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\title{On equivariant bundles and their moduli spaces}
\alttitle{Sur torseurs équivariants et leur espaces des modules}

\author{\firstname{Chiara} \lastname{Damiolini}}
\address{Department of Mathematics, University of Pennsylvania, Philadelphia, USA}
\address{Department of Mathematics, University of Texas at Austin, Austin, USA}
\email{chiara.damiolini@gmail.com}

\begin{abstract}
Let $G$ be an algebraic group and $\Gamma$ a finite subgroup of automorphisms of $G$. Fix also a possibly ramified $\Gamma$-covering $\widetilde{X} \to X$. In this setting one may define the notion of $(\Gamma,G)$-bundles over $\widetilde{X}$ and, in this paper, we give a description of these objects in terms of $\mathcal{H}$-bundles on $X$, for an appropriate group $\mathcal{H}$ over $X$ which depends on the local type of the $(\Gamma,G)$-bundles we intend to parametrize. This extends, and along the way clarifies, an earlier work of Balaji and Seshadri.
\end{abstract}

\begin{altabstract}
Soit $G$ un groupe algébrique et $\Gamma$ un sous-groupe fini d'automorphismes de $G$. Nous fixons également un $\Gamma$-revêtement éventuellement ramifié $\widetilde{X} \to X$. Dans ce cadre, on peut définir la notion de $(\Gamma, G)$-fibré sur $\widetilde{X}$ et, dans cet article, nous donnons une description de ces objets en termes de $\mathcal{H}$-fibrés sur $X$, pour un groupe $\mathcal{H}$ sur $X$ qui dépend du type local des $(\Gamma, G)$-fibrés que nous avons l'intention de paramétrer. Ceci étend, et en même temps clarifie, un travail antérieur de Balaji et Seshadri.
\end{altabstract}

\dateposted{2024-02-02}
\begin{document}
\maketitle



\section{Introduction}

The moduli space $\Bun_{\SL_r}(X)$ parametrizing vector bundles of rank $r$ and trivial determinant over a smooth curve $X$ is a central object in algebraic geometry, with deep connections to representation theory and conformal field theory~\cite{ramanan:1973:moduli, BeauvilleLaszlo1994Conformal, beauville:1993:CFT, sorger1996formule}. One can extend this notion by replacing $\SL_r$ with another simple algebraic group $G$ over the base field or, if there are marked points on the curve, by considering \emph{parabolic} $G$-bundles~\cite{KumarNR1994Grassmannian, mehta.seshadri:1980:moduli}. In all these cases there are connections to conformal field theory through generalized theta functions and conformal blocks~\cite{Pauly1996Parabolic, LS1997PicardBunG}. An effective way to describe all the previous instances is to use \emph{parahoric} bundles, that is principal bundles on $X$ with respect to a parahoric Bruhat--Tits group $\cH$ defined over the curve $X$ itself~\cite{BruhatTitsII, heinloth2010uniformization}. A natural way to produce such groups, and which gives rise to every \emph{split} parahoric Bruhat--Tits group, is from Galois coverings of curves~\cite{balaji2011moduli}. In this circumstance as well, links to conformal field theory through twisted conformal blocks have been established~\cite{damiolini:2020:conformal,zelaci:2019:moduli,hong2021conformal}.

We explain how groups can be constructed from coverings. Let $\Gamma$ be a finite group and let $\pi \colon \Xt \to X$ be a possibly ramified $\Gamma$-covering of smooth curves. Assume that $G$ is an algebraic group which is equipped with a group homomorphism $\rho \colon \Gamma \to \Aut(G)$. Then the group $\pi_*(\Xt \times G)$ has a natural action of $\Gamma$, and so one may construct the subgroup of $\Gamma$-invariant elements $\cH:= (\pi_*(\Xt \times G))^\Gamma$, which is a parahoric Bruhat--Tits group over $X$. The construction of a group starting with a covering as above still holds without assuming that $\Xt$ and $X$ are curves. In this paper we aim to characterize $\cH$-bundles in terms of certain $G$-bundles on $\Xt$. A particular instance of our main result (Theorem~\ref{thm-equivstacks}) can be stated as follows:

\begin{theo*}
The functor $\pi_*({\,\cdot\,})^\Gamma$ induces an equivalence between $\Bun^{\triv}_{(\Gamma,G)}(\Xt)$ and $\Bun_\cH(X)$.
\end{theo*}

Here $\Bun_{(\Gamma,G)}(\Xt)$ is the groupoid whose objects are $(\Gamma, G)$-bundles on $\Xt$, that is $G$-bundles on $\Xt$ which are equipped with an induced action of $\Gamma$ which lifts the action of $\Gamma$ on $\Xt$ and is compatible with the action of $\Gamma$ on $G$ (see Definition~\ref{defi:GammaGbundle}). Given every $(\Gamma,G)$-bundle $\Pp$, the groupoid $\Bun^{\Pp}_{(\Gamma,G)}(\Xt)$ is given by those $(\Gamma,G)$-bundles which have the same \emph{local type} as $\Pp$. When $\Pp$ is the trivial $(\Gamma, G)$-bundle, then we use the notation $\Bun^{\triv}_{(\Gamma,G)}(\Xt)$ instead. A fundamental ingredient to understand our main result is the concept of \emph{local types} of $(\Gamma,G)$-bundles (Definition~\ref{defi:localtype}). Essentially two $(\Gamma,G)$-bundles have the same local type exactly when they are locally isomorphic not only as $G$-bundles, but as $(\Gamma, G)$-bundles, i.e. taking into account also the action of $\Gamma$.

From another point of view, $\Bun^{\triv}_{(\Gamma,G)}(\Xt)$ can be also interpreted as the biggest subcategory of $\Bun_{(\Gamma,G)}(\Xt)$ where we can apply the functor $\pi_*({\,\cdot\,})^\Gamma$ and land in $\Bun_\cH$. In Example~\ref{eg:S4} we show that $\Bun^{\triv}_{(\Gamma,G)}(\Xt)$ does not coincide with $\Bun_{(\Gamma,G)}(\Xt)$, emphasizing in this way the fact that not all $(\Gamma,G)$-bundles are locally trivial, and that fixing local types is a necessary condition for our theorem to hold.

\subsection*{Comparison with~\cite{balaji2011moduli}}

The inspiration for working with bundles associated with groups arising from coverings, and to give their description in terms of $(\Gamma,G)$-bundles comes from Balaji's and Seshadri's paper~\cite{balaji2011moduli}, where the authors give a description of the moduli space of parahoric torsors over a smooth curve. In particular, in~\cite[Theorem~4.1.6]{balaji2011moduli} a similar statement to our main result is asserted with two main differences.

The first, and perhaps most important, concerns the concept of local type. More precisely Theorem~4.1.6 states that $\pi_*({\,\cdot\,})^\Gamma$ is an equivalence between $\Bun_{(\Gamma,G)}(\Xt)$ and $\Bun_\cH(X)$. However, as illustrated by Example~\ref{eg:S4}, this is not true in general. As shown in Proposition~\ref{prop-PisHbundle}, in order to address this problem, we need to introduce and use the concept of local types.

The second difference lies in how $\Gamma$ acts on $G$: in~\cite{balaji2011moduli} the authors let $\Gamma$ act on $G$ via inner automorphisms only. Under this hypothesis, and assuming that $\Xt$ and $X$ are smooth curves, $\cH = \pi_*({\,\cdot\,})^\Gamma$ is a \emph{split} parahoric Bruhat--Tits group. We do not impose any condition on how $\Gamma$ acts on $G$, obtaining in this case a wider class of groups $\cH$ arising from coverings.

In conclusion, from the above considerations, it follows that our main result not only extends~\cite[Theorem~4.1.6]{balaji2011moduli} beyond the case of inner automorphisms, but can be seen also as a way to clarify the need of fixing local types of $(\Gamma,G)$-bundles for that statement to hold true.

\subsection*{Further projects}

As aforementioned, one of the main motivations of this work comes from the study of parahoric Bruhat--Tits groups over a curve, so we will assume in this section that $\Xt$ and $X$ are smooth curves. In this setting, \cite[Theorem~5.2.7]{balaji2011moduli} states that all \emph{split} parahoric Bruhat--Tits can be recovered from coverings, provided that the Galois group $\Gamma$ acts on $G$ via inner automorphisms only. However it is natural to include also automorphism which are not necessarily inner (see for instance~\cite[Example~2.3]{damiolini:2020:conformal}). It follows that the category of groups $\cH$ which can be constructed through coverings, and to which we can apply our main result, is much larger than the one studied in~\cite{balaji2011moduli}. We can then ask whether all parahoric Bruhat--Tits groups arise in this fashion. If the answer to this question is affirmative, then Theorem~\ref{thm-equivstacks} implies that the study of all parahoric bundles over a curve $X$ can be translated into the study of $\Gamma$-equivariant bundles over $\Xt$ for an appropriate $\Gamma$-covering $\Xt \to X$. Observe that if this were true, then~\cite{damiolini:2020:conformal, hong2021conformal} would provide the notion of conformal blocks associated to every parahoric Bruhat--Tits group.

Another element under investigation is the classification of the possible local types. In the case of inner automorphisms, one can describe local types via conjugation classes in $G$.

\subsection*{New results}

After the first version of this paper appeared, the author and other researchers have contributed towards answering the questions above. In particular in two independent works, \cite{PR:2022,DH:2022}, the moduli stacks of bundles under Bruhat--Tits group scheme and equivariant torsors coming from coverings are related. Furthermore, in~\cite{DH:2022} the present author and Hong have given an explicit classification of local types for more general automorphisms. Independently, a cohomological description of the local types introduced in this paper have been given by~\cite{PR:2022}, where Pappas and Rapoport also provide a counterexample to~\cite[Lemma~4.1.5]{balaji2011moduli} in the context of parahoric bundles.

\subsection*{Plan of the paper} We begin the paper by fixing some notation which will be used throughout. We then introduce the main ingredients of the paper: we explain what we mean by $(\Gamma,G)$-bundles in Definition~\ref{defi:GammaGbundle} and, after Example~\ref{eg:S4}, we introduce the notion of local type in Definition~\ref{defi:localtype}. This leads to Proposition~\ref{prop-PisHbundle} which identifies the correct subcategory of $\Bun_{(\Gamma,G)}$ to which we can apply the functor $\pi_*({\,\cdot\,})^\Gamma$ and obtain $\cH$-bundles. Finally, in Section~\ref{sec:mainthm} we state and prove our main result, Theorem~\ref{thm-equivstacks} by explicitly constructing the map providing the inverse to $\pi_*({\,\cdot\,})^\Gamma$ (Proposition~\ref{prop-inversedef}).

\section{\texorpdfstring{$(\Gamma,G)$}{(Gamma-G)}-bundles and local types} \label{sec:loctype}

Throughout this paper, we will fix a finite group $\Gamma$. All the schemes will be over a field $k$ whose characteristic does not divide the order of $\Gamma$. We will further fix the following data:
\begin{itemize}
\item An affine and smooth algebraic group $G$ over $k$ which is endowed with a group homomorphism $\rho \colon \Gamma \to \Aut(G)$ sending $\gamma$ to $\gamma_G$.
\item A (ramified) $\Gamma$-covering $\pi \colon \Xt \to X$ over $k$, that is
\begin{itemize}
\item $\pi$ is a finite flat morphism;
\item the group of automorphisms of $\Xt$ over $X$ is isomorphic to $\Gamma$. The automorphism of $\Xt$ associated with $\gamma \in \Gamma$ is denoted $\gamma_{\Xt}$;
\item $\Xt$ is a generically étale $\Gamma$-torsor over $X$ via $\pi$.
\end{itemize}
\end{itemize}

The \emph{ramification locus} of $\pi$ is the subscheme of $\Xt$ where $\pi$ is not étale. Its image in $X$ is denoted by $\Rr$ and called the \emph{branch locus} of $\pi$.

\begin{defi}
Given a $G$-bundle $\Pp$ on $\Xt$ we denote by $\Go$ the automorphisms group scheme $\Isom_G(\Pp,\Pp)$.
For any $G$-bundle $\Pp'$ on $\Xt$ the scheme $\IoP:=\Isom_G(\Pp,\Pp')$ is a $\Go$-bundle.
\end{defi}

Recall (see for instance~\cite[Section~7.6]{bosch1990neron}) that the Weil restriction $\pi_*\IoP$ of $\IoP$ along $\pi$ is defined by the equality
\[
\pi_*\IoP(T) := \Hom_\Xt(T \times_X \Xt, \IoP)
\]
for every $T$ over $X$. It follows from~\cite[Section~7.6]{bosch1990neron} that $\pi_*\IoP$ is representable by a smooth scheme over $X$ and that $\pi_*\Go$ has the structure of an algebraic group.

The following statement is a version of~\cite[Lemma~4.1.4]{balaji2011moduli}, which we add for completeness.

\begin{lemm}\label{lem-bundle}
Let $\Pp'$ be a $G$-bundle over $\Xt$, then $\pi_*\IoP$ is a $\pi_*\Go$-bundle.
\end{lemm}

\begin{proof} Since taking fibred products and Weil restrictions commute, $\pi_*\Go$ still acts on $\pi_*\IoP$. Similarly we have that $\pi_*\IoP \times_X \pi_*\Go \cong \pi_*\IoP \times_X \pi_*\IoP$ via the map $(f,g) \mapsto (f, fg)$, thus we are left to prove that for every point $x \in X(\bar{k})$ there exists an étale neighbourhood $U$ of $x$ such that $(\pi_*\IoP)(U) \neq \emptyset$. Since $\pi$ is finite we know that $\pi^{-1}\lbrace x \rbrace$ is a finite scheme over $\Spec(\bar{k})$ over which both $\Pp$ and $\Pp'$ are trivial. It follows that the map $q \colon \pi_*\IoP \to X$ is surjective. We conclude that $q$ is smooth and surjective. Applying~\cite[Corollaire~17.16.3]{EGAIV4} for every $x \in X$ there exists an étale neighbourhood $U$ of $x$ such that $(\pi_*\IoP)(U) \neq \emptyset$.
\end{proof}

We will be interested in those $G$-bundles over $\Xt$ which are equipped with an action of $\Gamma$, compatible with the action of $\Gamma$ on $\Xt$.

\begin{defi}\label{defi:GammaGbundle}
A \emph{$(\Gamma, G)$-bundle} on $\Xt$ is a right $G$-bundle $\Pp$ together with a left action of $\Gamma$ on its total space lifting the action of $\Gamma$ on $\Xt$ and which is compatible with the action of $\Gamma$ on $G$. The automorphism of $\Pp$ lifting $\gamma_\Xt$ will be denoted $\gamma_\Pp$. The compatibility with the action of $\Gamma$ on $G$ means that the equality
\begin{equation}\label{eq:GammaG}
\gamma_{\Pp}(p g)=\gamma_{\Pp}(p) \, \gamma_G(g)
\end{equation}
holds for all $p \in \Pp$ and $g \in G$. We say that a $(\Gamma,G)$-bundle $\Pp$ on $\Xt$ is \emph{trivial} if it is isomorphic to the trivial $G$-bundle and $\gamma_\Pp = \gamma_G$ for every $\gamma \in \Gamma$.
\end{defi}

The goal of this paper is to describe $(\Gamma,G)$-bundles over $\Xt$ in terms of $\cH$ bundles over $X$ for an appropriate group scheme $\cH$ over $X$. We show how to attach to every $(\Gamma, G)$ bundle $\Pp$, a group scheme $\cH_\Pp$ on $X$. Define the action of $\Gamma$ on $\Go$ lifting the action of $\Gamma$ on $\Xt$ via the map $\Gamma \to \Aut(\Go)$ sending the element $\gamma$ to the automorphism $\gamma_{\Go}$ defined by
\begin{equation*}
\gamma_{\Go} (\phi) := \gamma_\Pp \circ \phi \circ \gamma_\Pp^{-1}
\end{equation*}
for all $\gamma \in \Gamma$ and $\phi \in \Go$. Moreover, the left actions of $\Gamma$ on $\Go$ and on $\Xt$ induce a left action of $\Gamma$ on the Weil restriction $\pi_*\Go$ given by
\[
(\gamma f)(t, x) :=\gamma_{\Go} \, f(t,\gamma_\Xt^{-1}(x))
\]
for every $(t,\widetilde{x}) \in T \times_X \Xt$ and $f \in \Hom(T \times_X \Xt, \Go)$. We then define $\cH_\Pp$ as the subgroup of $\Gamma$-invariant elements of $\pi_*(\Go)$, that is $\cH_\Pp:=(\pi_*(\Go))^\Gamma$. By~\cite[Proposition~3.4]{Edixhoven1992Neron}, $\cH_\Pp$ is a smooth group over $X$.

We now want to identify which $(\Gamma,G)$-bundles on $\Xt$ can be described in terms of $\cH_\Pp$ bundles over $X$ and, conversely, whether $\cH_\Pp$-bundles on $X$ determine a $(\Gamma,G)$-bundle on $\Xt$.

For any $(\Gamma, G)$-bundle $\Pp'$, the scheme $\IoP$ is a $(\Gamma, \Go)$-bundle where the action of $\Gamma$ is given by
\[
(\gamma, \phi) \mapsto \gamma_{\Pp'} \circ \phi \circ \gamma_{\Pp}^{-1}
\]
for all $\gamma \in \Gamma$ and $\phi \in \IoP$. Moreover, as for $\pi_*\Go$, also $\pi_*\IoP$ is equipped with a left action of $\Gamma$, hence it makes sense to consider the subsheaf of $\Gamma$-invariant elements $(\pi_*\IoP)^\Gamma$.

As suggested in~\cite[Lemma~4.1.5]{balaji2011moduli}, the candidate $\cH_\Pp$-bundle corresponding to the $(\Gamma,G)$-bundle $\Pp'$ should be $(\pi_*\IoP)^\Gamma$. However, as we see in Example~\ref{eg:S4}, this is not always the case. In fact, the group schemes $\cH_\Pp$ heavily depend on how $\Gamma$ acts on $\Pp$ and not merely on the action of $\Gamma$ on $G$ given by $\rho$.

\begin{exam}\label{eg:S4}
Let $\Gamma=\ZZ/2\ZZ=\{ 1, \gamma\}$ and $G=\mathfrak{S}_4$, the symmetric group on four elements which acts on $G$ via $\gamma_G(\alpha)=(34)(12)\alpha(12)(34)$. Consider the $\Gamma$-covering given by $\Xt=\Spec(k[t]) \to \Spec(k[t^2])=X$ and let $x \in X$ be the only ramification point of the covering. Let $\widetilde{G}=G \times \Xt$ be the trivial $(\Gamma,G)$-bundle on $\Xt$. Let $\Pp'$ be the $(\Gamma, G)$-bundle on $\Xt$ which is trivial as a $G$-bundle, but with the action of $\Gamma$ given by $\gamma_{\Pp'}(\alpha)=(34)(\alpha) (12)(34)$ for every $\alpha \in \mathfrak{S}_4$. Observe that both $\Pp$ and $\Pp'$ are $(\Gamma,G)$-bundles on $\Xt$ since equation~\eqref{eq:GammaG} is satisfied. It follows that
\[
(\pi_*\widetilde{G})^\Gamma(x)=\{\alpha \in \mathfrak{S}_4 \, | \, \alpha=(34)(12)\alpha(12)(34)\} \neq \emptyset,
\]
since for instance the identity is an element of that set. We can see that
\[
(\pi_*\Pp')^\Gamma(x)=\{\alpha \in \mathfrak{S}_4 \, | \, \alpha=(34)\alpha(12)(34)\} = \emptyset
\]
for parity reasons (see also Lemma~\ref{lem:reduced}).
\end{exam}

The example above shows that Equation~\eqref{eq:GammaG} does not automatically imply that all $(\Gamma, G)$-bundles are locally isomorphic to the trivial $(\Gamma,G)$-bundle.

\begin{rema}
Observe that from~\cite[Lemma~4.1.5]{balaji2011moduli} it should follow that $\pi_*(\Pp')^\Gamma$ is a $\pi_*(\widetilde{G})^\Gamma$-bundle, but Example~\ref{eg:S4} provides a counterexample to that statement. To correct this problem we introduce the concept of local type, generalizing the idea introduced in~\cite{balaji2011moduli}.
\end{rema}

\begin{defi}\label{defi:localtype}
Let $\Pp_1$ and $\Pp_2$ be two $(\Gamma,G)$-bundles on $\Xt$. Then they have the \emph{same local type} at $x \in X(\bar{k})$ if
\[
\Isom_G\left(\Pp_1 \times \pi^{-1}\{x\}, \Pp_2 \times \pi^{-1} \{x\} \right)^\Gamma \neq \emptyset.
\]
We say that $\Pp_1$ and $\Pp_2$ have the \emph{same local type}, and we write $\Pp_1 \sim \Pp_2$, if they have the same local type at any geometric point of $X$.
\end{defi}

Although the two following lemmas will not be used to show the main result, we report them as a tool to the reader who wishes to compute local types of $(\Gamma,G)$ bundles in explicit cases. A more exhaustive treatment of local types can be found in~\cite{DH:2022,PR:2022}.

\begin{lemm}\label{lem:reduced}
Two $(\Gamma, G)$-bundles $\Pp_1$ and $\Pp_2$ have the same local type if and only if
\[
\Isom_G\left(\Pp_1 \times (\pi^{-1}\{x\})_{red}, \Pp_2 \times (\pi^{-1}\{x\})_{red}\right)^\Gamma \neq \emptyset
\]
for all $x \in X(\bar{k})$.
\end{lemm}

\begin{proof} It is enough to prove that if $\Isom_G\left(\Pp_1 \times (\pi^{-1}\{x\})_{red}, \Pp_2 \times (\pi^{-1}\{x\})_{red}\right)^\Gamma$ is not empty, then $\Pp_1$ and $\Pp_2$ have the same local type at $x$. Let $\pi^{-1}\{ x \}=\Spec(A)$ where $A$ is a finite Artin $k$-algebra. Let $\mathfrak{m}$ be its maximal nilpotent ideal, so that $\left(\pi^{-1}\{ x \}\right)_{red}= \Spec(A/\mathfrak{m})$. Let $\Spec(B)= \Isom_G(\Pp_1, \Pp_2)$ and by assumption there exists $\varphi_{0} \colon B \to A/\mathfrak{m}$ which is $\Gamma$-invariant and makes the diagram commute:
\[
\xymatrix{ && B \ar[dll]_{\varphi_{0}}\\ A/\mathfrak{m}&& \ar[ll] \ar[u]A \rlap{~.}}
\]
The aim is to lift $\varphi_{0}$ to a $\Gamma$-equivariant morphism $B \to A$ and we can reduce to the case $\mathfrak{m}^2=0$. Since $B$ is smooth over $A$ we know that $\varphi_{0}$ admits a lift $\varphi \colon B \to A$. For any $\gamma \in \Gamma$ the element $\gamma(\varphi)$ is another lift of $\varphi_{0}$, so the association $\gamma \mapsto \varphi-\gamma(\varphi)$ defines a map $h \colon \Gamma \to \Der_A(B,\mathfrak{m})$. Since $h$ satisfies the cocyle condition, we have that $h$ can be seen as an element of $\mathrm{H}^1(\Gamma, \Der_A(B, \mathfrak{m}))$, which however is zero because the characteristic of $k$ does not divide the order of $\Gamma$. This means that there exists a derivation $\partial \in \Der_A(B, \mathfrak{m})$ such that $h(\gamma)=\gamma(\partial)-\partial$ for every $\gamma \in \Gamma$. This implies that the lift $\varphi_1 :=\varphi + \partial$ is a $\Gamma$-invariant lift of $\varphi_{0}$ and concludes the proof.
\end{proof}

\begin{lemm}\label{lem:loctypeR}
Let $\Pp_1$ and $\Pp_2$ be two $(\Gamma,G)$-bundles on $\Xt$. Then $\Pp_1$ and $\Pp_2$ have the same local type if and only if they have the same local type at any geometric point of the branch locus $\Rr \subset X$.
\end{lemm}

\begin{proof} It is sufficient to show that any two $(\Gamma, G)$ bundles have the same local type on $X \setminus \Rr$. As $\pi$ is étale on $U:= X \setminus \Rr$ we can chose an étale covering $V\to U$ such that $\pi^{-1}(V)=\Gamma \times V$, i.e. the covering becomes trivial over $V$. We are then left to demonstrate that any $(\Gamma,G)$-structure defined on $\Gamma \times V \times G$ via automorphisms $\gamma_\Pp$ satisfying Equation~\eqref{eq:GammaG} is isomorphic to the trivial one induced by $\gamma_G$. Any isomorphism between these $(\Gamma,G)$-bundles is uniquely determined by tuples $(\alpha_\gamma)_{\gamma \in \Gamma}$ of isomorphisms of $V \times G$ satisfying
\[
\alpha_{\gamma \sigma} \circ \gamma_G = \gamma_\Pp \circ \alpha_\sigma
\]
for every $\gamma,\sigma \in \Gamma$. Since
\begin{itemize}
\item $\alpha_\gamma$ is uniquely determined by the element $\alpha_\gamma(1) \in G(V)$, and
\item every automorphism $\gamma_\Pp$ on $\Gamma \times V \times G$ satisfying~\eqref{eq:GammaG} is uniquely determined by the element $\gamma_\Pp(1) \in G(V)$,
\end{itemize}
we deduce that defining $\alpha_\gamma(1):=\gamma_\Pp(1)$ gives rise to the wanted isomorphism.
\end{proof}

The following Proposition tells us that $\cH_\Pp$ can only detect those $(\Gamma,G)$-bundles having the same local type as $\Pp$.

\begin{prop}\label{prop-PisHbundle}
Let $\Pp$ be a $(\Gamma,G)$-bundle over $\Xt$. Then the sheaf $(\pi_*\IoP)^\Gamma$ is an $\cH_{\Pp}$-bundle if and only if $\Pp'$ has the same local type as $\Pp$.
\end{prop}

\begin{proof} We have already proved in Lemma~\ref{lem-bundle} that $\pi_*\IoP$ is a $\pi_*\Go$-bundle, so that we have the isomorphism
\[
\Psi \; \colon \; \pi_*\IoP \times_{X} \pi_*\Go \; \cong \; \pi_*\IoP \times_{X} \pi_*\IoP
\]
induced from $\IoP \times_{\Xt} \Go \cong \IoP \times_{\Xt} \IoP$ which sends $(f,g)$ to $(f, fg)$. The group $\Gamma$ acts diagonally on both source and target of $\Psi$ and with respect to this action $\Psi$ is $\Gamma$-equivariant. Thus it induces an isomorphism
\[
\Psi^\Gamma \; \colon \; \left(\pi_*\IoP\right)^\Gamma \times_X \cH_{\Pp} \; \cong \; \left(\pi_*\IoP\right)^\Gamma \times_X \left(\pi_*\IoP\right)^\Gamma.
\]
In order to conclude we need to check that $\left(\pi_*\IoP\right)^\Gamma$ admits local sections if and only if $\Pp'$ has the same local type as $\Pp$. Suppose that for every point $x \in X$ there exists an étale neighbourhood $(u,U) \to (x,X)$ of $x$ such that there exists $\phi \in \left(\pi_*\IoP\right)^\Gamma(U)$. This implies that the composition $\phi u$ is an element in $\left(\pi_*\IoP \right)^\Gamma(x)$ which means that $\Pp$ and $\Pp'$ have the same local type at $x$. Conversely, assume that $\Pp'$ and $\Pp$ have the same local type. By definition this means that $\left(\pi_*{\IoP}\right)^\Gamma(x) \neq \emptyset$ for every geometric point $x$. It follows that the map $q \colon \left(\pi_*\IoP \right)^\Gamma \to X$ is surjective on geometric points and since it is smooth, then $q$ is surjective. Invoking~\cite[Corollaire~17.16.3]{EGAIV4} we can then conclude that for every $x \in X$, the map $q$ admits a section on an étale neighbourhood $U$ of $x$, and so $\left(\pi_*\IoP\right)^\Gamma(U) \neq \emptyset$.
\end{proof}

\section{The equivalence between \texorpdfstring{$\Bun_{\cH_\Pp}$}{Bun H Pp} and \texorpdfstring{$\Bun_{(\Gamma,G)}^{\Pp}$}{Bun(Gamma,G) Pp}} \label{sec:mainthm}

In this section we state and prove our main result, Theorem~\ref{thm-equivstacks}. Let $\Bun_{(\Gamma,G)}^{\Pp}$ be the groupoid whose objects are $(\Gamma, G)$-bundles on $\Xt$ which have the same local type as $\Pp$ and let $\Bun_{\cH_\Pp}$ be the groupoid whose objects are $\cH_\Pp$-bundles over $X$. Using this terminology, Proposition~\ref{prop-PisHbundle} implies that
\[
\pi_*\I_{\Pp}({\,\cdot\,})^\Gamma \colon \Bun_{(\Gamma,G)}^{\Pp} \to \Bun_{\cH_{\Pp}}
\]
is a well defined map. We now construct the inverse following the proof of~\cite[Theorem~4.1.6]{balaji2011moduli}. Note that the inclusion of $\cH_\Pp$ inside $\pi_*\Go$ induces, by adjunction, the map $\pi^*\cH_\Pp \to \Go$. It follows that $\Pp$, which is naturally a left $\Go$-bundle, has an induced left action of $\pi^*\cH_\Pp$. This enables us to associate to every $\cH_\Pp$-bundle $\Ff$, the $G$-bundle $\pi^*(\Ff) \times^{\pi^*{\cH_\Pp}} \Pp$ on $X$. We can say more:

\begin{prop}\label{prop-inversedef}
The assignment $\Ff \mapsto \pi^*(\Ff) \times^{\pi^*{\cH_\Pp}} \Pp$ defines the map
\[
\pi^*({\,\cdot\,}) \times^{\pi^*{\cH_\Pp}} \Pp \colon \Bun_{\cH_{\Pp}} \to \Bun_{(\Gamma,G)}^{\Pp}.
\]
\end{prop}

\begin{proof} We first show that for any $\cH_\Pp$-bundle $\Ff$ over $X$, the scheme $\Ff^\Pp:=\pi^*(\Ff) \times^{\pi^*{\cH_\Pp}} \Pp$ is indeed a $(\Gamma, G)$-bundle. Observe that it has a natural right action of $G$ and a left action of $\Gamma$ induced by the ones on $\Pp$. Let $\gamma \in \Gamma$ and $g \in G$ and consider $(f,p) \in \Ff^{\Pp}$. The chain of equalities
\[
\gamma_{\Ff^\Pp}((f,p)g))=\gamma_{\Ff^\Pp}(f,pg)=(f, \gamma_\Pp(pg))=(f,\gamma_\Pp(p) \gamma_G(g))=\left(\gamma_{\Ff^\Pp} (f,p)\right) \gamma_G(g)
\]
tell us that $\Ff^{\Pp}$ is a $(\Gamma, G)$-bundle on $\Xt$.

We now check that $\Ff^{\Pp}$ has the same local type as $\Pp$. Let $x$ be a geometric point of $X$. Then there is an isomorphism
\begin{align*}
\Ff^{\Pp} \times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace &= \left(\pi^*\Ff \times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace\right) \times^{\pi^*\cH_\Pp \times \pi^{-1}\lbrace x \rbrace}\left(\Pp \times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace\right) \\
&= \pi^*\left(\Ff|_x\right)\times^{\pi^*(\cH_\Pp|_x)}\left(\Pp\times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace\right) \\
&\cong \pi^*\left(\cH_\Pp|_x \right)\times^{\pi^*(\cH_\Pp|_x)}\left(\Pp\times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace\right) \\
&= \Pp\times_{\widetilde{X}} \pi^{-1}\lbrace x \rbrace,
\end{align*}
which is $\Gamma$-invariant because it is induced by the isomorphism between $\Ff|_x$ and $\cH_\Pp|_x$ on which $\Gamma$ acts trivially.
\end{proof}

We have gathered all the ingredients to state and prove our main result.

\begin{theo}\label{thm-equivstacks}
The maps $\pi_*\I_{\Pp}({\,\cdot\,})^\Gamma$ and $\pi^*({\,\cdot\,}) \times^{\pi^*{\cH_\Pp}} \Pp$ are each other inverses, defining an equivalence between $\Bun_{(\Gamma,G)}^{\Pp}$ and $\Bun_{\cH_{\Pp}}$.
\end{theo}

\begin{proof} Let $\Ff$ be an $\cH_\Pp$-bundle. We first show that $\pi_*\I_{\Pp}(\Ff^\Pp)^\Gamma$ is naturally isomorphic to $\Ff$, where we have simplified our notation, as in the proof of Proposition~\ref{prop-inversedef}, by writing $\Ff^\Pp$ in place of $\pi^*(\Ff) \times^{\pi^*{\cH_\Pp}} \Pp$. The assignment $f \mapsto [\phi_f \colon p \mapsto (f,p)]$ defines a morphism from $\pi^*\Ff$ to $\I_\Pp\left(\Ff^{\Pp}\right)$. By pushing it down to $X$ and taking $\Gamma$ invariants we obtain a map
\[
\Ff \to \pi_*\I_\Pp\left(\Ff_{\Pp}\right)^\Gamma = \pi_*\I_\Pp\left(\pi^*\Ff \times^{\pi^*{\cH_\Pp}}\Pp\right)^\Gamma.
\]
Locally we can check that this map is $\cH_\Pp$-equivariant, and hence it must be an isomorphism since both source and target are $\cH_\Pp$-bundles.

Conversely, let $\Pp'$ be a $(\Gamma, G)$-bundle with the same local type as $\Pp$. Applying $\left(\pi_*\I_\Pp({\,\cdot\,})\right)^\Gamma$ and then $\pi^*({\,\cdot\,})\times^{\pi^*\cH_\Pp} \Pp$ to $\Pp'$ we obtain
\[
\pi^*\left(\pi_*(\IoP)^\Gamma\right) \times^{\pi^*\cH_\Pp} \Pp.
\]
The inclusion $\left(\pi_*(\IoP\right)^\Gamma \subseteq \pi_*(\IoP)$ induces, by adjunction, the map of $\pi^*\cH_\Pp$-bundles
\[
\pi^*\left(\pi_*(\IoP)^\Gamma\right) \to \IoP, \quad f \mapsto \alpha_f
\]
which extends to a map of $G$-bundles
\[
\alpha \colon \pi^*\left(\pi_*(\IoP)^\Gamma\right) \times^{\pi^*\cH_\Pp} \Pp \to \IoP \times^{\pi^*\cH_\Pp} \Pp, \quad (f,p) \mapsto (\alpha_f, p).
\]
The evaluation map $\beta \colon \IoP \times^{\pi^*\cH_\Pp} \Pp \to \Pp'$ allows us to obtain the morphism
\[
\beta \alpha \colon \pi^*\left(\pi_*(\IoP)^\Gamma\right) \times^{\pi^*\cH_\Pp} \Pp \to\Pp'
\]
which we are left to show to be equivariant with respect to the actions of $\Gamma$ and $G$. Since both $\alpha$ and $\beta$ are $G$-equivariant, their composition is as well. The $\Gamma$-invariance translates to showing that $\alpha_f\left(\gamma_{\Pp}(p)\right)$ and $\gamma_{\Pp'}(\alpha_f(p))$ coincide, which holds because $\alpha_f$ is $\Gamma$-equivariant.
\end{proof}

\subsection{Special case}

When $\Pp$ is the trivial $G$-bundle on $\Xt$ with the action of $\Gamma$ given $\rho$, then we have that $\Go \cong \Xt \times G$. If we apply Theorem~\ref{thm-equivstacks} in this context, we obtain the theorem stated in the introduction.

\subsection*{Acknowledgments}

I am deeply indebted to V. Balaji and C. S. Seshadri for their inspirational work and perspective on parahoric bundles. Thanks to J. Heinloth, A. Gibney and J. Hong for helpful conversations. I am grateful to the anonymous referee for their suggestions.

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