t_0], x)\mbox{
    to     }([h, \frac{t_0}{k}], x).$ The composition
$((\mbox{ })_k)_r=(\mbox{ })_{kr}.$

The functor $(\mbox{                  })_k$ gives a well-defined
map $$K_{orb}(\Lambda(X/\!\!/G))\longrightarrow K_{orb}(\Lambda^k
(X/\!\!/G))$$ by pullback of bundles.  We still use the symbol
$(\mbox{                  })_k$ to denote it when there is no
confusion.  For any $\Lambda(X/\!\!/G)-$vector bundle
$\mathcal{V}$,
 $S^1$ acts on $(\mathcal{V})_k$ via
\begin{align*}q^{\frac{1}{k}}: \mathbb{R}/k\mathbb{Z} &\longrightarrow U(1)\\ a&\mapsto e^{\frac{2\pi ia}{k}}.\end{align*}
If $\mathcal{V}$ has the decomposition
$\mathcal{V}=\bigoplus_{j\in\mathbb{Z}}V_jq^j,$ then
\begin{equation}(\mathcal{V})_k=\bigoplus_{j\in\mathbb{Z}}V_jq^{\frac{j}{k}}.\label{lpok}\end{equation}



\textbf{The Functor $(\mbox{ })_k^{\Lambda}$}

Let $\Lambda^{var}_{(\underline{g}, \sigma)}(X)$ be the groupoid
with the same objects as $\Lambda^1_{(\underline{g}, \sigma)}(X)$
and morphisms
$$\coprod\limits_k\coprod_{(i_1, \cdots i_k), (j_1, \cdots
j_k)}\Lambda^k_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots
g_{j_1})\times X^{g_{i_k}\cdots g_{i_1}},$$ where $(i_1, \cdots
i_k)$ and $(j_1, \cdots j_k)$ go over all the $k-$cycles of
$\sigma$ respectively.

We can define a similar functor
\begin{equation}(\mbox{  })_k^{\Lambda}: \Lambda^{var}_{(\underline{g},
\sigma)}(X)\longrightarrow \Lambda^1_{(\underline{g},
\sigma)}(X)\label{rescalelambda}\end{equation} that is identity on
objects and sends each $[g, t]\in \Lambda^k_G(g_{i_k}\cdots
g_{i_1}, g_{j_k}\cdots g_{j_1})$ to $[g, \frac{t}{k}]\in
\Lambda^1_G(g_{i_k}\cdots g_{i_1}, g_{j_k}\cdots g_{j_1})$.

And we use the same symbol $(\mbox{  })_k^{\Lambda}$ to denote the
pull back \begin{equation}K_{orb}(\Lambda^1_{(\underline{g},
\sigma)}(X))\longrightarrow K_{orb}(\Lambda^{var}_{(\underline{g},
\sigma)}(X)).\label{rescalekpower}\end{equation}

%I still use the same symbol to denote the restriction of $\coprod\limits_k(\mbox{ })_k$ to the subgroupoid.



\bigskip


\textbf{The external product $\boxtimes$ }%\label{externalproductdgroupoid}

Let $X$ be a $G$-space and $Y$ an $H$-space. Let  $(\underline{g},
\sigma)\in G\wr\Sigma_n$ and $(\underline{h}, \tau)\in
G\wr\Sigma_m$.


In Section \ref{propertiesqec} we  introduce the K\"{u}nneth map
and the tensor product $\otimes_{\mathbb{Z}[q^{\pm}]}$ of
quasi-elliptic cohomology. To simplify the symbol in this section,
for $x \in K^{*}_{\Lambda_G(g)}(X^g)$ and $y\in
K^*_{\Lambda_H(h)}(Y^h)$, we use $$x\boxtimes y$$ to denote their
external product $x\otimes_{\mathbb{Z}[q^{\pm}]} y$ in
$K^{*}_{\Lambda_{G\times H}(g, h)}(X\times Y)^{(g, h)}$; and for
$\widetilde{x}\in QEll^*_G(X)$ and $\widetilde{y}\in QEll^*_H(Y)$,
we  use
$$\widetilde{x}\boxtimes \widetilde{y}$$ to denote their external product as
$\mathbb{Z}[q^{\pm}]-$algebras in $QEll^*_{G\times H}(X\times Y)$.

\bigskip


Similar to the K\"{u}nneth map (\ref{ku}) of quasi-elliptic
cohomology, we can define the K\"{u}nneth map
$$K^*_{orb}(d_{(\underline{g},
\sigma)}(X))\otimes_{\mathbb{Z}[q^{\pm}]}K^*_{orb}(d_{(\underline{h},
\tau)}(X))\longrightarrow K^*_{orb}(d_{(\underline{g},
\sigma)}(X)\times_{\mathbb{T}}d_{(\underline{h}, \tau)}(X)),$$
which is compatible with that of quasi-elliptic cohomology in the
sense of (\ref{kudqellc}).

Each $K^*_{orb}(d_{(\underline{g}, \sigma)}(X))$ is a
$\mathbb{Z}[q^{\pm}]-$algebra, as shown at the end of Section
\ref{setupad}. The external product in the theory
$K^*_{orb}(d_{(\underline{g}, \sigma)}(-))$ is the tensor product
of $\mathbb{Z}[q^{\pm}]-$algebras.

Let $d_{(\underline{g},
\sigma)}(X)\times_{\mathbb{T}}d_{(\underline{h}, \tau)}(X)$ denote
the fibred product of  the two groupoids. It is a subgroupoid of
$d_{(\underline{g}, \underline{h}, \sigma\tau)}(X)$ when we
consider $(\underline{g}, \underline{h}, \sigma\tau)$ in
$G\wr(\Sigma_{n}\times\Sigma_m)$ as an element in
$G\wr\Sigma_{n+m}$. The groupoids $d_{(\underline{g},
\sigma)}(X)\times_{\mathbb{T}}d_{(\underline{h}, \tau)}(X)$  and
$d_{(\underline{g}, \underline{h}, \sigma\tau)}(X)$ have the same
objects.

We have the K\"{u}nneth map
\begin{equation}K^*_{orb}(d_{(\underline{g},
\sigma)}(X))\otimes_{\mathbb{Z}[q^{\pm}]}K^*_{orb}(d_{(\underline{h},
\tau)}(X))\longrightarrow K^*_{orb}(d_{(\underline{g},
\sigma)}(X)\times_{\mathbb{T}}d_{(\underline{h}, \tau)}(X)).\label{kunnethdc1}\end{equation} %make explicitly?
It is compatible with the K\"{u}nneth map (\ref{ku})  of the
quasi-elliptic cohomology in the sense that the diagram below
commutes.
\begin{equation}\begin{CD}[cols=0.1em]K^*_{orb}(d_{(\underline{g},
\sigma)}(X))\!\underset{\mathbb{Z}[q^{\pm}]}\otimes\!
K^*_{orb}\!(d_{(\underline{h}, \sigma)}(X)) @>>>
K^*_{orb}(d_{(\underline{g},
\sigma)}(X)\!\times_{\mathbb{T}}\!d_{(\!\underline{h},
\sigma\!)}(X))\\
@V{f_{(\underline{g},
\sigma)}^*\otimes_{\mathbb{Z}[q^{\pm}]}f_{(\underline{h},
\sigma)}^*}VV    @V{f_{(\underline{(g, h)}, \sigma)}^*}VV
\\ K^{*}_{\Lambda_{G\wr\Sigma_n}\!(\underline{g},
\sigma)}\!((X^n)^{(\underline{g}, \sigma)})
\!\!\!\underset{\mathbb{Z}[q^{\pm}]}\otimes\!\!\!
K^*_{\Lambda_{H\wr\Sigma_n}\!(\underline{h},
\sigma)}\!((Y^m)^{(\underline{h}, \sigma)}) @>>>
K^{*}_{\!\Lambda_{(\!G\!\times
\!H\!)\wr\Sigma_n}\!\!(\underline{(g, h)},
\sigma)}\!((X\!\times\!Y)^n)^{(\underline{(g, h)},
\sigma)}\end{CD}\label{kudqellc}\end{equation} where the
horizontal maps are K\"{u}nneth maps.



\bigskip


%\subsubsection{The external product $\boxtimes$}
In addition, if we have a vector bundle
$E=\coprod\limits_k\coprod\limits_{(i_1, \cdots
i_k)}E_{g_{i_k}\cdots g_{i_1}}$ over $\Lambda^1_{(\underline{g},
\sigma)}(X)$,  the external product
$$\boxtimes_k\boxtimes_{(i_1, \cdots
i_k)}E_{g_{i_k}\cdots g_{i_1}}$$ is a %$d_{(\underline{g}, \sigma)}(X)-$
vector bundler over $d_{(\underline{g}, \sigma)}(X)$. This defines
a map \begin{equation} K_{orb}(\Lambda^1_{(\underline{g},
\sigma)}(X))\longrightarrow K_{orb}(d_{(\underline{g},
\sigma)}(X))\label{lambdaproductd}\end{equation}







\bigskip

Composing all the relevant functors, we get the explicit formula of $\mathbb{P}_{(\underline{g}, \sigma)}$ %defined in (\ref{poweroperationqec})
\begin{equation}\mathbb{P}_{(\underline{g},
\sigma)}(\mathcal{V})=f^*_{(\underline{g},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(\mathcal{V}_{g_{i_k}\cdots
g_{i_1}})_k).\label{ee}\end{equation}%natural?


%Both $f^*_{(\underline{g}, \sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
%i_k)}(\mathcal{V}_{g_{i_k}\cdots g_{i_1}})_k)$ and $\boxtimes_{k}\boxtimes_{(i_1,\cdots
%i_k)}(\mathcal{V}_{g_{i_k}\cdots g_{i_1}})_k$ have the structure of a $\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$-space, since
%$f_{(\underline{g}, \sigma)}$ is a $\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)-$equivariant homeomorphism.


$\mathbb{P}_{(\underline{g}, \sigma)}$ is natural. And by the
proof of Theorem \ref{iprm}, if $(\underline{g}, \sigma)$ and
$(\underline{h}, \tau)$ are conjugate in $G\wr\Sigma_n$, then
$\mathbb{P}_{(\underline{g}, \sigma)}(E)$ and
$\mathbb{P}_{(\underline{h}, \tau)}(E)$ are isomorphic.

\begin{theorem}The family of maps $$\mathbb{P}_n=\prod_{(\underline{g},
\sigma)\in
(G\wr\Sigma_n)_{conj}}\mathbb{P}_{(\underline{g},\sigma)}:
QEll^*_G(X)\longrightarrow QEll^*_{G\wr\Sigma_n}(X^{\times n}),$$
defined in (\ref{ee}) satisfy

(i) $\mathbb{P}_1=$Id, $\mathbb{P}_0(x)=1$.


(ii) Let $x\in QEll^*_G(X)$, $(\underline{g}, \sigma)\in
G\wr\Sigma_n$ and $(\underline{h}, \tau)\in G\wr\Sigma_m$. The
external product of two power operations
$$\mathbb{P}_{(\underline{g}, \sigma)}(x)\boxtimes \mathbb{P}_{(\underline{h}, \tau)}(x)=res|^{\Lambda_{G\wr\Sigma_{m+n}}(\underline{g},\underline{h}; \sigma\tau)}
_{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}\Lambda_{G\wr\Sigma_m}(\underline{h},
\tau)} \mathbb{P}_{(\underline{g},\underline{h};
\sigma\tau)}(x).$$


(iii) The composition of two power operations is
$$\mathbb{P}_{((\underline{\underline{h}, \tau}); \sigma)}(\mathbb{P}_m(x))=res|^{\Lambda_{G\wr\Sigma_{mn}}(\underline{\underline{h}}, (\underline{\tau}, \sigma))}
_{\Lambda_{(G\wr\Sigma_m)\wr\Sigma_n}((\underline{\underline{h},
\tau}); \sigma)}\mathbb{P}_{(\underline{\underline{h}},
(\underline{\tau}, \sigma))}(x)$$ where
$(\underline{\underline{h}, \tau})\in (G\wr \Sigma_m)^{\times n}$,
and $\sigma\in\Sigma_n$. $(\underline{\tau}, \sigma)$ is in
$\Sigma_m\wr\Sigma_n$, thus, can be viewed as an element in
$\Sigma_{mn}$.


(iv) $\mathbb{P}$ preserves external product. For $\underline{(g,
h)}=((g_1, h_1), \cdots (g_n, h_n))\in (G\times H)^{\times n}$,
$\sigma\in \Sigma_n$,
$$\mathbb{P}_{(\underline{(g, h)},\sigma)}(x\boxtimes y)=res|^{\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)\times_{\mathbb{T}}\Lambda_{H\wr\Sigma_n}(\underline{h}, \sigma)}
_{\Lambda_{(G\times H)\wr\Sigma_n}(\underline{(g, h)},
\sigma)}\mathbb{P}_{(\underline{g}, \sigma)}(x)\boxtimes
\mathbb{P}_{(\underline{h}, \sigma)}(y).$$ \label{main1p}
\end{theorem}
\begin{proof}

We check each one respectively.

(i) When $n=1$, all the cycles of a permutation is 1-cycle.
$(\mbox{ })_1$ and the homeomorphism $f_{(g, (1))}$ are both
identity maps. Directly from the formula (\ref{ee}),
$\mathbb{P}_1(x)=x.$

\bigskip
(ii) \begin{align*}&\mathbb{P}_{(\underline{g},
\sigma)}(x)\boxtimes \mathbb{P}_{(\underline{h}, \tau)}(x)\\
=&f^*_{(\underline{g},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(x_{g_{i_k}\cdots g_{i_1}})_k)\boxtimes f^*_{(\underline{h},
\tau)}(\boxtimes_{j}\boxtimes_{(r_1,\cdots
r_j)}(x_{h_{r_j}\cdots h_{r_1}})_j)\\
=&res|^{\Lambda_{G\wr\Sigma_{m+n}}(\underline{g},\underline{h};
\sigma\tau)} _{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}\Lambda_{G\wr\Sigma_m}(\underline{h},
\tau)} f^*_{(\underline{g}, \underline{h}; \sigma
\tau)}((\boxtimes_{k}\boxtimes_{(i_1,\cdots i_k)}(x_{g_{i_k}\cdots
g_{i_1}})_k)\\ &\boxtimes (\boxtimes_{j}\boxtimes_{(r_1,\cdots
r_j)}(x_{h_{r_j}\cdots h_{r_1}})_j)).\end{align*}where
$(i_1,\cdots i_k)$ goes over all the $k$-cycles of $\sigma$ and
$(r_1,\cdots r_j)$ goes over all the $j$-cycles of $\tau$ and
$(\mbox{                  })_k$ is the map cited in (\ref{lpok}).
The second step is from Proposition \ref{sim} (iii).

$$f^*_{(\underline{g}, \underline{h}; \sigma
\tau)}((\boxtimes_{k}\boxtimes_{(i_1,\cdots i_k)}(x_{g_{i_k}\cdots
g_{i_1}})_k)\boxtimes (\boxtimes_{j}\boxtimes_{(r_1,\cdots
r_j)}(x_{h_{r_j}\cdots h_{r_1}})_j))$$ is exactly
$$\mathbb{P}_{(\underline{g},\underline{h};
\sigma\tau)}(x).$$

\bigskip
(iii) Recall that for an element $(\underline{\tau},
\sigma)=(\tau_1, \cdots \tau_n, \sigma)\in\Sigma_{mn}$, it acts on
the set with $mn$ elements
$$\{(i, j)| 1\leq i\leq n, 1\leq j \leq m\}$$ in this way: $$(\underline{\tau}, \sigma)\cdot(i, j)=(\sigma(i),
\tau_{\sigma(i)}(j)).$$ That also shows how to view it as an
element in $\Sigma_{mn}$.

Then for any integer $q$, \begin{equation}(\underline{\tau},
\sigma)^q\cdot (i, j)=(\sigma^q(i),
\tau_{\sigma^q(i)}\tau_{\sigma^{q-1}(i)}\cdots
\tau_{\sigma(i)}(j)).\label{not}\end{equation}
\smallskip

To find all the cycles of $(\underline{\tau}, \sigma)$ is exactly
to find all the orbits of the action by $(\underline{\tau},
\sigma)$. If $i$ belongs to an $s$-cycle of $\sigma$ and $j$
belongs to a $r$-cycle of
$\tau_{\sigma^s(i)}\tau_{\sigma^{s-1}(i)}\cdots \tau_{\sigma(i)}$,
then the orbit containing $(i, j)$ has $sr$ elements by
(\ref{not}). In other words, $(i_1, \cdots i_s)$ is an $s$-cycle
of $\sigma$ and $(j_1, \cdots j_r)$ is a $r$-cycle of
$\tau:=\tau_{i_s}\cdots \tau_{i_1}$ if and only if
\begin{align*}\bigg(&(i_1, \tau_{i_1}(j_{r-1}))(i_2, \tau_{i_2}\tau_{i_1}(j_{r-1}))\cdots (i_s, j_r)\\
&(i_1, \tau_{i_1}(j_{r-2}))(i_2, \tau_{i_2}\tau_{i_1}(j_{r-2}))\cdots (i_s, j_{r-1})\\
&\cdots\\
&(i_1, \tau_{i_1}(j_{1}))(i_2,
\tau_{i_2}\tau_{i_1}(j_{1}))\cdots (i_s, j_2)\\
&(i_1, \tau_{i_1}(j_{r}))(i_2, \tau_{i_2}\tau_{i_1}(j_{r}))\cdots
(i_s, j_1)\bigg)\\\end{align*} is an $sr$-cycle of
$(\underline{\tau}, \sigma)$.

\begin{align*}&\mathbb{P}_{((\underline{\underline{h}, \tau});
\sigma)}(\mathbb{P}_m(x))\\ =&f^*_{((\underline{\underline{h},
\tau}); \sigma)}[\boxtimes_k\boxtimes_{(i_1,\cdots
i_k)}(\mathbb{P}_{((\underline{h}_{i_k},\tau_{i_k})\cdots(\underline{h}_{i_1},\tau_{i_1}))}(x))_k]\\
=&f^*_{((\underline{\underline{h}, \tau});
\sigma)}[\boxtimes_k\boxtimes_{(i_1, \cdots
i_k)}[f^*_{((\underline{h}_{i_k},\tau_{i_k})\cdots(\underline{h}_{i_1},\tau_{i_1}))}(\boxtimes_r\boxtimes_{(j_1,
\cdots
j_r)}(x_{H_{\underline{i}\underline{j}}})_r)]_k]\\
=&(f^*_{((\underline{\underline{h}, \tau}); \sigma)}\circ
\prod_{k, (i_1, \cdots
i_k)}f^*_{((\underline{h}_{i_k},\tau_{i_k})\cdots(\underline{h}_{i_1},\tau_{i_1}))})[\boxtimes_{k,
(i_1,\cdots i_k)}\boxtimes_{r, (j_1, \cdots
j_r)}(x_{H_{\underline{i},
\underline{j}}})_{kr}]\\
=&f^*_{(\underline{\underline{h}}, (\underline{\tau},
\sigma))}[\boxtimes_{k, (i_1,\cdots i_k)}\boxtimes_{r, (j_1,
\cdots j_r)}(x_{H_{\underline{i},
\underline{j}}})_{kr}]\end{align*} where
\begin{align*}H_{\underline{i}\underline{j}}:=\mbox{  }&h_{i_k, j_1}h_{i_{k-1}, \tau^{-1}_{i_k}(j_1)}\cdots h_{i_1, (\tau_{i_k}\cdots\tau_{i_2})^{-1}(j_1)}\\
&h_{i_k, j_2}h_{i_{k-1}, \tau^{-1}_{i_k}(j_2)}\cdots h_{i_1, (\tau_{i_k}\cdots\tau_{i_2})^{-1}(j_2)}\\
&\cdots\\
&h_{i_k, j_r}h_{i_{k-1}, \tau^{-1}_{i_k}(j_r)}\cdots h_{i_1,
(\tau_{i_k}\cdots\tau_{i_2})^{-1}(j_r)}\\=\mbox{  }&h_{i_k,
j_1}h_{i_{k-1},
\tau_{i_{k-1}}\cdots\tau_{i_2}\tau_{i_1}(j_1)}\cdots h_{i_1,
\tau_{i_1}(j_r)}\\
&h_{i_k, j_2}h_{i_{k-1},
\tau_{i_{k-1}}\cdots\tau_{i_2}\tau_{i_1}(j_2)}\cdots h_{i_1,
\tau_{i_1}(j_1)}\\
&\cdots\\
&h_{i_k, j_r}h_{i_{k-1},
\tau_{i_{k-1}}\cdots\tau_{i_2}\tau_{i_1}(j_{r-1})}\cdots h_{i_1,
\tau_{i_1}(j_{r-1})}
\end{align*}
where $(i_1, \cdots i_k)$ goes over all the $k$-cycles of
$\sigma\in\Sigma_m$ and $(j_1, \cdots j_r)$ goes over all the
$r$-cycles of $\tau_{i_k}\cdots\tau_{i_1}\in\Sigma_n$. The last step %in (\ref{exha})
is by Proposition 4.11 in \cite{Gan07}.

$$f^*_{(\underline{\underline{h}}, (\underline{\tau},
\sigma))}[\boxtimes_{k, (i_1,\cdots i_k)}\boxtimes_{r, (j_1,
\cdots j_r)}(x_{H_{\underline{i}, \underline{j}}})_{kr}]$$ is the
same space as $\mathbb{P}_{(\underline{\underline{h}},
(\underline{\tau}, \sigma))}(x)$, %by the computation (\ref{kew}),
but the action is restricted by
$$res|^{\Lambda_{G\wr\Sigma_{mn}}(\underline{\underline{h}}, (\underline{\tau}, \sigma))}
_{\Lambda_{(G\wr\Sigma_m)\wr\Sigma_n}((\underline{\underline{h},
\tau}); \sigma)}.$$


\bigskip
(iv)We have
\begin{align*}&\mathbb{P}_{(\underline{(g,h)},\sigma)}(x\boxtimes
y)=f^*_{(\underline{(g, h)},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}((x\boxtimes y)_{(g_{i_k}\cdots g_{i_1}, h_{i_k}\cdots h_{i_1})})_k)\\
=&f^*_{(\underline{(g, h)},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(x_{g_{i_k}\cdots g_{i_1}})_k\boxtimes (y_{h_{i_k}\cdots
h_{i_1}})_k)\\ =&f^*_{(\underline{(g, h)},
\sigma)}(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(x_{g_{i_k}\cdots
g_{i_1}})_k)\boxtimes(\boxtimes_{j}\boxtimes_{(r_1,\cdots r_j)}
(y_{h_{r_j}\cdots h_{r_1}})_j)\\
=&res|^{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}\Lambda_{H\wr\Sigma_n}(\underline{h},
\sigma)} _{\Lambda_{(G\times H)\wr\Sigma_n}(\underline{(g, h)},
\sigma)}(f^*_{(\underline{g}, \sigma)}\times f^*_{(\underline{h},
\sigma)})(\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(x_{g_{i_k}\cdots g_{i_1}})_k)\\
&\boxtimes(\boxtimes_{j}\boxtimes_{(r_1,\cdots r_j)}
(y_{h_{r_j}\cdots h_{r_1}})_j)\\
=&res|^{\Lambda_{G\wr\Sigma_n}(\underline{g},
\sigma)\times_{\mathbb{T}}\Lambda_{H\wr\Sigma_n}(\underline{h},
\sigma)} _{\Lambda_{(G\times H)\wr\Sigma_n}(\underline{(g, h)},
\sigma)}f^*_{(\underline{g},\sigma)}[\boxtimes_{k}\boxtimes_{(i_1,\cdots
i_k)}(x_{g_{i_k}\cdots g_{i_1}})_k]\\ &\boxtimes
f^*_{(\underline{h},\sigma)}[\boxtimes_{j}\boxtimes_{(r_1,\cdots
r_j)} (y_{h_{r_j}\cdots h_{r_1}})_j],\end{align*}where $(i_1,
\cdots i_k)$ goes over all the $k-$cycles of $\sigma$ and $(r_1,
\cdots r_j)$ goes over all the $j-$cycles of $\sigma$.  It equals
to
$$res|^{\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)\times_{\mathbb{T}}\Lambda_{H\wr\Sigma_n}(\underline{h}, \sigma)}
_{\Lambda_{(G\times H)\wr\Sigma_n}(\underline{g},
\sigma)}\mathbb{P}_{(\underline{g}, \sigma)}(x)\boxtimes
\mathbb{P}_{(\underline{h}, \sigma)}(y).$$


\end{proof}

\begin{example}Let $G$ be the trivial group and $X$ a space. Let $\sigma\in\Sigma_n$.  Then $QEll^*_G(X)=
K^*_{\mathbb{T}}(X)$. And $f_{(\underline{1}, \sigma)}$ is the
homeomorphism
$$(X^{\times n})^{(\underline{1}, \sigma)}\cong
\prod_k\prod_{(i_1, \cdots i_k)}X,$$ where the second direct
product goes over all the $k$-cycles of $\sigma$. By (\ref{ee}),
the power operation is
$$\mathbb{P}_{(\underline{1}, \sigma)}(x)=\boxtimes_k\boxtimes_{(i_1, \cdots
i_k)}(x)_k.$$

When $n=2$, $\mathbb{P}_{(\underline{1}, (1)(1))}(x)=x\boxtimes x$
and $\mathbb{P}_{(\underline{1}, (12))}(x)=(x)_2$.


When $n=3$, $\mathbb{P}_{(\underline{1}, (1)(1)(1))}(x)=x\boxtimes
x\boxtimes x$, $\mathbb{P}_{(\underline{1},
(12)(1))}(x)=(x)_2\boxtimes x$, and $\mathbb{P}_{(\underline{1},
(123))}(x)=(x)_3$.

When $n=4$, $\mathbb{P}_{(\underline{1}, (1)(1)(1)(1))}(x)=
x\boxtimes x\boxtimes x\boxtimes x$, $\mathbb{P}_{(\underline{1},
(12))}(x)=(x)_2\boxtimes x\boxtimes x$,
$\mathbb{P}_{(\underline{1}, (123))}(x)=(x)_3\boxtimes x$,
$\mathbb{P}_{(\underline{1}, (1234))}(x)=(x)_4$, and
$\mathbb{P}_{(\underline{1}, (12)(34))}(x)=(x)_2\boxtimes (x)_2.$
Note that there is a $\Sigma_2-$action permuting the two $(x)_2$
in $\mathbb{P}_{(\underline{1}, (12)(34))}(x)$.



\label{pointsymmpower}

\end{example}


\begin{remark}
We have the relation between equivariant Tate K-theory and
quasi-elliptic cohomology
\begin{equation}QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]}\mathbb{Z}((q))\cong K_{Tate}(X/\!\!/G).\end{equation}

It extends uniquely to  a power operation for Tate K-theory
$$
QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]}\mathbb{Z}((q))\longrightarrow
QEll_{G\wr\Sigma_n}(X^{\times
n})\otimes_{\mathbb{Z}[q^{\pm}]}\mathbb{Z}((q))
$$
which  is the stringy power operation $P^{string}_n$ constructed
in \cite{Gan07}. It is elliptic in the sense of \cite{AHS04}.
\label{stringyrelation}

\end{remark}




\section{Orbifold quasi-elliptic cohomology and some basic
constructions}\label{orbifoldquasibeforepower}

The elliptic cohomology of orbifolds involves a rich interaction
between the orbifold structure and the elliptic curve. Ganter
explores this interaction in the case of the Tate curve in
\cite{Gan13}, describing $K_{Tate}$ for an orbifold $X$ in term of
the equivariant K-theory and the groupoid structure of $X$. In
Section \ref{orbifoldquasibeforepower} we give a description of
orbifold quasi-elliptic cohomology in Definition \ref{orbqecnora}.



In Section \ref{sb2} we recall the symmetric power of groupoid,
which is introduced in \cite{Gan13}. In addition,  we  discuss the
inertia groupoid of symmetric power and the groupoids needed for
the construction of the power operation in Section \ref{s2}.

\subsection{Definition}\label{sb1} First we recall some relevant
constructions and notations. The main reference is \cite{Gan13}.

Consider the category of groupoids $\mathcal{G}pd$ as a 2-category
with small topological groupoids as the objects and with
$$\mbox{1Hom}(X, Y)=Fun(X, Y),$$ %and continuous natural  isomorphisms between them.
the groupoid of continuous functors from $X$  to $Y$.


The center of a groupoid $X$ is defined to be the group
$$\mbox{Center}(X):=\mbox{2Hom}(\mbox{Id}_X, \mbox{Id}_X)=\mathcal{N}at(\mbox{Id}_X, \mbox{Id}_X)$$ of natural
transformations from Id$_x$ to Id$_x$.

Let $\mathcal{G}pd^{cen}$ denote the 2-category whose objects are
pairs $(X,\xi)$ with $\xi$ a center element of $X$, and the set of
morphisms from $(X, \xi)$ to $(Y, \nu)$ is $$\mbox{1Hom}((X, \xi),
(Y, \nu))\subset Fun(X, Y)$$ with $$f\xi=\nu f$$ for each morphism
$f$. We will assume all the center elements have finite order.

\begin{example}
If $G$ is a finite group, $\mbox{Center}(pt/\!\!/G)$ is the center
of the group $G$.


\end{example}


\begin{example}The inertia groupoid $I(X)$  of a groupoid $X$, which is
defined in Definition \ref{inertiagroupoid}, is isomorphic to
$$Fun(\mbox{pt}/\!\!/\mathbb{Z}, X).$$ Each object of
$I(X)$ can be viewed as pairs $(x, g)$ with $x\in \mbox{ob}(X)$
and $g\in aut(x)$, $gx=x$. A morphism from $(x_1, g_1)$ to $(x_2,
g_2)$ is a morphism $h: x_1\longrightarrow x_2$ in $X$ satisfying
$h\circ g_1=g_2\circ h$ in $X$. So in $I(X)$,
$$\mbox{Hom}((x_1, g_1), (x_2, g_2))=\{h: x_1\longrightarrow x_2| h\circ g_1=g_2\circ
h\}.$$

Let $\xi^k$ denote the center element of $I(X)$ sending $(x, g)$
to $(x, g^k)$. We will use $\xi$ to denote $\xi^1$. For any $k\in
\mathbb{Z}$, we have the 2-functor
\begin{align*}\mathcal{G}pd &\longrightarrow \mathcal{G}pd^{cen}\\
X&\mapsto (I(X), \xi^k).
\end{align*}

\end{example}

\begin{example}In the global quotient case,   as indicated in Example
\ref{torsionquotient}, $I(X/\!\!/G)$ is isomorphic to
$\prod\limits_{g\in G_{conj}}X^g/\!\!/C_G(g)$. And the center
element $\xi^k|_{X^g}=g^k$.

\end{example}

Let $\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\xi}I(X)$ denote the
groupoid
$$(\mbox{pt}/\!\!/\mathbb{R})\times I(X)/\sim$$ with
$\sim$ generated by $1\sim\xi$.

\begin{definition}For any topological groupoid $X$, the quasi-elliptic
cohomology $QEll^*(X)$ is the orbifold K-theory
\begin{equation}K^*_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi}I(X)).\label{orbqecnora}\end{equation}




In other words, %we can define $QEll^*$ in this way. %\begin{definition}\label{qkrro}
for a topological groupoid $X$, $QEll(X)$ is defined to be a
subring of  $K_{orb}(X)\llbracket q^{\pm\frac{1}{|\xi|}}
\rrbracket$ that is the Grothendieck group of finite sums
$$\sum_{a\in\mathbb{Q}} V_a q^a$$ satisfying: %the \textit{rotation condition} with respect to $\xi$:
$$\mbox{for each }a\in\mathbb{Q} \mbox{, the coefficient }V_a \mbox{ is an }e^{2\pi ia}-\mbox{eigenbundle of }\xi.$$

In the global quotient case,
$$QEll^*(X/\!\!/G)=QEll^*_G(X).$$ \label{orbqecnoradef}\end{definition}

\begin{remark}
For any topological groupoid $X$, we can also consider the
category $$Loop_1(X):=Bibun(S^1/\!\!/\ast, X)$$ and formulate
$Loop_1^{ext}(X)$ by adding the rotation action by circle, as in
Section \ref{orbifoldloop}. And then, we can formulate the
subgroupoid $\Lambda(X)$ of  $Loop_1^{ext}(X)$ consisting of the
constant loops, which is isomorphic to
$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \xi}I(X)$.
\end{remark}



\subsection{Symmetric powers of orbifolds and its inertia
groupoid}\label{sb2}

In this section  we  introduce the total symmetric power of
groupoids, the inertia groupoid of it, the total symmetric power
of the inertia groupoid, and other groupoids that are relevant in
the construction of the power operation. In addition, in Lemma
\ref{eq1}, Lemma \ref{eq2} and Lemma \ref{eq3} we show the
relation between these groupoids.


$\mathcal{G}pd$ is a symmetric bimonoidal category with the
monoidal structures given by $(\sqcup, \emptyset)$ and $(\times,
\mbox{pt})$. Each groupoid $X\in \mbox{ob}(\mathcal{G}pd)$ is a
monoid with respect to $(\sqcup, \emptyset)$, via the folding map
$$d: X\sqcup X\longrightarrow X.$$ $X$ is a monoid with respect to $(\times, \mbox{pt})$ if and
only if $X$ itself is a symmetric monoidal category.

\begin{definition}[Total Symmetric Power]
We introduced the $n-$th symmetric power $X\wr\Sigma_n$ of a
groupoid $X$ in Definition \ref{symmetricpowergroupoid}. In
addition,  the total symmetric power of $X$ is defined to be
\begin{equation}S(X):=\coprod\limits_{n\geq 0}
X\wr\Sigma_n.\end{equation} $S$ is  a functor from $\mathcal{G}pd$
to $\mathcal{G}pd$.


It is exponential. We have $$S(X\sqcup Y)=\coprod\limits_{n\geq 0}
(X\sqcup Y)\wr\Sigma_n= (\coprod\limits_{n\geq 0}
X\wr\Sigma_n)\times (\coprod\limits_{m\geq 0} Y\wr\Sigma_m)=
S(X)\times S(Y).$$ In other words, the functor
$$S:(\mathcal{G}pd, \sqcup, \emptyset)\longrightarrow
(\mathcal{G}pd,\times,\mbox{pt})$$ is monoidal.

Moreover, $S$ maps monoids to monoids. $S(X)$ is a monoid with
respect to $(\ast, (\mbox{   }))$ where $$\ast:=S(d),$$ i.e. the
concatenation, and the unit $(\mbox{   })$ is the unique object in
$X\wr\Sigma_0$. The monoidal structure is symmetric.

The triple $$(S(X), \ast, (\mbox{   }))$$ is "free" symmetric
monoidal category on $X$ in the sense that the functor $$X\mapsto
(S(X), \ast, (\mbox{   }))$$ is left adjoint to the forgetful
functor from the category of monoids in $\mathcal{G}pd$ to
$\mathcal{G}pd$.
\end{definition}


%\subsection{The Inertia groupoid of the total symmetric power} \label{sb3}


\begin{example}[The inertia groupoid of the total symmetric power]
Now we consider the groupoid $I(S(X))$. It is the disjoint union
$$\coprod\limits_{n\geq 0}I(X\wr\Sigma_n).$$ Each object of
it is of the form $(\underline{x}, \underline{g}, \sigma, n)$
where $n\geq 0$ is an integer, $\underline{x}\in
ob(X\wr\Sigma_n)$, $(\underline{g}, \sigma)\in G\wr\Sigma_n$.
Since $(\underline{g}, \sigma)$ is an automorphism of
$\underline{x}$ in $X\wr\Sigma_n$, each arrow $g_i$ is a morphism
from $x_i$ to $x_{\sigma^{-1}(i)}$. The information of each object
in $I(S(X))$ is contained in $(\underline{g}, \sigma)$.

The groupoid $I(S(X))$ inherits a monoidal structure from $S(X)$:
$$(\underline{g}, \sigma)\ast(\underline{h}, \tau)=(\underline{g}, \underline{h}, \sigma\sqcup\tau).$$

An object $(\underline{g}, \sigma)$ of $I(S(X))$ is indecomposable
with respect to $\ast$ if and only if $n>0$ and
$\sigma\in\Sigma_n$ has only one orbit.\end{example}



Moreover, for groupoids  like
$\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X$, instead of
$S(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$, we consider a
subgroupoid
$$S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$$ of
it.

\begin{definition}[The groupoid $S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$]
Let $$\rho_k:
\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X\longrightarrow
\mbox{pt}/\!\!/(\mathbb{R}/\mathbb{Z})$$ be the functor sending
all the objects to the single point, and an arrow $$[g, t]$$ to
$$t\mbox{ mod }\mathbb{Z}.$$

Let
$\times_{\mathbb{R}}(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$
denote the limit of the diagram of groupoids
$$\xymatrix{\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X \ar[r]^{\rho_k}
&\mbox{pt}/\!\!/(\mathbb{R}/\mathbb{Z})
&\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X \ar[l]_{\rho_k}}.$$

And let
$$\times^n_{\mathbb{R}}(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$$
denote the limit of $n$ morphisms $\rho_k$s. It inherits a
$\Sigma_n-$action on it by permutation from that on the product
$(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)^{\times n}$.


Let $S_n^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$ denote
the groupoid with the same objects as
$$\times^n_{\mathbb{R}}(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$$
and morphisms of the form $([g_1, t_1], \cdots [g_n, t_n];
\sigma)$ with $([g_1, t_1], \cdots [g_n, t_n])$ a morphism in
$\times^n_{\mathbb{R}}(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$
and $\sigma\in\Sigma_n$. This new groupoid
$S_n^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$ is a
subgroupoid of $$(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)\wr
\Sigma_n.$$

Define
\begin{equation}S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X):=\coprod\limits_{n\geq
0}S_n^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X).\end{equation}

The triple $$(S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X),
\ast, (\mbox{  }))$$ is  a symmetric monoid.
\end{definition}


$S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}X)$ is the
symmetric product that we will use to formulate the power
operation of quasi-elliptic cohomology.


\begin{definition}[The groupoid $\Phi(X)$]
Let $\Phi_k(X)$ denote the full subgroupoid of $I(S(X))$ whose
objects are $(\underline{x}, \underline{g}, \varsigma_k)$ with
$\varsigma_k$ a generater in the cyclic group $\langle (1 2 \cdots
k)\rangle$. And let $\phi_k\in$Center$(\Phi_k)$ denote the
restriction of
$S_k(\xi)$ to $\Phi_k$. %%%

Let $\Phi(X):=\coprod\limits_{k\geq 1}\Phi_k(X)$. Then
$$\phi:=\coprod\limits_{k\geq 1}\phi_k\in\mbox{Center}(\Phi)$$ is the
restriction of $S(\xi)$ to $\Phi$.

Thus, the essential image of $\Phi(X)$ in $I(SX)$ is the
subgroupoid consisting of indecomposable objects. And the functor
$\Phi$ is additive: $\Phi(\emptyset)=\emptyset$ and it preserves
$\sqcup$.
\end{definition}

\begin{lemma}For each integer $k\geq 1$, there is an equivalence between $$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi_k}\Phi_k(X)$$ and the groupoid
$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}]$ which identifies
$\phi_k$ with $\xi^{\frac{1}{k}}$. Here $\xi^{\frac{1}{k}}$ is an
added element such that the composition of $k$
$\xi^{\frac{1}{k}}$s is $\xi$. \label{eq1}\end{lemma}


%\begin{proof}This is a straightforward corollary of Lemma 3.5 in \cite{Gan13}.\end{proof}

\begin{proof}%Lemma \ref{eq1} is a special case of Theorem \ref{iptm} that the element $\sigma\in\Sigma_k$ has only one cycle.

We can define a functor $$A_k:
\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi_k}\Phi_k(X)\longrightarrow
\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}]$$ by sending an object
$(\underline{x}, \underline{g}, (1 2 \cdots k))$ to $(x_1,
g_k\cdots g_1)$ and sending a morphism $[\underline{h}, (1 2\cdots
k)^m, t]$ to $$[h_kg^{-1}_{1-m}\cdots g^{-1}_{k-1}g^{-1}_k,
m+t].$$

%Let's denote this functor by $A_k$.
Recall $h_kg^{-1}_{1-m}\cdots g^{-1}_{k-1}g^{-1}_k$ conjugates
$g_k\cdots g_1$ to itself. It is the element
$$\beta^{\underline{h}, \mbox{Id}}_{(1 2\cdots k), (1 2\cdots
k)}$$ defined in (\ref{betadef}). The functor $A_k$ is an
isomorphism, as implied in the proof of Theorem \ref{iptm}.

\end{proof}

\begin{remark}
As an immediate corollary,  we have the equivalence between the
disjoint union of groupoids $\coprod\limits_k
\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi_k}\Phi_k(X)$ and
$\coprod\limits_k\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}]$ where $k$ goes over all
the integers.

Moreover, the symmetric power
$S(\coprod\limits_k\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi_k}\Phi_k(X))$
is equivalent to the groupoid
$S(\coprod\limits_k\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}])$. %Why we need $S^R$ here but not $S$ is because the existence of the $t$ part.
The formula of the equivalence is straightforward to obtain from
the functors $A_k$s.
\end{remark}


%\begin{remark}$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi}\Phi(X)$ is a "global" version of $A_{\underline{g}, \sigma}(X)$ in \cite{HZ}.
%And the equivalence in Lemma \ref{eq1} is a "global" version of Theorem 7 in \cite{HZ}. \end{remark}


Theorem \ref{iptm} can be reinterpreted as the lemma below.
%, where the $t$ part makes a difference and we need the product $S^R$.


\begin{lemma}
The groupoid
$S^R(\coprod\limits_k\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}])$ is equivalent to
$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)} I(S(X))$.
\label{eq2}
\end{lemma}

%Most details in the proof of Lemma \ref{eq2} come from that of Theorem \ref{iptm}.

\begin{proof}
We define a functor $F$ from
$$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)} I(S(X))$$ to
$$S^R(\coprod\limits_k\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X)[\xi^{\frac{1}{k}}]).$$

Let $$(\underline{x}, \underline{g}, \sigma)$$ be an object in the
 category $\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
S(\xi)} I(S(X))$ where $\sigma\in \Sigma_n$. $F$ sends it to
$$\prod_k\prod_{(i_1, \cdots i_k)}(x_{i_1}, g_{i_k}\cdots g_{i_1})$$ where the second product
goes over all the $k-$cycles of $\sigma$.


Let $$[\underline{h}, \tau; t]$$ be a morphism starting at the
object $(\underline{x}, \underline{g}, \sigma)$.

Let $\tau$ send the $k-$cycle $i=(i_1, \cdots i_k)$ of $\sigma$ to
a $k-$cycle $j=(j_1, \cdots j_k)$ of $\sigma$ and
$\tau(i_1)=j_{1+m_i}$.

$F$ sends $[\underline{h}, \tau; t]$ to
$$\times_k\times_{i\in\theta_k}([\beta^{\underline{h},
\tau}_{\tau(1), 1}, m_1+t], \cdots [\beta^{\underline{h},
\tau}_{\tau(M^{\sigma}_i), M^{\sigma}_i}, m_{M^{\sigma}_i}+t],
\tau|_{W^{\sigma}_i})$$ where $\tau|_{W^{\sigma}_i}$ denotes the
permutation induced by $\tau$ on the set
$W^{\sigma}_i=\{\alpha^i_1, \alpha^i_2, \cdots
\alpha^i_{M^{\sigma}_i}\}$, $\tau^{-1}(j)$ is short for $\tau^{-1}
(\alpha^i_j)$, and $\tau(j_l)=\tau(j)_{l+m_j}$.

$F$ is fully faithful and essentially surjective. The proof is
similar to that of Theorem \ref{iptm}.
%And it's straightforward to check that $F$ is  monoidal.
\end{proof}




\begin{lemma}We have an equivalence of %monoidal
 groupoids $$Q^R: S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \phi}\Phi(X))\longrightarrow \mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)}
I(S(X)),$$ which is natural in $X$ and satisfies
$$Q^RS^R(\phi)=S(\xi) Q^R.$$\label{eq3}\end{lemma} %check

\begin{proof}Let $I$ be the inclusion $$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \phi}\Phi(X)\longrightarrow \mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)}
I(S(X)).$$ Let $\epsilon$ be the counit of the adjunction
$(S,\ast, (\mbox{  }))\dashv \mbox{forget}$. Let $Q$ be the
composition
$$S(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \phi}\Phi(X))\buildrel{S(I)}\over\longrightarrow S(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)}
I(S(X)))\buildrel{\epsilon}\over\longrightarrow\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
S(\xi)} I(S(X)).$$

Let $Q^R$ be the restriction of $Q$ to the subgroupoid
$S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \phi}\Phi(X))$, i.e.
the composition \begin{align*}Q^R:
S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\phi}\Phi(X))&\buildrel{S^R(I)}\over\longrightarrow
S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)} I(S(X)))\\
&\buildrel{\mbox{restriction of
}\epsilon}\over\longrightarrow\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
S(\xi)} I(S(X)).\end{align*}

The essential image of $I$ consists exactly of the indecomposable
objects of $\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\xi)}
I(S(X))$, thus, both $Q$ and $Q^R$ are essentially surjective.

%Since $\sigma$ and $\tau$ are conjugate in $\Sigma_n$ if and only if their cycle decompositions are congruent. It follows that

$Q$ is not fully faithful but $Q^R$ is. This is why we need the
product $S^R$ instead of $S$.
%The proof for this fact in detail is analogous to that of Theorem 7 in \cite{HZ}, where the groupoid $X$ is the loop space, the group
%$\Lambda_{G\wr\Sigma_n}(\underline{g}, \sigma)$ is the set of morphisms starting at an object $(\underline{\gamma}, \underline{g}, \sigma)$ when $(\underline{g}, \sigma)$ is fixed.


\end{proof}



%\begin{corollary}\label{eq0} Moreover, we have an equivalence of monoidal groupoids $$L^R: S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
%\phi}\Phi(X))\longrightarrow \mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \phi}S\Phi(X).$$ $\phi$ is sent to itself.\end{corollary}
%\begin{proof} By Lemma 3.4 in \cite{Gan13}, $S\Phi(X)$ is equivalent to $\Lambda(SX)$ as monoidal groupoids. Composing with the
%equivalence in Lemma \ref{eq3}, we get the conclusion. \end{proof}


\section{Power Operation for orbifold quasi-elliptic
cohomology}\label{s2}

In this section we define power operations for orbifold
quasi-elliptic cohomology, which satisfy the axioms that Ganter
formulated in \cite{Gan13} for power operations for orbifold
theories.


We first recall the definition of power operation for cohomology
theories for orbifolds and construct Atiyah's power operation for
quasi-elliptic cohomology in Example \ref{atiyahell}. Then we
construct the total power operation for the orbifold
quasi-elliptic cohomology
%$$P^{Ell}: K^R_{rot}(X,\xi)\longrightarrow K^R_{rot}(S(\Phi(X)), S(\phi))$$
$$P^{Ell}: QEll(X)\longrightarrow QEll(SX)$$
%K^*_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \xi}I^{tors}(X))\longrightarrow
%K^*_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim S(\phi)}I^{tors}(S(\Phi(X))))$$
for any orbifold groupoid $X$, as shown in (\ref{main2porb}). %$K^R_{rot}$ is defined in Definition \ref{krro}.
The power operation we constructed in Section \ref{s2complete} is
a special case of it for $G-$spaces.

%\subsection{Atiyah's Power Operation}\label{atiyahpower}

%Definition \ref{ng1} and Example \ref{ng2} are from Nora Ganter's paper \cite{Gan13}.

First we recall the definition of the total power operation for
orbifold theories given in Definition 3.8, \cite{Gan13}.

\begin{definition}\label{ng1}
Let $E$ be a cohomology theory for orbifolds with product and let
$S$ be the symmetric power product on the orbifolds. A total power
operation for $E$ is a (non-linear) natural transformation

$$P: E\Rightarrow E\circ S$$ satisfying

$\bullet$ Comodule property: $P$ makes $E$ a comodule over the
comonad $(-)\circ S.$


$\bullet$ Exponentiality: The map $$P: E(\emptyset)\longrightarrow
E(\mbox{pt})$$ sends 0 to 1, and $$P : E(X \sqcup Y
)\longrightarrow E(SX\times SY)$$ sends $(a, b)$ to the external
product $P(a)\otimes  P(b)$.

We can write $P$ as $P=(P_n)_{n\geq 0},$ with $P_n:=l_n^*P$. Then
$P_n$ is called the $n$th power operation of $P$. \end{definition}


\begin{example}
We recall the Atiyah's power operations for equivariant K-theories
defined in \cite{Ati66}. In real or complex K-theory, let $x=[V]$
be represented by a $G$-equivariant vector bundle $V$ over a
$G-$space $X$. Then the $n$th exterior product $P_n(V) :=
V^{\boxtimes n}$ over $X^{\times n}$ has an obvious
$G\wr\Sigma_n-$structure.

Then we consider the case when $X$ is a topological orbifold. Let
$V$ be an orbifold vector bundle over $X$. Then
$P_n(V):=V^{\boxtimes n}$ is an orbifold vector bundle over
$X^{\times n}$. $P=(P_n)_{n\geq 0}$ satisfies the axioms of a
total power operation.

 \label{ng2}\end{example}



\begin{example}\label{atiyahell}
We consider the case when $E$ is quasi-elliptic cohomology. Let
$X$ be a topological orbifold. As defined in Section \ref{sb1},
$\xi^k$ is the center element on $I(X)$ sending $(x,g)$ to
$(x,g^k)$. Let $V$ be an orbifold vector bundle over the orbifold
$$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \xi}I(X),$$ thus, $V$
represents an element in $QEll(X)$.

Recall $QEll(X)$ is a $\mathbb{Z}[q^{\pm}]-$algebra via the
inclusion $$\mathbb{Z}[q^{\pm}]\cong
QEll(\mbox{pt})\longrightarrow QEll(X),$$ and the exterior product
$\boxtimes$ in quasi-elliptic cohomology is defined as the tensor
product of $\mathbb{Z}[q^{\pm}]-$algebras. Then $$P_n(V):=
V^{\boxtimes n}$$ is an orbifold vector bundle over
$$S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \xi}I(X)),$$ which
 is a groupoid equivalent to
$$\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim \xi}SI(X).$$ So $P_n(V)$
is in $QEll^*(S(X))$.

$P=(P_n)_{n\geq 0}$ satisfies the axioms of a total power
operation.


\label{apel}\end{example}



%\subsection{Power operation in $QEll^*$}\label{powerorbifold}
%Recall $K^R_{rot}(X,\xi)= K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\xi}I^{tors}(X)).$

Before the construction of the power operation of $QEll$, we
introduce several maps necessary for the construction of the power
operation.

Let $X$ be an orbifold groupoid %and $\xi$ the element of center defined in Section \ref{sb1},
and $k\geq 1$ an integer. We define
the map
\begin{align}
s_k:
K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\xi}I(X))&\longrightarrow
K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}I(X))\\
[\sum V_aq^a]&\mapsto[\sum V_a q^{\frac{a}{k}}]\end{align} and
\begin{equation}\coprod\limits_k s_k:
K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\xi}I(X))\longrightarrow
K_{orb}(\coprod\limits_k(\mbox{pt}/\!\!/\mathbb{R}\times_{k\sim\xi}I(X))).\end{equation}



The functor $$(\mbox{  })_k: \Lambda_{(\underline{g},
\sigma)}(X)\longrightarrow \Lambda^1_{(\underline{g},
\sigma)}(X)$$ defined in (\ref{lpok})  %in   Section \ref{lov}
is a special local case of $s_k$ when $X$ is a $G-$space and
$(\underline{g}, \sigma)$ is fixed.



\bigskip

Let $\theta: QEll(X)\longrightarrow
K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi}\Phi(X))$    %K^R_{rot}(\Phi(X),   \phi)$
 be the additive operation whose $k-$th component is
$A_k^*\circ s_k$, where $A_k$ is the equivalence defined in Lemma
\ref{eq1}.


\bigskip

Now we are ready to define the total power operation $P^{Ell}$ of
$QEll^*$ as the composition below:

\begin{equation}\xymatrix{ QEll(X)\ar[r]^>>>>>{\theta}
&K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim\phi}\Phi(X))\ar[r]^>>>>>{P}
&K_{orb}(S^R(\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\phi}\Phi(X)))\ar[d]^>>>>>>{(Q^{R *})^{-1}} \\ &&QEll(SX).}
\label{main2porb}\end{equation}

\begin{theorem}$P^{Ell}$ satisfies the axioms of a total power
operation in Definition \ref{ng1}.\end{theorem}

\begin{proof}
From the definition of $P^{Ell}$, we can see it is a well-defined
natural transformation $QEll\Rightarrow QEll\circ S$ and is a
comodule over the comonad $(-)\circ S$.

In addition, the functor $\theta$ has the property of additivity
\begin{align*}\theta: QEll(X\sqcup Y)&\longrightarrow QEll(\Phi(X)\sqcup\Phi(Y))\\ (a, b)&\mapsto (\theta(a), \theta(b)). \end{align*}
And the power operation $P$ defined in Example \ref{atiyahell} has
the exponential property. Therefore, $P^{Ell}$ has the property of
exponential. So $P^{Ell}$ is a total power operation.
%since The exponential property follows immediately from that of $P$ and from additivity of $\theta$.
\end{proof}


\begin{remark}
Let $X/\!\!/G$ be a quotient orbifold. The power operation we
constructed in Section \ref{111} for quotient orbifolds is in fact
the one below, if expressed totally:

$\mathbb{P}: QEll^*(X/\!\!/G)\buildrel{\coprod\limits_k
s_k}\over\longrightarrow K^*_{orb}(\coprod\limits_k
\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X/\!\!/G)[\xi^{\frac{1}{k}}])\buildrel{P}\over\longrightarrow
\\ K^*_{orb}(S^R(\coprod\limits_k
\mbox{pt}/\!\!/\mathbb{R}\times_{1\sim
\xi^{\frac{1}{k}}}I(X/\!\!/G)[\xi^{\frac{1}{k}}]))\buildrel{J^*}\over\longrightarrow
QEll^*(S(X/\!\!/G))$ where $J$ is constructed from the functors
$J_{(\underline{g}, \sigma)}$ in the proof of Theorem \ref{iptm}.


For global quotient orbifolds, $P^{Ell}$ and $\mathbb{P}$ are the
same up to isomorphism. The diagram

$$\xymatrix{QEll^*(X/\!\!/G)\ar[d]^{\theta} &QEll^*(S(X/\!\!/G)) \\ K_{orb}(\mbox{pt}/\!\!/\mathbb{R}\underset{1\sim\phi}\times
I(\Phi(X/\!\!/G)))\ar[r]^{P}
&K_{orb}(S^R(\mbox{pt}/\!\!/\mathbb{R}\underset{1\sim
\phi}\times\Phi(X/\!\!/G)))\ar[u]^>>>>>{(Q^{R*})^{-1}}
\\
 K_{orb}(\coprod\limits_k \mbox{pt}/\!\!/\mathbb{R}\!\!\underset{1\sim
\xi^{\frac{1}{k}}}\times\!\!
I(X/\!\!/G)[\xi^{\frac{1}{k}}])\ar[r]^>>>>>{P}\ar[u]^{\coprod\limits_k
A_k^*} &K_{orb}(S^R(\coprod\limits_k
\mbox{pt}/\!\!/\mathbb{R}\!\!\underset{1\sim
\xi^{\frac{1}{k}}}\times\!\!
I(X/\!\!/G)[\xi^{\frac{1}{k}}]))\ar[u]^{S^R(\coprod\limits_k
A_k^*)} }$$ commutes. The vertical maps $\coprod\limits_k A_k^*$
and $S^R(\coprod\limits_k A_k^*)$ are both equivalences of
groupoids. And the horizontal maps are the power operation defined
in Example \ref{atiyahell}.

\end{remark}

\section{Finite Subgroups of the Tate Curve}%$N-$torsion points on Tate curves}
\label{stricklandtorsion}

Strickland showed in \cite{Str98} that the quotient of the Morava
E-theory of the symmetric group by a certain transfer ideal can be
identified with the product of rings $\prod\limits_{k\geq 0}R_k$
where each $R_k$ classifies subgroup-schemes of degree $p^k$  in
the  formal group associated
to $E^0\mathbb{C}P^{\infty}$.  % what's this interpretation
In this section we prove similar conclusions for Tate K-theory and
quasi-ellitpic cohomology. The main conclusion for Section
\ref{stricklandtorsion} is Theorem \ref{stricktheorem}.

\subsection{Background}\label{tatecurve}

In this section we introduce the Tate curve and its finite
subgroups. The main references are Section 2.6 in \cite{AHS} and
Section 8.7, 8.8 in \cite{KM85}.

An elliptic curve over the complex numbers $\mathbb{C}$ is a
connected Riemann surface, i.e. a connected compact 1-dimensional
complex manifold, of genus 1. By the uniformization theorem every
elliptic curve over $\mathbb{C}$ is analytically isomorphic to a
1-dimensional complex torus, and can be expressed as
$$\mathbb{C}^*/q^{\mathbb{Z}}$$ with $q\in\mathbb{C}$ and
$0<|q|<1$, where $\mathbb{C}^*$ is the multiplicative group
$\mathbb{C}\backslash\{0\}.$

The Tate curve  $Tate(q)$  is the elliptic curve $$E_q:
y^2+xy=x^3+a_4x+a_6$$ whose coefficients are given by the formal
power series in $\mathbb{Z}((q))$
$$a_4=-5\sum_{n\geqslant 1} n^3q^n/(1-q^n)   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,   a_6=-\frac{1}{12}\sum_{n\geqslant 1}(7n^5+5n^3)q^n/(1-q^n).$$

%Let $A\subset\mathbb{Z}((q))$ be the subring consisting of power series which absolutely converge on the open unit disk $$\{q\in\mathbb{C} | |q|<1\}.$$
%The Tate curve Tate$(q)$ is the generalized elliptic curve over $spec\mathbb{Z}((q))$ obtained by change of base along the inclusion $A\subset \mathbb{Z}((q))$.

%Tate$(q)\otimes_{\mathbb{Z}((q))}\mathbb{C}$ is analytically
%isomorphic to $\mathbb{C}^*/q^{\mathbb{Z}}$.


Before we talk about the torsion part of $Tate(q)$, we recall a
smooth one-dimensional commutative group scheme $T$ over
$\mathbb{Z}[q^{\pm}]$. It sits in a short exact sequence of
group-schemes over $\mathbb{Z}[q^{\pm}]$
$$0\longrightarrow \mathbb{G}_m\longrightarrow T\longrightarrow \mathbb{Q}/\mathbb{Z}\longrightarrow 0.$$

 The $N-$division points
$T[N]$ of it is the disjoint union of $N$ schemes $T_0[N]$,
$\cdots$ $T_{N-1}[N]$, where
$$T_i[N]=\mbox{Spec}(\mathbb{Z}[q^{\pm}][x]/(x^N-q^i)).$$ It fits
into a short exact sequence $$0\longrightarrow
\mu_N\buildrel{a_N}\over\longrightarrow
T[N]\buildrel{b_N}\over\longrightarrow
\mathbb{Z}/N\mathbb{Z}\longrightarrow 0,$$ %cite the e_N-pairing?
The canonical extension structure on $T(N)$ is compatible with an
alternating paring of $\mathbb{Z}[q^{\pm}]-$group schemes $e_N:
T(N)\times T(N)\longrightarrow \mu_N$ in the sense that
$$e_N(a_N(x), y)= x^{b_N(y)}\mbox{,   for any  }\mathbb{Z}[q^{\pm}]-\mbox{algebra } R\mbox{   and  any }x\in\mu_N(R).$$

And we have the conclusion below, which is Theorem 8.7.5 in
\cite{KM85}.

\begin{theorem}
There exists a faithfully flat $\mathbb{Z}[q^{\pm}]-$algebra $R$,
an elliptic curve $E/R$, and an isomorphism of ind-group-schemes
over $R$ $$T_{torsion}\otimes_{\mathbb{Z}[q^{\pm}]}
R\buildrel\sim\over\longrightarrow E_{tors},$$ such that for every
$N\geq 1$, the isomorphism on $N-$division points $T[N]\otimes
R\buildrel\sim\over\longrightarrow E[N]$ is compatible with
$e_N-$pairings. \label{torsionT}
\end{theorem}

Thus, we have the unique isomorphism of ind-group-schemes on
$\mathbb{Z}((q))$ $$T_{torsion}\otimes_{\mathbb{Z}[q^{\pm}]}
\mathbb{Z}((q))\buildrel\sim\over\longrightarrow Tate(q)_{tors}.$$
The isomorphism is compatible with the canonical extension
structure: for each $N\geq 1$, $$\xymatrix{0 \ar[r]
&\mu_N\ar[d]_{=}\ar[r]
&T[N] \ar[r]\ar[d]_{\cong} &\mathbb{Z}/N\mathbb{Z} \ar[r] \ar[d]_= &0\\
0\ar[r] &\mu_N \ar[r] &Tate(q)[N] \ar[r]
&\mathbb{Z}/N\mathbb{Z}\ar[r] &0}$$




Therefore, $Tate(q)[N]$  %, the scheme of points of order  $N$ in Tate$(q)$,  %is defined to be $$\mbox{Spec}(K_{Tate}(\mbox{pt}/\!\!/C_N)).$$ It
is isomorphic to the disjoint union
$$\coprod\limits_{k=0}^{N-1}\mbox{Spec}(\mathbb{Z}((q))[x]/(x^N-q^k)).$$


In addition, we have the question how to classify all the finite
subgroups of $Tate(q)$.  As shown in Proposition 6.5.1 in
\cite{KM85}, the ring $O_{Sub_n}$ that classifies subgroups of
$Tate(q)$ of order $n$ exists.


To give a description of $O_{Sub_n}$, first we describe the
isogenies for the analytic Tate curve over $\mathbb{C}$.
%The answer is already known, as shown below.

To give a subgroup for each order $N$, pick a pair of integers
$(d, e)$ and a nonzero complex number $q'$ such that $N=de$ and
$d, e\geq 1$. Let $q'$ be a nonzero complex number such that
$q^d=q'^e$. Consider the map
\begin{align*}\psi_d: \mathbb{C}^*/q^{\mathbb{Z}}&\longrightarrow \mathbb{C}^*/q'^{\mathbb{Z}}\\x&\mapsto x^d. \end{align*}
It is well-defined since $\psi_d(q^{\mathbb{Z}})\subseteq
q'^{\mathbb{Z}}$.

We can check that Ker$\psi_d$ has order $N$. Explicitly, it is
$$\{\mu_d^nq^{\frac{m}{e}}q^{\mathbb{Z}}| n, m\in\mathbb{Z}\}$$ where $\mu_d$ is a $d-$th primitive root of 1 and $q^{\frac{1}{e}}$ is a $e-$th primitive root of $q$.
In fact $$\{\mbox{Ker}\psi_d| \mbox{   } d \mbox{  divides
}N\mbox{  and  }d\geq 1\}$$ gives all the subgroups of
$\mathbb{C}^*/q^{\mathbb{Z}}$ of order $N$.


\begin{proposition}For each pair of number $(d, e)$, there exists an isogeny
$$\Psi_{d, e}: Tate((q))\longrightarrow Tate((q'))$$ of the elliptic curves over $O_{Sub_n}$ such that
its kernel is the universal subgroup.  \end{proposition}


We have
$$O_{Sub_n}\otimes\mathbb{C}=\prod_{N=de}\mathbb{C}((q))[q']/\langle
q^d-q'^e \rangle.$$ Moreover, we have  the conclusion below.


\begin{proposition} The finite subgroups of the
Tate curve are the kernels of isogenies. \end{proposition}


\subsection{Formulas for Induction} \label{inductionqell} %transfer?

%To prove Theorem \ref{stricktheorem}, we need the formulas of induction in Tate K-theory, which is given in \cite{Gan13}. I give
%the induction formula in quasi-elliptic cohomology.

Before the main conclusion,  we introduce the induction formula
for quasi-elliptic cohomology. The induction formula for Tate
K-theory is constructed in Section 2.3.3 in \cite{Gan13}.



Let $H\subseteq G$ be an inclusion of compact Lie groups and $X$
be a $G-$space. Then we have the inclusion of the groupoids
$$j:X/\!\!/H\longrightarrow X/\!\!/G.$$

%The induced transfer $\mathcal{I}^G_H$ for  $QEll$ is constructed in a similar way.
Let $a'=\prod\limits_{\sigma\in H_{conj}}a'_{\sigma}$ be an
element in
$$QEll(X/\!\!/H)=\prod_{\sigma\in H_{conj}}K_{\Lambda_H(\sigma)}(X^{\sigma})$$ where $\sigma$ goes over all the conjugacy
classes in $H$.

We can define the finite covering map
$$f': \Lambda(G\times_H X/\!\!/G)\longrightarrow  \Lambda(X/\!\!/G)$$   sending  an object $(\sigma, [g, x])$ to  $(\sigma, gx)$ and
a morphism  $([g', t], (\sigma, [g, x]))$ to $([g', t], (gx,
\sigma))$.



The transfer of  quasi-elliptic cohomology $$\mathcal{I}_H^G:
QEll(X/\!\!/H)\longrightarrow QEll(X/\!\!/G)$$ is defined to be
the composition \begin{equation}
QEll(X/\!\!/H)\buildrel\cong\over\longrightarrow
QEll((G\times_HX)/\!\!/G)\longrightarrow
QEll(X/\!\!/G)\label{qelltransfer}\end{equation} where the first
map is the change-of-group isomorphism and the second is the
finite covering.


Thus
$$\mathcal{I}^G_H(a')_{g}=\sum_{r}r\cdot a'_{r^{-1}gr}$$ where $r$ goes
over a set of representatives of $(G/H)^{g}$, in other words,
$r^{-1}gr$ goes over a set of representatives of conjugacy classes
in $H$ conjugate to $g$ in $G$.

\begin{equation}\mathcal{I}^G_H(a')_{g}=\begin{cases}Ind^{\Lambda_G}_{\Lambda_H}(a'_{g}) &\mbox{if  }g\mbox{ is conjuate to some element  }h \mbox{  in
}H;\\ 0 &\mbox{if there is no element conjugate to }g\mbox{  in
}H.
\end{cases}\end{equation}

\bigskip
There is another way to describe the transfer, which is shown in
\cite{Rez11} for quasi-elliptic cohomology. %I'll show it below.
The transfer of Tate K-theory can be described similarly. %type the construction here



\subsection{The main theorem}\label{proofstrict}

Theorem \ref{stricktheorem} gives a classification of finite
subgroups of the Tate curve and a similar conclusion for the
quasi-elliptic cohomology. In Section \ref{proofstrict} we show
the proof of Theorem \ref{stricktheorem}. We assume the readers
are familiar with the transfer ideal $I_{tr}$ of equivariant
K-theory. References for that include Chapter II, \cite{LMSM} and
Section 1.8, \cite{Rez06}.

Let $N$ be an integer. Analogous to the transfer ideal $I_{tr}$ of
equivariant K-theory,  we can define the transfer ideal for Tate
K-theory
\begin{equation}I^{Tate}_{tr}:= \sum_{\substack{i+j=N,\\
N>j>0}}\mbox{Image}[I^{\Sigma_N}_{\Sigma_i\times\Sigma_j}:
K_{Tate}(\mbox{pt}/\!\!/\Sigma_i\times\Sigma_j)\longrightarrow
K_{Tate}(\mbox{pt}/\!\!/\Sigma_N)]\label{transferidealtatek}\end{equation}
where $I^G_H$ is the transfer map of $K_{Tate}$ along
$H\hookrightarrow G$ defined in Proposition 2.23 in \cite{Gan13},
and the transfer ideal for quasi-elliptic cohomology
\begin{equation}\mathcal{I}^{QEll}_{tr}:= \sum_{\substack{i+j=N,\\
N>j>0}}\mbox{Image}[\mathcal{I}^{\Sigma_N}_{\Sigma_i\times\Sigma_j}:
QEll(\mbox{pt}/\!\!/\Sigma_i\times\Sigma_j)\longrightarrow
QEll(\mbox{pt}/\!\!/\Sigma_N)]\label{transferidealqec}\end{equation}
with $\mathcal{I}^G_H$ the transfer map of $QEll$ along
$H\hookrightarrow G$ defined in (\ref{qelltransfer}).

\begin{theorem}\label{stricktheorem} The Tate K-theory of symmetric groups modulo the transfer
ideal $I^{Tate}_{tr}$ classifies the finite subgroups of the Tate
curve. Explicitly,
\begin{equation}K_{Tate}(\mbox{pt}/\!\!/\Sigma_N)/I^{Tate}_{tr}\cong
\prod_{N=de}\mathbb{Z}((q))[q']/\langle q^d-q'^e
\rangle,\label{tatec}\end{equation} where  $q'$ is the image of
$q$ under the power operation $P^{Tate}$ constructed in Definition
3.15, \cite{Gan13}. The product goes over all the ordered pairs of
positive integers $(d, e)$ such that $N=de$.


\bigskip

We have the analogous conclusion for quasi-elliptic cohomology.
\begin{equation}QEll(\mbox{pt}/\!\!/\Sigma_N)/\mathcal{I}^{QEll}_{tr}\cong
\prod_{N=de}\mathbb{Z}[q^{\pm}][q']/\langle q^d-q'^e
\rangle,\label{ellc}\end{equation} where $q'$ is the image of $q$
under the power operation $\mathbb{P}_N$ constructed in Section
\ref{s2complete}. The product goes over all the ordered pairs of
positive integers $(d, e)$ such that $N=de$.

\end{theorem}





We show the proof of (\ref{ellc}). The proof of (\ref{tatec}) is
similar.



\begin{proof}[Proof of (\ref{ellc})]


We divide the elements in $\Sigma_N$ into two cases.


\textbf{Case I:}

The decomposition of $\sigma$ has cycles of different length. For
example, the element $$(1\mbox{  } 2)(3\mbox{  } 4)(5\mbox{  }
6)(7 \mbox{ }8\mbox{  } 9 \mbox{ }10)(11\mbox{ } 12\mbox{  }
13\mbox{ } 14)(15 \mbox{ }16\mbox{ } 17)\in\Sigma_{17}$$ is in
this case and $(1\mbox{   }2)(3\mbox{   }4)(5\mbox{   }6)$,
$(1\mbox{ }2\mbox{   }3\mbox{   }4\mbox{   }5)(6\mbox{   }7\mbox{
}8\mbox{ }9\mbox{   }10)$ are not.

Most elements in $\Sigma_N$ belong to Case I. $\sigma$ is not in
this case if and only if it consists of cycles of the same length,
such as $(1\mbox{  }2)(3\mbox{  }4)$, $(1\mbox{  }2\mbox{
}3)$, $1$, $(1\mbox{  }2\mbox{  }3)(4\mbox{  }5\mbox{  }6)$.


For those $\sigma$ that belong to Case I,
$\Lambda_{\Sigma_N}(\sigma)=\Lambda_{\Sigma_r\times\Sigma_{N-r}}(\sigma)$,
so
$Ind^{\Lambda_{\Sigma_N}(\sigma)}_{\Lambda_{\Sigma_r\times\Sigma_{N-r}}(\sigma)}$
is the identity map, so
$K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})$ is equal to
$Ind^{\Lambda_{\Sigma_N}(\sigma)}_{\Lambda_{\Sigma_r\times\Sigma_{N-r}}(\sigma)}
K_{\Lambda_{\Sigma_r\times\Sigma_{N-r}}(\sigma)}(\mbox{pt})$.
Thus, the summand corresponding to $\sigma$ in
$QEll(\mbox{pt}/\!\!/\Sigma_N)$ is completely cancelled. % by the image of the transfer.




\bigskip


\textbf{Case II:}


$\sigma$ consists of cycles of the same length. In other words, it
consists of $d$ $e-$cycles with $N=de$. %For example, $(1\mbox{
%}2)(3\mbox{  }4)\in \Sigma_4$ and $(1\mbox{ }2\mbox{  }3)(4\mbox{ }5\mbox{  }6)(7\mbox{ }8\mbox{ }9)(10\mbox{ }11\mbox{
%}12)\in\Sigma_{12}$ are both in this case.

The centralizer $C_{\Sigma_N}(\sigma)\cong C_e\wr \Sigma_d$, where
$C_e=\mathbb{Z}/e\mathbb{Z}$ is the cyclic group with order $e$.            % and is also $C_{\Sigma_e}(12\cdots e)$.
We have $$\Lambda_{\Sigma_N}(\sigma)\cong
\Lambda_{\Sigma_e}(12\cdots e)\wr_{\mathbb{T}}\Sigma_d$$ is the
subgroup of $\Lambda_{\Sigma_e}(12\cdots e)\wr\Sigma_d$ with
elements of the form $$([a_1,t], [a_2, t], \cdots [a_d, t];
\tau),\mbox{   with    }a_1, \cdots a_d\in C_e,\mbox{   } \tau\in
\Sigma_d, \mbox{  }t\in\mathbb{R}.$$
$K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})$ is the representation
ring $R\Lambda_{\Sigma_N}(\sigma)$. According to Theorem
\ref{repfibwr}, as a $\mathbb{Z}[q^{\pm}]-$module, it has the
basis
\begin{align*}\{&Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_{(d)}}(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_1}\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}(q^{\frac{a_r}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_r}\otimes
D_{\tau}\mbox{ }|\\&(d)=(d_1, d_2, \cdots d_r)\mbox{ is a
partition of }d. \\ &a_1, a_2, \cdots a_r\mbox{ are in
 }\{0, 1, \cdots e-1\}. \mbox{    And }\tau\in R\Sigma_{(d)}\mbox{  is  irreducible}.\}\end{align*}
where for each $a\in\mathbb{Z}$, $q^{\frac{a}{e}}:
\Lambda_{C_e}((12\cdots e))\longrightarrow U(1)$ is the map
\begin{equation}q^{\frac{a}{e}}([(12\cdots e)^j, t])=e^{2\pi
ia\frac{j+t}{e}}.\end{equation} Namely, it is the map $x_1^a$ in
the sense of Example \ref{ppex}. %1.7 in \cite{Rez11}.

For each partition $(d)$ of $d$, if it has more than one cycle, %it's not $(d)$,
$\Sigma_{(d)}$ is a subgroup of some $\Sigma_{d_1}\times
\Sigma_{d-d_1}$ for some positive integer $0<d_1<d$. So for each
$$Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_{(d)}}(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_1}\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}(q^{\frac{a_r}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_r}\otimes
D_{\tau}$$ with $r\geq 2$, it is equal to
\begin{align*}Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_{d_1}\times\Sigma_{d-d_1})}&(Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_{d_1}\times\Sigma_{d-d_1})}_{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_{(d)}}(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_1}\otimes_{\mathbb{Z}[q^{\pm}]}\\
&\cdots\otimes_{\mathbb{Z}[q^{\pm}]}(q^{\frac{a_r}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_r}\otimes
D_{\tau})\end{align*} by the property of induced representation.
Note that $$\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_{d_1}\times\Sigma_{d-d_1})\cong
\Lambda_{\Sigma_{d_1e}\times\Sigma_{N-d_1e}}(\sigma).$$ So
$$Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_{d_1}\times\Sigma_{d-d_1})}_{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_{(d)}}(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_1}\otimes_{\mathbb{Z}[q^{\pm}]}\cdots\otimes_{\mathbb{Z}[q^{\pm}]}(q^{\frac{a_r}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d_r}\otimes
D_{\tau}$$ is in
$K_{\Lambda_{\Sigma_{d_1e}\times\Sigma_{N-d_1e}}(\sigma)}(\mbox{pt})$,
Thus, each
base element with $r\geq 2$ is contained in the transfer ideal. %in the image of transfer.

When $r=1$, consider $$(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}\otimes D_{\tau}$$ with $\tau\in R\Sigma_d$. %We know
As indicated in Proposition 1.1 and Corollary 1.5 in \cite{Ati66},
each $\tau$, except the trivial representation of $\Sigma_d$, can
be induced from a representation $\tau'$ in some
$R(\Sigma_i\times\Sigma_{d-i})$ with $d>i>0$. %Since
%$(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}$ is a 1-dimensional representation,
%Note that there is a bijection between the sets of the right cosets $$\Sigma_d\setminus (\Sigma_i\times\Sigma_{d-i})\longrightarrow \big(\Lambda_{\Sigma_e}(12\cdots
%e)\wr_{\mathbb{T}}\Sigma_d\big)\setminus  \big(\Lambda_{\Sigma_{e}}(12\cdots e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})\big)$$ sending each
%$ (\Sigma_i\times\Sigma_{d-i}) \alpha$ to $\big(\Lambda_{\Sigma_{e}}(12\cdots e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})\big) \alpha$.
%By computing the character, we can see %compute the characters

 \textbf{Claim: }the representation $$Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})}
(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}i}\otimes_{\mathbb{Z}[q^{\pm}]}
(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}(d-i)} \otimes
D_{\tau'}$$ is isomorphic to
$$(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}\otimes
D_{Ind^{\Sigma_d}_{\Sigma_i\times\Sigma_{d-i}}\tau'},$$ which is
$$(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}\otimes D_{\tau}.$$


To prove this, we consider a set
$\{\tau_{\alpha}\}_{\alpha\in\Sigma_d/\Sigma_i\times\Sigma_{d-i}}$
of coset representatives. Then $$\{\eta_{\alpha}:=(1, \cdots 1;
\tau_{\alpha})\}_{\alpha\in\Sigma_d/\Sigma_i\times\Sigma_{d-i}}$$
is a set of coset representatives of
$$\big(\Lambda_{\Sigma_e}(12\cdots e)\wr_{\mathbb{T}}\Sigma_d  \big) /\big( \Lambda_{\Sigma_{e}}(12\cdots e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})\big).$$

Let $W$ be a representation space of
$\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})$, Then
$$Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})}W$$ is the direct
product of $[\Sigma_d :\Sigma_i\times\Sigma_{d-i}]$ copies of $W$.
For any element $$H=(g_1, \cdots g_d; \beta)\in
\Lambda_{\Sigma_e}(12\cdots e)\wr_{\mathbb{T}}\Sigma_d,$$ and each
$\alpha\in\Sigma_d/\Sigma_i\times\Sigma_{d-i}$, there is a unique
$\alpha'\in\Sigma_d/\Sigma_i\times\Sigma_{d-i}$ and a unique $$
J_{\alpha}=(g'_1, \cdots g'_d; \gamma_{\alpha})\in
\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})$$ such that
$H\eta_{\alpha}=\eta_{\alpha'}J_{\alpha}$. Note that $$g'_1,
\cdots g'_d$$ is a permutation of $$g_1, \cdots g_d.$$ So
$(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g'_1, \cdots
g'_d)= (q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g_1,
\cdots g_d).$ In addition,
$\beta\tau_{\alpha}=\tau_{\alpha'}\gamma_{\alpha}$. Let
$$\prod_{\alpha}w_{\alpha}$$ be an element in
$$Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})}W.$$ We have
\begin{align*}&\bigg(Ind^{\Lambda_{\Sigma_e}(12\cdots
e)\wr_{\mathbb{T}}\Sigma_d}_{\Lambda_{\Sigma_{e}}(12\cdots
e)\wr_{\mathbb{T}}(\Sigma_i\times\Sigma_{d-i})}
(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}i}\otimes_{\mathbb{Z}[q^{\pm}]}
(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d-i}
\otimes D_{\tau'}\bigg)(H)(\prod_{\alpha}w_{\alpha})\\
&=\prod_{\alpha}J_{\alpha}w_{\beta(\alpha)}
=\prod_{\alpha}(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g_1, \cdots g_d)D_{\tau'}(1,\cdots 1;\gamma_{\alpha})(w_{\beta\alpha})\\
&=(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g_1, \cdots g_d)\prod_{\alpha}\tau'(\gamma_{\alpha})(w_{\beta\alpha})\\
&=(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g_1, \cdots g_d)(Ind^{\Sigma_d}_{\Sigma_i\times\Sigma_{d-i}}\tau')(\beta)(\prod_{\alpha}w_{\alpha})\\
&=(q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}(g_1, \cdots g_d; \beta)D_{Ind^{\Sigma_d}_{\Sigma_i\times\Sigma_{d-i}}\tau'}(g_1, \cdots g_d; \beta)(\prod_{\alpha}w_{\alpha})\\
&= \big((q^{\frac{a_1}{e}})^{\otimes_{\mathbb{Z}[
q^{\pm}]}d}\otimes
D_{Ind^{\Sigma_d}_{\Sigma_i\times\Sigma_{d-i}}\tau'}\big)(g_1,
\cdots g_d; \beta)(\prod_{\alpha}w_{\alpha})\end{align*}


So the claim is proved.
\bigskip


%atiyah
Since $$\{Ind^{\Sigma_d}_{\Sigma_i\times\Sigma_{d-i}}\tau'\mbox{ }
|   \mbox{   }\tau'\in R(\Sigma_i\times\Sigma_{d-i})\mbox{ and }
i=1, 2, \cdots d-1.\}$$ contains all the irreducible
representation of $\Sigma_d$ except the trivial representation,
which is corresponding to the partition $(d)$,  thus, by the
claim, $K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})$ modulo the
image of the transfer, is a $\mathbb{Z}[q^{\pm}]-$module generated
by the equivalent classes represented by
\begin{equation}\{((q^{\frac{a}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d})^{\sim}\mbox{  }|\mbox{  } a=0, 1, \cdots
e-1\}.\label{get}\end{equation}

For any $a$, $(q^{\frac{a}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}$
is $(q^{\frac{1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}$ to the
$a-$th power. Note that, by (\ref{ee}),
$(q^{\frac{1}{e}})^{\otimes_{\mathbb{Z}[q^{\pm}]}d}$ is
$$q':=\mathbb{P}_{\sigma}(q).$$

%So $K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})$ modulo the image of the transfer is the polynomial ring $\mathbb{Z}[q^{\pm}][q']/(q'^e-q^d).$
%$q'$ is the only solution which is not in $\mathbb{Z}[q, \q^{-1}$ that solves the equation $x^e-q^d=0$, as
%indicated in Lemma \ref{uniq} below.

\bigskip

To get the isomorphism (\ref{ellc}), consider a map $$\Psi:
\mathbb{Z}[q^{\pm}][x]\longrightarrow
K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})/\mathcal{I}^{QEll}_{tr}$$
by sending $q$ to $q$ and $x$ to $q'$, which is a well-defined
$\mathbb{Z}[q^{\pm}]-$homomorphism.

Since $q'^e= q^d$,
$K_{\Lambda_{\Sigma_N}(\sigma)}(\mbox{pt})/\mathcal{I}^{QEll}_{tr}$
is a $\mathbb{Z}[q^{\pm}]-$module generated by $$1, q', \cdots
q'^{e-1}.$$ So any element in it can be expressed as
$$\sum_{j=0}^{e-1}f_j(q)q'^j$$ where each $f_j(q)$ is in the
polynomial ring $\mathbb{Z}[q^{\pm}]$. It is the image of
$$\sum_{j=0}^{e-1}f_j(q)x^j$$ in $\mathbb{Z}[q^{\pm}][x]$. So $\Psi$ is surjective.



Then we study its kernel. If $$F:=\sum_{j=0}^{e-1}f_j(q)q'^j$$
is in $\mathcal{I}^{QEll}_{tr}$, then it is in $\mathbb{Z}[q^{\pm}]$. % a better way to express??
So we can assume $F=0$.

%what conditions should the polynomials $f_j$  satisfy?
%Since $f_j$ are polynomials with finite terms, we can assume each $f_j\in\mathbb{Z}[q]$.

For each element $[(a_1, \cdots a_d; \beta), t]$ in
$\Lambda_{\Sigma_N}(\sigma)$ with $(a_1, \cdots a_d; \beta)\in
C_{\Sigma_N}(\sigma),$
\begin{equation}q([(a_1, \cdots a_d;
\beta), t])=e^{2\pi it},\end{equation} % might be correct..checked...correct
\begin{equation}q'([(a_1, \cdots a_d; \beta),
t])=e^{\frac{2\pi i(a_1+\cdots a_d+dt)}{e}}.\end{equation}

\begin{align*}F([(a_1, \cdots a_d; \beta),
t])&=\sum_{j=0}^{e-1}f_j(q)q'^j([(a_1, \cdots a_d; \beta),
t])\\=\sum_{j=0}^{e-1}f_j(e^{2\pi it})e^{\frac{2\pi ij(a_1+\cdots
+a_d+dt)}{e}} &=\sum_{j=0}^{e-1}f_j(e^{2\pi it})e^{\frac{2\pi
ijdt}{e}}e^{\frac{2\pi ij(a_1+\cdots +a_d)}{e}}.\end{align*}


Let $$F_j(t): =f_j(e^{2\pi it})e^{\frac{2\pi ijdt}{e}}$$ be the
complex-valued function in the variable $t$. And let $\alpha$
denote the number $e^{\frac{2\pi i}{e}}$. The integers
$$(a_1+\cdots +a_d)$$ go over $0, 1, \cdots e-1$. Consider the
$e$ equations
$$\sum_{j=0}^{e-1}F_j(t)\alpha^{jk}=0\mbox{, for }k=0, 1, \cdots e-1.$$

In other words, \[ \left( \begin{array}{ccccc}
1 & 1 & 1 &\cdots &1 \\
1 & \alpha & \alpha^2 &\cdots &\alpha^{e-1} \\
1 & \alpha^2 & \alpha^4 &\cdots &\alpha^{2(e-1)}\\
\vdots&\vdots&\vdots&&\vdots\\
1 &\alpha^{e-1} &\alpha^{2(e-1)} &\cdots &\alpha^{(e-1)^2}
\end{array} \right)\left(
\begin{array}{c}
F_0(t) \\
F_1(t) \\
F_2(t) \\
\vdots \\
F_{e-1}(t)\end{array} \right)=0\]

The determinant of the Vandermonde matrix \[\left(
\begin{array}{ccccc}
1 & 1 & 1 &\cdots &1 \\
1 & \alpha & \alpha^2 &\cdots &\alpha^{e-1} \\
1 & \alpha^2 & \alpha^4 &\cdots &\alpha^{2(e-1)}\\
\vdots&\vdots&\vdots&&\vdots\\
1 &\alpha^{e-1} &\alpha^{2(e-1)} &\cdots &\alpha^{(e-1)^2}
\end{array} \right)\] is \begin{equation}\prod_{j=0}^{e-2}\prod_{k=j+1}^{e-1}(\alpha^{k}-\alpha^{j}).\label{det}\end{equation}

%For example, when $e=2$, the determinant is $\alpha-1$; when $e=3$, it's $(\alpha^2-\alpha)(\alpha^2-\alpha)(\alpha-1)$; when
%$e=4$, it's $(\alpha^3-\alpha^2)(\alpha^3-\alpha)(\alpha^3-1)(\alpha^2-\alpha)(\alpha^2-1)(\alpha-1).$


When $\alpha =e^{\frac{2\pi i}{e}}$, each
$(\alpha^{k}-\alpha^{j})$ in the product (\ref{det}) is nonzero,
so for any $e$, the determinant is nonzero and the matrix is
non-singular. So we get $F_j(t)=0$ for any $t\in\mathbb{R}$ and
$j=0, 1, 2, \cdots e-1$.

So each $f_j(q)$ in $F$ is the zero polynomial.


The kernel of $\Psi$ is the ideal generated by $q'^e-q^d$.

\end{proof}




From the power operation of quasi-elliptic cohomology, we can
construct a new operation for quasi-elliptic cohomology.
\begin{proposition}The composition
\begin{align*}\overline{P}_N: &QEll_{G}(X) \buildrel{\mathbb{P}_N}\over\longrightarrow QEll_{G\wr\Sigma_N}(X^{\times N})
\buildrel{res}\over\longrightarrow QEll_{G\times
\Sigma_N}(X^{\times N})\\  &\buildrel{diag^*}\over\longrightarrow
 QEll_{G\times\Sigma_N}(X) \cong QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_{\Sigma_N}(\mbox{pt}) \\
  &\longrightarrow QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]} QEll_{\Sigma_N}(\mbox{pt})/\mathcal{I}^{QEll}_{tr} \\ &\cong
 QEll_G(X)\otimes_{\mathbb{Z}[q^{\pm}]}\prod_{N=de}\mathbb{Z}[q^{\pm}][q']/\langle q^d-q'^e
\rangle\end{align*} defines a ring homomorphism, where $res$ is
the restriction map by the inclusion
$$G\times\Sigma_N\hookrightarrow G\wr\Sigma_N\mbox{,    } (g, \sigma)\mapsto (g, \cdots g;
\sigma),$$ $diag$ is the diagonal map $$X\longrightarrow X^{\times
N}\mbox{,    } x\mapsto (x, \cdots x)$$ and the last map is the
isomorphism (\ref{ellc}).\label{adamsqell}
\end{proposition}

\begin{proof}

Let $V=\prod\limits_{g\in G_{conj}}V_{g}\in QEll_G(X)$. Apply the
explicit formula of the power operation in (\ref{ee}), the
composition $diag^*\circ res\circ\mathbb{P}_N$ sends $V$ to
$$\prod_{\substack{g\in G_{conj} \\ \sigma\in
{\Sigma_N}_{conj}}} \otimes_k\otimes_{(i_1, \cdots
i_k)}V_{g^k}q^{\frac{1}{k}}$$ where $(i_1, \cdots i_k)$ goes over
all the $k-$cycles of $\sigma$, and the tensor products are those
of the $\mathbb{Z}[q^{\pm}]-$algebras. Then, as shown in the proof
of (\ref{ellc}), after taking the quotient by the transfer ideal
$\mathcal{I}^{QEll}_{tr}$, all the factors in $diag^*\circ
res\circ\mathbb{P}_N(V)$ are cancelled except those corresponding
to the elements in ${\Sigma_N}_{conj}$ with cycles of the same
length.  And for the factor corresponding to the element
$\sigma\in {\Sigma_N}_{conj}$ with $d$ $e-$cycles and $de=N$, the
nontrivial part is $V_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d,
e}$ where  $V_{g^e, d}$ is the fixed point space of
$V_{g^e}^{\otimes_{\mathbb{Z}[q, q^{-1}]}d}$ by the permutations
$\Sigma_d$ and $q'_{d, e}= \mathbb{P}_{\sigma}(q) =
(q^{\frac{1}{e}})^{\otimes_{\mathbb{Z}[q, q^{-1}]}d}$.
%The restriction of $q'_{d, e}$ to the subgroup $C_{\Sigma_N}(\sigma)$ is trivial.

Thus, \begin{equation}\overline{P}_N(V)= \prod\limits_{\substack{g\in G_{conj} \\
N=de}}V_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d,
e}.\end{equation}


Let $V, W$ be two elements in $QEll_G(X)$. We have $$(V\oplus
W)_{g^e, d}=V_{g^e, d}\oplus W_{g^e, d}\mbox{   and    }(V\otimes
W)_{g^e, d}=V_{g^e, d}\otimes W_{g^e, d}.$$
\begin{align*}\overline{P}_N(V\oplus W)&=\prod\limits_{\substack{g\in G_{conj} \\ N=de}}
(V\oplus W)_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]}
q'_{d, e}\\ %& = \prod\limits_{\substack{g\in G_{conj} \\ N=de}}\bigoplus_{a=0}^{ e-1}(V_{g^e, q'^a_{d,
%e}}\oplus W_{g^e, q'^a_{d, e}})\otimes_{\mathbb{Z}[q^{\pm}]} q'^a_{d, e}\\
&=\bigg( \prod\limits_{\substack{g\in G_{conj} \\
N=de}}V_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d, e}\bigg)
\oplus
\bigg(\prod\limits_{\substack{g\in G_{conj} \\
N=de}} W_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d, e}\bigg)
\\ &=\overline{P}_N(V)\oplus\overline{P}_N(W) .\end{align*}
Similarly, \begin{align*}\overline{P}_N(V\otimes
W)&=\prod\limits_{\substack{g\in G_{conj} \\ N=de}} (V\otimes
W)_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]}
q'_{d, e}\\ %& = \prod\limits_{\substack{g\in G_{conj} \\ N=de}}\bigoplus_{a=0}^{ e-1}(V_{g^e, q'^a_{d,
%e}}\oplus W_{g^e, q'^a_{d, e}})\otimes_{\mathbb{Z}[q^{\pm}]} q'^a_{d, e}\\
&=\bigg( \prod\limits_{\substack{g\in G_{conj} \\
N=de}}V_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d, e}\bigg)
\otimes
\bigg(\prod\limits_{\substack{g\in G_{conj} \\
N=de}} W_{g^e, d}\otimes_{\mathbb{Z}[q^{\pm}]} q'_{d, e}\bigg)
\\ &=\overline{P}_N(V)\otimes\overline{P}_N(W) .\end{align*}
\end{proof}






%    Text of article.

%    Bibliographies can be prepared with BibTeX using amsplain,
%    amsalpha, or (for "historical" overviews) natbib style.
\bibliographystyle{amsplain}


%    Insert the bibliography data here.



\begin{thebibliography}{a}

\bibitem{AHS}Matthew Ando, Michael J. Hopkins, and Neil P. Strickland: \textsl{Elliptic spectra, the Witten
genus and the theorem of the cube}, Invent. Math. ,
146(3):595-687, 2001.


\bibitem{AHS04}Matthew Ando, Michael J. Hopkins, and Neil P. Strickland:
\textsl{The sigma orientation is an $H_{\infty}$ map}, Amer. J.
Math., 126(2):247-334, 2004.


\bibitem{ALRuan}Alejandro Adem, Johann Leida, Yongbin Ruan: \textsl{Orbifolds and stringy
topology}, Cambridge Tracts in Mathematics, 171. Cambridge
University Press, Cambridge, 2007.

\bibitem{Ati89}M.F.Atiyah: \textsl{K-theory}, second edition,
Addison-Wesley Publishing Company Advanced Book Program, Redwood
City, CA, 1989, Notes by D.W.Anderson.

\bibitem{Ati66}M.F.Atiyah: \textsl{Power operations in K-theory}, Quart. J. Math. Oxford Ser. (2) 17 1966 pp. 165�193.

%\bibitem{KR}Atiyah, M.F.: \textsl{K-theory and reality}, Quart. J. Math. Oxford Ser. (2) 17 1966 367-386.

\bibitem{BIE}Atiyah, M.F.: \textsl{Bott Periodicity and the Index of Elliptic
Operators},  Quart. J. Math. Oxford Ser. (2) 19 1968 113�140.


\bibitem{KC}Atiyah, M.F. and Segal, G.B.:
\textsl{Equivariant K-theory and completion}, J. Differential
Geometry 3 1969 1-18.


\bibitem{SSCP}Tobias Barthel, Nathaniel Stapleton: \textsl{The character of the total power operation}, available at arXiv:1502.01987.

\bibitem{BM}Clemens Berger, Ieke Moerdijk: \textsl{On an extension of the notion of Reedy
category}, Mathematische Zeitschrift, December 2011, Volume 269,
Issue 3, pp 977-1004.

%\bibitem{BG}Bohmann, Anna Marie: \textsl{Global orthogonal spectra},  Homology, Homotopy and Applications. Vol 16 (2014), No 1, 313-332.

\bibitem{Bredon} G. Bredon: \textsl{Equivariant cohomology
theories}, Springer Lecture Notes in Mathematics Vol 34. 1967.

\bibitem{BT}Theodor Br$\ddot{o}$cker, Tammo tom Dieck: \textsl{Representation of Compact Lie Groups},  Springer GTM \textbf{98} 1985.

%\bibitem{BMMS86}R.R.Bruner, J.P.May, J.E.McClure, and M.Steinberger: \textsl{$H_{\infty}$ ring spectra and their
%applications}, volume 1176 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986.

\bibitem{Clifford1937}Clifford, A.H.: \textsl{Representations induced in an invariant subgroup}, Ann. of Math. 38(1937), 533-550.



\bibitem{CurtisReiner62}Curtis, C.W., and Reiner, I.: \textsl{Representation Theory of Finite Groups and Associative Algebras}, Pure and Applied Mathematics, Vol. XI,
John Wiley \& Sons, New York-London-Sydney, 1962, xiv + 689 pp.



\bibitem{DR}P. Deligne and M. Rapoport: \textsl{Les sch$\acute{e}$mas de modules de courbes
elliptiques}, In Modular functions of one variable, II (Proc.
Internat. Summer School, Univ. Antwerp, Antwerp, 1972) , pages
143-316. Lecture Notes in Math., Vol. 349. Springer, Berlin, 1973.

%\bibitem{DS}W.G.Dwyer and J.Spalinski: \textsl{Homotopy theories and model categories}, In: Handbook of algebraic topology, ed.I.M.James,
%Elsevier (1995), 73-126.


\bibitem{Dev96}Jorge A. Devoto: \textsl{Equivariant elliptic homology and finite
groups}, Michigan Math. J. , 43(1):3-32, 1996.

%\bibitem{Elmendorfthesis}Anthony D.Elmendorf:   \textsl{Maps between Classifying Spaces and Equivariant Homotopy and Fixed Point Sets},
%Thesis (Ph.D.)�The University of Chicago. 1979.

\bibitem{FHT13}Daniel S. Freed, Michael J.Hopkins, and Constantin
Teleman, \textsl{Loop groups and twisted K-theory II},
J.Amer.Math.Soc 26(2013), no. 3, 595-644.
\bibitem{FH}William Fulton, Joe Harris: \textsl{Representation Theory, a first course},  Springer GTM \textbf{129} 1991.


\bibitem{Gan06}Nora Ganter: \textsl{Orbifold genera, product
formulas and power operations}, Advances in Mathematics, 2006.

\bibitem{Gan07}Nora Ganter: \textsl{Stringy power operations in Tate K-theory},
(2007), available at arXiv: math/0701565.


%\bibitem{Gan13}Nora Ganter: \textsl{Power operations in orbifold Tate
%K -theory}, (2013), availabel at arXiv:  math/13012754.


\bibitem{Gan13}Nora Ganter: \textsl{Power operations in orbifold Tate
K-theory},  Homology Homotopy Appl. 15 (2013), no. 1, 313-342.

\bibitem{Gepnerorbi}David Gepner, Andre Henriques: \textsl{Homotopy Theory of Orbispaces},     arXiv:math/0701916.

\bibitem{GKV}V. Ginzburg, M. Kapranov, E. Vasserot:
\textsl{Elliptic algebras and equivariant elliptic cohomology I},



%\bibitem{GP}J.P.C. Greenlees and J.P. May: \textsl{Localization and completion theorems for MU-
%module spectra}, Ann. of Math. (2), 146(3) (1997), 509-544. MR 1491447.

\bibitem{Groj}I. Grojnowski: \textsl{Delocalised equivariant elliptic cohomology (1994)}, in Elliptic cohomology, volume 342 of London Math. Soc. Lecture
Note Ser., pages 114-121. Cambridge Univ. Press, Cambridge, 2007.

%\bibitem{GM}Bert Guillou: \textsl{A short note on models for equivariant homotopy theory}, 2006.

%\bibitem{Hmodel}P. S. Hirschhorn: \textsl{Model categories and their localizations}, Mathematical Surveys and Monographs, 99. American
%Mathematical Society, Providence, RI, 2003. xvi+457 pp.


\bibitem{HKR}M J Hopkins, N J Kuhn, D C Ravenel:  \textsl{Generalized group characters and complex
oriented cohomology theories}, J. Am. Math. Soc. 13 (2000) 553594.


%\bibitem{Hoveymodel}M. Hovey: \textsl{Model Categories}, Mathematical Surveys and Monographs 63 (American Mathematical
%Society, Providence, RI, 1999).

\bibitem{JK81}Gordon James, Adalbert Kerber: \textsl{Representation theory of the symmetric
group}, Encyclopedia of mathematics and its applications v. 16,
Reading, Mass. : Addison-Wesley Pub. Co., Advanced Book
Program/World Science Division, 1981.

%\bibitem{JK}Joachim, Michael: \textsl{Higher coherences for equivariant K-theory}, Structured ring spectra, 87-114, London Math. Soc.
%Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004.

\bibitem{KM85}Nicholas M. Katz and Barry Mazur: \textsl{Arithmetic moduli of elliptic curves}, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ,
1985.

\bibitem{LProceeding}Landweber, P. S.: \textsl{Elliptic Curves and Modular Forms in Algebraic Topology: proceedings of a conference held at the Institute for Advanced Study}, Princeton, September 1986,
Lecture Notes in Mathematics.
\bibitem{LerStack}Eugene Lerman: \textsl{Orbifolds as stacks}, 2008. arXiv:0806.4160v1.
\bibitem{LMSM}Lewis, L. G., Jr.,  May, J. P.,  Steinberger, M.,  McClure, J. E.: \textsl{Equivariant stable homotopy theory}, With contributions by
J. E. McClure. Lecture Notes in Mathematics, 1213.
Springer-Verlag, Berlin, 1986.


\bibitem{LU01} Ernesto Lupercio and Bernardo Uribe: \textsl{Inertia orbifolds, configuration spaces and the ghost loop
space}, Quarterly Journal of Mathematics 55, Issue 2, pp. 185-201,
arxiv/math.AT/0210222; Loop groupoids, gerbes, and twisted sectors
on orbifolds, in: Orbifolds in Mathematics and Physics, Madison,
WI, 2001, in: Contemp. Math. 310, Amer. Math. Soc., Providence,
RI, 2002, pp. 163�184, MR2004c:58043, math.AT/0110207

\bibitem{LUloop} E. Lupercio and B. Uribe: \textsl{Loop groupoids, gerbes, and twisted sectors on orbifolds}, in Orbifolds in Mathematics
and Physics, A. Adem J. Morava and Y. Ruan, eds., Contemporary
Math 310, AMS (2002), 163�184,

\bibitem{Lurie}Jacob Lurie: \textsl{A Survey of Elliptic Cohomology}, in Algebraic Topology Abel Symposia Volume 4, 2009, pp 219-277



%\bibitem{MM}M.A.Mandell, J.P.May: \textsl{Equivariant orthogonal spectra and
%S-modules}, Mem.,Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.



%\bibitem{MMSS}M.A.Mandell, J.P.May, S.Schwede, and B.Shipley: \textsl{Model categories of diagram spectra}, Proc. London Math.
%Soc. 82(2001), 441-512.



%\bibitem{OP}J.P.May: \textsl{$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra}, With contributions
%by F. Quinn, N. Ray, and J.Tornehave. Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin-New York, 1977. 268 pp.

\bibitem{Alaska}J.P.May: \textsl{Equivariant homotopy and cohomology
theory}, CBMS Regional Conference Series in Mathematics, vol. 91,
Published for the Conference Board of the Mathematical Sciences,
Washington, DC, 1996 (with contributions by M. Cole, G.
Comeza�a, S. Costenoble, A.D. Elmendorf, J.P.C. Greenlees, L.G.
Lewis, Jr., R.J. Piacenza, G. Triantafillou, S. Waner).


%\bibitem{MCII}John Milnor: \textsl{Construction of Universal Bundles, II}, Annals of Mathematics, Vol. 63, No. 3, May 1956, Printed in U.S.A.


\bibitem{MV}Mitter, P., Viallet, C.:  \textsl{On the bundle of connections and the
gauge orbit manifold in Yang � Mills theory}, Commun. Math. Phys.
79 (1981) 457.
\bibitem{Moe02}I. Moerdijk: \textsl{Orbifolds as groupoids: an introduction}, Orbifolds in mathematics and
physics, Madison, WI, 2001, Contemparary Mathematics, vol. 310,
American Mathematics Society, Providence, RI, 2002, pp. 205-222.

\bibitem{Rez06}Charles Rezk: \textsl{Lectures on power
operations}.

\bibitem{Rez10}Charles Rezk: \textsl{Elliptic $\Lambda-$Rings}, 2010.

\bibitem{Rez11}Charles Rezk: \textsl{Quasi-elliptic cohomology}, 2011.

%\bibitem{Rez14} Charles Rezk: \textsl{Global Homotopy Theory and Cohesion}, 2014.

\bibitem{Rez16}Charles Rezk: \textsl{Loopy Stuff}, 2016.

\bibitem{MMSS}M.A.Mandell, J.P.May, S.Schwede, B.Shipley:  \textsl{Model
categories of diagram spectra}, Proc. London Math. Soc. (3) 82
(2001), no. 2, 441-512.

\bibitem{SSST}Tomer M. Schlank, Nathaniel Stapleton: \textsl{A transchromatic proof of Strickland's theorem}, available at  arXiv:1404.0717.

\bibitem{SP2GRP}Schommer-Pries, Christopher J.:
 \textsl{Central extensions of smooth 2-groups and a finite-dimensional
string 2-group},  Geom. Topol. 15 (2011), no. 2, 609�676.


%\bibitem{SS0}Stefan Schwede: \textsl{Global Homotopy Theory}, v.0.23/ April 30, 2015, Preliminary and incomplete version,
%\textit{http:/\!\!/www.math.uni-bonn.de/people/schwede/global.pdf}.


%\bibitem{SS}Stefan Schwede: \textsl{Global Homotopy Theory}, v.0.25/ February 24 2016, Preliminary and incomplete version,
%\textit{http:/\!\!/www.math.uni-bonn.de/people/schwede/global.pdf}.

\bibitem{SegalequiK}Segal, Graeme: \textsl{Equivariant K-theory}, Inst. Hautes \'{E}tudes Sci. Publ. Math. No. 34 1968 129�151.

\bibitem{STGCM}Nathaniel Stapleton:\textsl{Transchromatic generalized character maps}, 2013, available at arXiv:1304.5194.
\bibitem{STTCM}Nathaniel Stapleton:\textsl{Transchromatic twisted character maps}, 2011, available at arXiv:1110.3346.


\bibitem{Str98}N. P. Strickland: \textsl{Morava E-theory of symmetric groups}, Topology 37 (1998), no. 4, 757779. MR 1607736 (99e:55008).

%\bibitem{Willson} S.J.Willson: \textsl{Equivariant homology theories on $G-$complexes}, Trans. Amer. Math. Soc. 212(1975), 155-271.

\bibitem{Yau10}Donald Yau: \textsl{Lambda-rings}, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.

\end{thebibliography}
\end{document}

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

%    Templates for common elements of an article; for additional
%    information, see the AMS-LaTeX instructions manual, instr-l.pdf,
%    included in every AMS author package, and the amsthm user's guide,
%    linked from http:/\!\!/www.ams.org/tex/amslatex.html .

%    Section headings
\section{}
\subsection{}

%    Ordinary theorem and proof
\begin{theorem}[Optional addition to theorem head]
% text of theorem
\end{theorem}

\begin{proof}[Optional replacement proof heading]
% text of proof
\end{proof}

%    Figure insertion; default placement is top; if the figure occupies
%    more than 75% of a page, the [p] option should be specified.
\begin{figure}
\includegraphics{filename}
\caption{text of caption} \label{}
\end{figure}

%    Mathematical displays; for additional information, see the amsmath
%    user's guide, linked from http:/\!\!/www.ams.org/tex/amslatex.html .

% Numbered equation
\begin{equation}
\end{equation}

% Unnumbered equation
\begin{equation*}
\end{equation*}

% Aligned equations
\begin{align}
  &  \\
  &
\end{align}

%-----------------------------------------------------------------------
% End of article-template.tex
%-----------------------------------------------------------------------

1612.00930.extracted_bib
{AHSMatthew Ando, Michael J. Hopkins, and Neil P. Strickland: \textsl{Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. , 146(3):595-687, 2001.
1612.00930.extracted_bib
{AHS04Matthew Ando, Michael J. Hopkins, and Neil P. Strickland: \textsl{The sigma orientation is an $H_{\infty$ map, Amer. J. Math., 126(2):247-334, 2004.
1612.00930.extracted_bib
{ALRuanAlejandro Adem, Johann Leida, Yongbin Ruan: \textsl{Orbifolds and stringy topology, Cambridge Tracts in Mathematics, 171. Cambridge University Press, Cambridge, 2007.
1612.00930.extracted_bib
{Ati89M.F.Atiyah: \textsl{K-theory, second edition, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989, Notes by D.W.Anderson.
1612.00930.extracted_bib
{Ati66M.F.Atiyah: \textsl{Power operations in K-theory, Quart. J. Math. Oxford Ser. (2) 17 1966 pp. 165�193. %
1612.00930.extracted_bib
{KRAtiyah, M.F.: \textsl{K-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 1966 367-386.
1612.00930.extracted_bib
{BIEAtiyah, M.F.: \textsl{Bott Periodicity and the Index of Elliptic Operators, Quart. J. Math. Oxford Ser. (2) 19 1968 113�140.
1612.00930.extracted_bib
{KCAtiyah, M.F. and Segal, G.B.: \textsl{Equivariant K-theory and completion, J. Differential Geometry 3 1969 1-18.
1612.00930.extracted_bib
{SSCPTobias Barthel, Nathaniel Stapleton: \textsl{The character of the total power operation, available at arXiv:1502.01987.
1612.00930.extracted_bib
{BMClemens Berger, Ieke Moerdijk: \textsl{On an extension of the notion of Reedy category, Mathematische Zeitschrift, December 2011, Volume 269, Issue 3, pp 977-1004. %
1612.00930.extracted_bib
{BGBohmann, Anna Marie: \textsl{Global orthogonal spectra, Homology, Homotopy and Applications. Vol 16 (2014), No 1, 313-332.
1612.00930.extracted_bib
{Bredon G. Bredon: \textsl{Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol 34. 1967.
1612.00930.extracted_bib
{BTTheodor Br$\ddot{o$cker, Tammo tom Dieck: \textsl{Representation of Compact Lie Groups, Springer GTM 98 1985. %
1612.00930.extracted_bib
{BMMS86R.R.Bruner, J.P.May, J.E.McClure, and M.Steinberger: \textsl{$H_{\infty$ ring spectra and their %applications, volume 1176 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986.
1612.00930.extracted_bib
{Clifford1937Clifford, A.H.: \textsl{Representations induced in an invariant subgroup, Ann. of Math. 38(1937), 533-550.
1612.00930.extracted_bib
{CurtisReiner62Curtis, C.W., and Reiner, I.: \textsl{Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, Vol. XI, John Wiley \& Sons, New York-London-Sydney, 1962, xiv + 689 pp.
1612.00930.extracted_bib
{DRP. Deligne and M. Rapoport: \textsl{Les sch$\acute{e$mas de modules de courbes elliptiques, In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) , pages 143-316. Lecture Notes in Math., Vol. 349. Springer, Berlin, 1973. %
1612.00930.extracted_bib
{DSW.G.Dwyer and J.Spalinski: \textsl{Homotopy theories and model categories, In: Handbook of algebraic topology, ed.I.M.James, %Elsevier (1995), 73-126.
1612.00930.extracted_bib
{Dev96Jorge A. Devoto: \textsl{Equivariant elliptic homology and finite groups, Michigan Math. J. , 43(1):3-32, 1996. %
1612.00930.extracted_bib
{ElmendorfthesisAnthony D.Elmendorf: \textsl{Maps between Classifying Spaces and Equivariant Homotopy and Fixed Point Sets, %Thesis (Ph.D.)�The University of Chicago. 1979.
1612.00930.extracted_bib
{FHT13Daniel S. Freed, Michael J.Hopkins, and Constantin Teleman, \textsl{Loop groups and twisted K-theory II, J.Amer.Math.Soc 26(2013), no. 3, 595-644.
1612.00930.extracted_bib
{FHWilliam Fulton, Joe Harris: \textsl{Representation Theory, a first course, Springer GTM 129 1991.
1612.00930.extracted_bib
{Gan06Nora Ganter: \textsl{Orbifold genera, product formulas and power operations, Advances in Mathematics, 2006.
1612.00930.extracted_bib
{Gan07Nora Ganter: \textsl{Stringy power operations in Tate K-theory, (2007), available at arXiv: math/0701565. %
1612.00930.extracted_bib
{Gan13Nora Ganter: \textsl{Power operations in orbifold Tate %K -theory, (2013), availabel at arXiv: math/13012754.
1612.00930.extracted_bib
{Gan13Nora Ganter: \textsl{Power operations in orbifold Tate K-theory, Homology Homotopy Appl. 15 (2013), no. 1, 313-342.
1612.00930.extracted_bib
{GepnerorbiDavid Gepner, Andre Henriques: \textsl{Homotopy Theory of Orbispaces, arXiv:math/0701916.
1612.00930.extracted_bib
{GKVV. Ginzburg, M. Kapranov, E. Vasserot: \textsl{Elliptic algebras and equivariant elliptic cohomology I, %
1612.00930.extracted_bib
{GPJ.P.C. Greenlees and J.P. May: \textsl{Localization and completion theorems for MU- %module spectra, Ann. of Math. (2), 146(3) (1997), 509-544. MR 1491447.
1612.00930.extracted_bib
{GrojI. Grojnowski: \textsl{Delocalised equivariant elliptic cohomology (1994), in Elliptic cohomology, volume 342 of London Math. Soc. Lecture Note Ser., pages 114-121. Cambridge Univ. Press, Cambridge, 2007. %
1612.00930.extracted_bib
{GMBert Guillou: \textsl{A short note on models for equivariant homotopy theory, 2006. %
1612.00930.extracted_bib
{HmodelP. S. Hirschhorn: \textsl{Model categories and their localizations, Mathematical Surveys and Monographs, 99. American %Mathematical Society, Providence, RI, 2003. xvi+457 pp.
1612.00930.extracted_bib
{HKRM J Hopkins, N J Kuhn, D C Ravenel: \textsl{Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc. 13 (2000) 553594. %
1612.00930.extracted_bib
{HoveymodelM. Hovey: \textsl{Model Categories, Mathematical Surveys and Monographs 63 (American Mathematical %Society, Providence, RI, 1999).
1612.00930.extracted_bib
{JK81Gordon James, Adalbert Kerber: \textsl{Representation theory of the symmetric group, Encyclopedia of mathematics and its applications v. 16, Reading, Mass. : Addison-Wesley Pub. Co., Advanced Book Program/World Science Division, 1981. %
1612.00930.extracted_bib
{JKJoachim, Michael: \textsl{Higher coherences for equivariant K-theory, Structured ring spectra, 87-114, London Math. Soc. %Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004.
1612.00930.extracted_bib
{KM85Nicholas M. Katz and Barry Mazur: \textsl{Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985.
1612.00930.extracted_bib
{LProceedingLandweber, P. S.: \textsl{Elliptic Curves and Modular Forms in Algebraic Topology: proceedings of a conference held at the Institute for Advanced Study, Princeton, September 1986, Lecture Notes in Mathematics.
1612.00930.extracted_bib
{LerStackEugene Lerman: \textsl{Orbifolds as stacks, 2008. arXiv:0806.4160v1.
1612.00930.extracted_bib
{LMSMLewis, L. G., Jr., May, J. P., Steinberger, M., McClure, J. E.: \textsl{Equivariant stable homotopy theory, With contributions by J. E. McClure. Lecture Notes in Mathematics, 1213. Springer-Verlag, Berlin, 1986.
1612.00930.extracted_bib
{LU01 Ernesto Lupercio and Bernardo Uribe: \textsl{Inertia orbifolds, configuration spaces and the ghost loop space, Quarterly Journal of Mathematics 55, Issue 2, pp. 185-201, arxiv/math.AT/0210222; Loop groupoids, gerbes, and twisted sectors on orbifolds, in: Orbifolds in Mathematics and Physics, Madison, WI, 2001, in: Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163�184, MR2004c:58043, math.AT/0110207
1612.00930.extracted_bib
{LUloop E. Lupercio and B. Uribe: \textsl{Loop groupoids, gerbes, and twisted sectors on orbifolds, in Orbifolds in Mathematics and Physics, A. Adem J. Morava and Y. Ruan, eds., Contemporary Math 310, AMS (2002), 163�184,
1612.00930.extracted_bib
{LurieJacob Lurie: \textsl{A Survey of Elliptic Cohomology, in Algebraic Topology Abel Symposia Volume 4, 2009, pp 219-277 %
1612.00930.extracted_bib
{MMM.A.Mandell, J.P.May: \textsl{Equivariant orthogonal spectra and %S-modules, Mem.,Amer. Math. Soc. 159 (2002), no. 755, x+108 pp. %
1612.00930.extracted_bib
{MMSSM.A.Mandell, J.P.May, S.Schwede, and B.Shipley: \textsl{Model categories of diagram spectra, Proc. London Math. %Soc. 82(2001), 441-512. %
1612.00930.extracted_bib
{OPJ.P.May: \textsl{$E_{\infty$ ring spaces and $E_{\infty$ ring spectra, With contributions %by F. Quinn, N. Ray, and J.Tornehave. Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin-New York, 1977. 268 pp.
1612.00930.extracted_bib
{AlaskaJ.P.May: \textsl{Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 (with contributions by M. Cole, G. Comeza�a, S. Costenoble, A.D. Elmendorf, J.P.C. Greenlees, L.G. Lewis, Jr., R.J. Piacenza, G. Triantafillou, S. Waner). %
1612.00930.extracted_bib
{MCIIJohn Milnor: \textsl{Construction of Universal Bundles, II, Annals of Mathematics, Vol. 63, No. 3, May 1956, Printed in U.S.A.
1612.00930.extracted_bib
{MVMitter, P., Viallet, C.: \textsl{On the bundle of connections and the gauge orbit manifold in Yang � Mills theory, Commun. Math. Phys. 79 (1981) 457.
1612.00930.extracted_bib
{Moe02I. Moerdijk: \textsl{Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics, Madison, WI, 2001, Contemparary Mathematics, vol. 310, American Mathematics Society, Providence, RI, 2002, pp. 205-222.
1612.00930.extracted_bib
{Rez06Charles Rezk: \textsl{Lectures on power operations.
1612.00930.extracted_bib
{Rez10Charles Rezk: \textsl{Elliptic $\Lambda-$Rings, 2010.
1612.00930.extracted_bib
{Rez11Charles Rezk: \textsl{Quasi-elliptic cohomology, 2011. %
1612.00930.extracted_bib
{Rez14 Charles Rezk: \textsl{Global Homotopy Theory and Cohesion, 2014.
1612.00930.extracted_bib
{Rez16Charles Rezk: \textsl{Loopy Stuff, 2016.
1612.00930.extracted_bib
{MMSSM.A.Mandell, J.P.May, S.Schwede, B.Shipley: \textsl{Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441-512.
1612.00930.extracted_bib
{SSSTTomer M. Schlank, Nathaniel Stapleton: \textsl{A transchromatic proof of Strickland's theorem, available at arXiv:1404.0717.
1612.00930.extracted_bib
{SP2GRPSchommer-Pries, Christopher J.: \textsl{Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Topol. 15 (2011), no. 2, 609�676. %
1612.00930.extracted_bib
{SS0Stefan Schwede: \textsl{Global Homotopy Theory, v.0.23/ April 30, 2015, Preliminary and incomplete version, %http:/\!\!/www.math.uni-bonn.de/people/schwede/global.pdf. %
1612.00930.extracted_bib
{SSStefan Schwede: \textsl{Global Homotopy Theory, v.0.25/ February 24 2016, Preliminary and incomplete version, %http:/\!\!/www.math.uni-bonn.de/people/schwede/global.pdf.
1612.00930.extracted_bib
{SegalequiKSegal, Graeme: \textsl{Equivariant K-theory, Inst. Hautes \'{Etudes Sci. Publ. Math. No. 34 1968 129�151.
1612.00930.extracted_bib
{STGCMNathaniel Stapleton:\textsl{Transchromatic generalized character maps, 2013, available at arXiv:1304.5194.
1612.00930.extracted_bib
{STTCMNathaniel Stapleton:\textsl{Transchromatic twisted character maps, 2011, available at arXiv:1110.3346.
1612.00930.extracted_bib
{Str98N. P. Strickland: \textsl{Morava E-theory of symmetric groups, Topology 37 (1998), no. 4, 757779. MR 1607736 (99e:55008). %
1612.00930.extracted_bib
{Willson S.J.Willson: \textsl{Equivariant homology theories on $G-$complexes, Trans. Amer. Math. Soc. 212(1975), 155-271.
1612.00930.extracted_bib
{Yau10Donald Yau: \textsl{Lambda-rings, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.