# Hecke operators for curves over non-archimedean local fields and related
finite rings

Alexander Braverman  ,  David Kazhdan  and  Alexander Polishchuk  Deparment of
Mathematics, University of Toronto, Ontario, Canada<EMAIL_ADDRESS>Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel<EMAIL_ADDRESS>Department of Mathematics,
University of Oregon, Eugene, OR 97403, USA; National Research University
Higher School of Economics<EMAIL_ADDRESS>
###### Abstract.

We study Hecke operators associated with curves over a non-archimedean local
field $K$ and over the rings $O/{\mathfrak{m}}^{N}$, where $O\subset K$ is the
ring of integers. Our main result is commutativity of a certain “small” local
Hecke algebra over $O/{\mathfrak{m}}^{N}$, associated with a connected split
reductive group $G$ such that $[G,G]$ is simple and simply connected. The
proof uses a Hecke algebra associated with $G(K(\\!(t)\\!))$ and a global
argument involving $G$-bundles on curves.

D.K. is partially supported by the ERC grant No 669655. A.P. is partially
supported by the NSF grant DMS-2001224, and within the framework of the HSE
University Basic Research Program and by the Russian Academic Excellence
Project ‘5-100’.

##  1\. Introduction

Let $C$ be a smooth proper curve over a non-archimedean local field $K$. In
this paper we study Hecke operators on certain vector spaces related to the
moduli space of $G$-bundles on $C$ and over the related curves over finite
rings $O/{\mathfrak{m}}^{n}$, under the assumption of existence of smooth
model $C_{O}$ over the ring of integers $O\subset K$.

We refer to [5] for a survey of conjectures and approaches to the analog of
the Langlands program in the case of curves over local fields, and to [10],
[11] for more precise conjectures and results in the archimedean case.

For a split reductive group $G$ we consider the moduli stack
$\underline{{\operatorname{Bun}}}_{G}$ of $G$-bundles over $C$. Applying the
construction of [15] we consider the Schwartz space

${\mathcal{W}}:=\SS({\operatorname{Bun}}_{G},|\omega|^{1/2})$

of half-densities on
${\operatorname{Bun}}_{G}=\underline{{\operatorname{Bun}}}_{G}(K)$. Hecke
operators associated with points of $C$ generate a commutative algebra $H(C)$
of endomorphisms of the $\operatorname{\mathbb{C}}$-vector space
${\mathcal{W}}$ (see Theorem 5.3).

######  Definition 1.1.

  1. (1)

For a homomorphism $s:H(C)\to{\mathbb{C}}$ we define
$\mathcal{W}_{s}(C)=\\{w\in\mathcal{W}(C)|hw=s(h)w\\}$.

  2. (2)

The spectrum $S(C)$ of $H(C)$ is the set of homomorphisms
$s:H(C)\to{\mathbb{C}}$ such that $\mathcal{W}_{s}(C)\neq\\{0\\}$.

######  Conjecture 1.2.

$\bigoplus_{s\in S(C)}\mathcal{W}_{s}(C)=\mathcal{W}(C)$.

######  Remark 1.3.

In the archimedean case, the analogs of Hecke operators could also be defined,
and the spectrum $S(C)$ has a conjectural description in terms of
$G^{\vee}$-opers on $C$ (see [10], [11]).

In the case when there exists a smooth proper model $C_{O}$ over $O$ one can
look for an approach to the above conjecture through the analysis of similar
objects for $C_{O}$ and its reductions over $O/{\mathfrak{m}}^{N}$, where
${\mathfrak{m}}\subset O$ is the maximal ideal. From now on we fix a smooth
proper model $C_{O}$ over $O$.

Let ${\operatorname{Bun}}^{O}_{G}\subset{\operatorname{Bun}}_{G}$ be the open
subgroupoid $G$-bundles on $C$ for which there exists an extension to $C_{O}$
(not to be confused with the groupoid ${\operatorname{Bun}}_{G}(O)$ of
$G$-bundles on $C_{O}$). We show that in the case when $[G,G]$ is simple and
simply connected and the characteristic of $O/{\mathfrak{m}}$ is sufficiently
large, ${\operatorname{Bun}}^{O}_{G}$ coincides with the subgroupoid of
generically trivial $G$-bundles (see Lemmas 4.11 and 4.1). We also show that
all Schwartz half-densities supported on ${\operatorname{Bun}}^{O}_{G}$ come
from smooth functions on ${\operatorname{Bun}}_{G}(O)$ and therefore depend on
the reduction modulo some power of the maximal ideal
${\mathfrak{m}}^{N}\subset O$ (see Prop. 2.11).

We then consider the reductions $C_{N}$ of $C$ over $O/{\mathfrak{m}}^{N}$. As
in [6], we consider a big local Hecke algebra (noncommutative for $N>1$),
defined in terms of distributions on the local group
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$. For every point $v\in
C_{N}(O/{\mathfrak{m}}^{N})$, there is an action of this Hecke algebra on the
space $\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$ of finitely
supported functions. In [6] we studied this action in the context of
automorphic representations. In this paper we define a small local Hecke
algebra ${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$ (a subalgebra
in the big Hecke algebra). Our main result, generalizing [6, Thm. 2.6], is
that this small Hecke algebra is commutative provided $[G,G]$ is simply
connected and simple, and the characteristic of $O/{\mathfrak{m}}$ is
sufficiently large (see Theorem 3.10).

Due to the compatibility between Hecke operators over $K$ and
$O/{\mathfrak{m}}^{N}$ (see Sec. 5.2), one can therefore reduce the
construction of $H(C)$-eigenvectors in ${\mathcal{W}}$ to the similar problem
for the curves $C_{N}$ over $O/{\mathfrak{m}}^{N}$. The latter problem is much
closer to the classical Langlands program for curves over finite fields (see
[6] for some partial results).

The key tool we use in this paper is the theory of representations of the ind-
pro-group ${\mathbb{G}}:=G(K(\\!(t)\\!))$ in pro-vector spaces developed in
[14], [15]. More precisely, similarly to [15, Sec. 3], we realize local Hecke
algebras as endomorphisms of functors of $G[\\![t]\\!]$-coinvariants. Such an
interpretation allows to define and analyze actions of these algebras on
spaces related to $G$-bundles.

######  Remark 1.4.

The local Hecke algebras we consider in this paper are associated with the
embedding of groups $G(K[\\![t]\\!])\subset G(K(\\!(t)\\!))$ (resp.,
$G(O/{\mathfrak{m}}^{N}[\\![t]\\!])\subset
G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$, in a version over
$O/{\mathfrak{m}}^{N}$). One can exchange the roles of the uniformizer in $O$
and of formal variable $t$, and consider instead the pair
$G(O(\\!(t)\\!))\subset G(K(\\!(t)\\!))$ (resp., $G(O[t]/(t^{N}))\subset
G(K[t]/(t^{N}))$). For example, the work [18] (and most of [4]) deals with the
Hecke algebras coming from pairs of the second kind.

In the functional case $K=k(\\!(x)\\!)$, when $O=k[\\![x]\\!]$ and
$O/{\mathfrak{m}}^{N}=k[x]/(x^{N})$, we can identify the pair of the first
kind associated to $K$ with the pair of the second kind associated to
$K^{\prime}=k(\\!(t)\\!)$ (with $O^{\prime}=k[\\![t]\\!]$):

$O/{\mathfrak{m}}^{N}(\\!(t)\\!)=K^{\prime}[x]/(x^{N}),\ \
O/{\mathfrak{m}}^{N}[\\![t]\\!]=O^{\prime}[x]/(x^{N}).$

We conjecture that in the case when the characteristic of $K$ is zero, small
Hecke algebras associated with pairs $G(O[t]/(t^{N}))\subset G(K[t]/(t^{N}))$
are still commutative but we do not address this problem in this paper (case
$N=2$ is proved in [18]).

The paper is organized as follows. In Sec. 2 we collect some general results
about Schwartz spaces of varieties and stacks over $K$ and over $O$. In Sec. 3
we discuss local Hecke algebras related to the loop group of $G$ over $K$ and
over related rings. First, we discuss the Hecke algebra over $K$, denoted by
${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$ (where $c$ is the level),
in the framework of $G(K(\\!(t)\\!))$-representations in pro-vector spaces. In
Sec. 3.3 we recall the commutative subalgebra in the Hecke algebra
${\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{crit}$ at the critical level
constructed in [4]. In Sec. 3.4 we consider versions of this algebra over $O$
and $O/{\mathfrak{m}}^{N}$ denoted by
${\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})$ and
${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$, respectively. We show that
the algebra ${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$ is isomorphic
to the algebra of $G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$-biinvariant
distributions with compact support on $G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$. In
Sec. 3.5 we define algebra homomorphisms between the Hecke algebras,

${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}\to{\mathcal{H}}(G_{O},{\mathbb{H}}_{O})\to{\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N}),$

and show their compatibility with the action on $G[\\![t]\\!]$-coinvariants.
In Sec. 3.6 we discuss coordinate-free versions of our local Hecke algebras
and define the small Hecke algebra
${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$.

In Sec. 4 we present auxiliary results on $G$-bundles. In Sec. 4.1 we recall
facts about the principal $\mathfrak{s}l_{2}$-subalgebra. In Sec. 4.2 and 4.3
we introduce and study the notion of a nice $G$-bundle on a curve with a fixed
point $p$: these are $G$-bundles for which we can control automorphisms over
$C-p$ with a fixed order $n$ of pole at $p$. One of the key results here is
the existence of curves over given finite fields (of sufficiently large
characteristic) with such nice $G$-bundles for arbitrarily large $n$ (see Cor.
4.9). In Sec. 4.5 we relate generic triviality of $G$-bundles over a curve $C$
over $K$ with existence of an extension to $C_{O}$ (a smooth proper model over
$O$).

In Sec. 5 we study Hecke operators on the Schwartz space ${\mathcal{W}}$ of
half-densities on ${\operatorname{Bun}}_{G}$ for a curve $C$ over $K$, and use
global arguments to prove our main result Theorem 3.10. In Sec. 5.1 we relate
the operators on ${\mathcal{W}}$ arising from the representation theory of
$G(K(\\!(t)\\!))$ in pro-vector spaces with those given by the Hecke
correspondences. We also prove commutation of Hecke operators in $H(C)$
corresponding to different $K$-points of a curve $C$ (see Theorem 5.3). In
Sec. 5.2 we discuss Hecke operators over $O$ and $O/{\mathfrak{m}}^{N}$
arising from elements of $H(C)$. Note that we can explicitly describe these
Hecke operators only for minuscule coweights $\lambda$. Finally in Sec. 5.3 we
give a global proof of Theorem 3.10 on commutativity of the small local Hecke
algebra over $O/{\mathfrak{m}}^{N}$, using Theorem 5.3 and nice $G$-bundles.

Notation:

$K$ is a non-archimedian local field, $O\subset K$ its ring of integers,
${\mathfrak{m}}\subset O$ the maximal ideal, $k=O/{\mathfrak{m}}$.

For a connected split reductive group $G$, $\Lambda$ denotes the lattice of
coweights of $G$, $\Lambda^{+}\subset\Lambda$ the semigroup of dominant
coweights, $\operatorname{Gr}_{G}=G(\\!(t)\\!)/G[\\![t]\\!]$ the affine
Grassmannian.

##  2\. Geometry over $K$ and $O$

###  2.1. Varieties over $K$ and $O$

Let us recall some standard general constructions.

######  Definition 2.1.

Let $X$ be a scheme of finite type over $K$.

  1. (1)

${\mathbb{C}}(X(K))$ is the space of locally constant ${\mathbb{C}}$-valued
functions on $X(K)$.

  2. (2)

$\mathcal{S}(X(K))\subset{\mathbb{C}}(X(K))$ is the subspace of compactly
supported functions.

  3. (3)

For a line bundle $\mathcal{L}$ on $X$ and a character
$c:K^{*}\to\operatorname{\mathbb{C}}^{*}$ (i.e., a continuous homomorphism),
we denote by $\mathcal{L}_{c}$ the associated complex local system on $X(K)$
(defined using push-out with respect to $c$). In particular, for a complex
number $z$, we denote by $|\mathcal{L}|^{z}:=\mathcal{L}_{|\cdot|^{z}}$ the
local system associated with the homomorphism
$|\cdot|^{z}:K^{*}\to\operatorname{\mathbb{C}}^{*}$.

  4. (4)

For any $\operatorname{\mathbb{C}}$-local system $L$ on $X(K)$, since
transition functions are locally constant, we can define the space
${\mathbb{C}}(X(K),L)$ of locally constant sections and the subspace
$\mathcal{S}(X(K),L)\subset{\mathbb{C}}(X(K),L)$ of compactly supported
sections. In particular, for a line bundle $\mathcal{L}$ on $X$, and a
character $c:K^{*}\to\operatorname{\mathbb{C}}^{*}$, this gives a complex
vector space $\mathcal{S}(X(K),\mathcal{L}_{c})$. 111See Section $2.1$ of [5].

Now consider the case when $X$ is scheme of finite type over $O$, with the
associated scheme $X_{K}$ over $K$. In this case, the space $X(O)$ is an open
subset of $X(K)=X_{K}(K)$. Furthermore, for any line bundle $\mathcal{L}$ on
$X$, the complex line bundle $|\mathcal{L}|^{z}$ is canonically trivialized
over $X(O)$. Hence, the space $\mathcal{S}(X(O),|\mathcal{L}|^{z})$ of
compactly supported locally constant sections of $|\mathcal{L}|^{z}$ can be
identified with the space $\mathcal{S}(X(O))$ of locally constant functions on
$X(O)$.

Thus, for any line bundle $\mathcal{L}$ on $X$, and every $n\geq 1$, we have a
natural map

$r_{n,|\mathcal{L}|^{z}}:\SS(X(O/{\mathfrak{m}}^{n}))\to\SS(X(O))\hookrightarrow\SS(X(K),|\mathcal{L}|^{z}),$
(2.1)

which sends a finitely supported function on $X(O/{\mathfrak{m}}^{n})$ to the
corresponding locally constant compactly supported function on $X(O)$, which
is then viewed as a locally constant section of $|\mathcal{L}|^{z}$ on $X(O)$.
Note that we have

$\SS(X(O))=\varinjlim\SS(X(O/{\mathfrak{m}}^{n})).$

Assume now that $X$ is smooth over $K$, and let $\omega_{X}$ be the canonical
line bundle. Then ${\mathbb{C}}(X(K),|\omega_{X}|)$ is the space of smooth
measures on $X(K)$. A nowhere vanishing section $\eta\in\Gamma(X,\omega_{X})$
defines a smooth complex valued measure $|\eta|$ on $X(K)$. For
$\mu\in\SS(X(K),|\omega_{X}|)$, the integral $\int_{X(K)}\mu$ is well defined.
Similarly, for a smooth morphism $f:X\to Y$, a line bundle $\mathcal{L}$ on
$Y$, and a character $c:K^{*}\to\operatorname{\mathbb{C}}^{*}$, we have a
well-defined push-forward map

$f_{!}:\SS(X(K),(f^{*}\mathcal{L})_{c}\otimes|\omega_{X/Y}|)\to\SS(Y(K),\mathcal{L}_{c}).$
(2.2)

More generally, if $X$ is Gorenstein and has rational singularities then there
is a well defined integration map

$\SS(X(K),|\omega_{X}|)\to\operatorname{\mathbb{C}}:\mu\mapsto\int_{X^{sm}(K)}\mu,$

where the integral is absolutely convergent (see [1, Sec. 3.4]). Now consider
a proper Gorenstein morphism $f:X\to Y$ of integral schemes of finite type
over $K$. Assume that there exists a proper rational map $\pi:\widetilde{X}\to
X$ with the following properties:

  * •

The map $\pi f:\widetilde{X}\to Y$ is smooth;

  * •

there is an isomorphism
$\pi^{*}\omega_{X/Y}\simeq\omega_{\widetilde{X}/Y}(-D)$ for some effective
Cartier divisor $D$ on $\widetilde{X}$;

  * •

locally in smooth topology over $Y$, there is an isomorphism
$(\widetilde{X},D)\simeq(Y\times F,Y\times E)$, where $F$ is smooth and
$E\subset F$ is a Cartier divisor.

Then for any line bundle ${\mathcal{L}}$ on $Y$ and a character
$c:K^{*}\to\operatorname{\mathbb{C}}^{*}$, we can define the push-forward map

$f_{!}:\SS(X(K),(f^{*}\mathcal{L})_{c}\otimes|\omega_{X/Y}|)\to\SS(Y(K),\mathcal{L}_{c})$

as the composition

$\SS(X(K),(f^{*}\mathcal{L})_{c}\otimes|\omega_{X/Y}|)\to\SS(\widetilde{X}(K),(\pi^{*}f^{*}{\mathcal{L}})_{c}\otimes|\omega_{\widetilde{X}/Y}(-D)|)\to\SS(Y(K),\mathcal{L}_{c}),$

where the second arrow is given by integration in fibers of $\widetilde{X}\to
Y$. The fact that the resulting section of $\mathcal{L}_{c}$ is locally
constant follows from our assumption on the local structure of
$(\widetilde{X},D)$.

The following result goes back to Weil [19].

######  Lemma 2.2.

Let $X$ be a smooth variety over $O$. Then for any function
$\phi\in\SS(X(O/{\mathfrak{m}}^{n}))$ one has

$\int r_{n,|\omega_{X}|}(\phi)=\frac{1}{|O/{\mathfrak{m}}^{n}|^{\dim
X}}\sum_{x\in X(O/{\mathfrak{m}}^{n})}\phi(x).$

###### Proof.

This is essentially in [19, Sec. 2.2]. It is enough to consider the case when
$\phi$ is the delta-function of a point $x_{0}\in X(O/{\mathfrak{m}}^{n})$.
Then we have the corresponding compact open neighborhood

$V(x_{0},n):=\\{x\in X(O)\ |\ x\equiv x_{0}\mod{\mathfrak{m}}^{n}\\},$

and our statement reduces to the statement that

$\mu(V_{x_{0},n})=\frac{1}{|O/{\mathfrak{m}}^{n}|^{\dim X}},$

where $\mu$ is the measure on $X(O)$ obtained from the canonical
trivialization of $|\omega_{X}|$ over $X(O)$. We can choose near $x_{0}$ an
étale map $f:X\to{\mathbb{A}}^{d}_{O}$ sending $x_{0}$ to the origin, where
$d=\dim X$. Then by Hensel lemma, $f$ maps $V_{x_{0},n}$ bijectively onto
${\mathbb{A}}^{d}({\mathfrak{m}}^{n})\subset{\mathbb{A}}^{d}(O)$, and the
result follows. ∎

###  2.2. Stacks over $K$ and $O$

We refer to [15] for details and proofs of the results below on stacks over
$K$.

######  Definition 2.3.

  1. (1)

For a smooth stack $\mathcal{Y}$ over $K$ we denote by $\omega_{\mathcal{Y}}$
the canonical bundle on $\mathcal{Y}$.

  2. (2)

For a smooth representable map $q:X\to\mathcal{Y}$ we denote by
$\omega_{q}=\omega_{X/\mathcal{Y}}$ the relative canonical bundle.

  3. (3)

We say that a smooth stack $\mathcal{Y}$ of finite type over $K$ is admissible
if it can be represented as a quotient $[X/H]$ where $X$ is a smooth variety
over $K$ and $H$ is a linear algebraic group acting on $X$.

  4. (4)

A stack $\mathcal{Y}$ over $K$ is called admissible222This notion is slightly
stronger than that of [15]. if it is the union $\cup_{i\geq 1}\mathcal{Y}_{i}$
of an increasing sequence $\mathcal{Y}_{1}\subset\mathcal{Y}_{2}\subset\ldots$
of open admissible substacks of finite type over $K$.

######  Claim 2.4.

Let $\mathcal{Y}$ of be an admissible stack finite type over $K$.

  1. (1)

A line bundle $\mathcal{L}$ on $\mathcal{Y}=[X/H]$ is canonically represented
by an $H$-equivariant line bundle $\tilde{\mathcal{L}}$ on $X$.

  2. (2)

The canonical bundle $\omega_{\mathcal{Y}}$ on $\mathcal{Y}=[X/H]$ is
represented by $\omega_{X}\otimes\Lambda^{d}(\mathfrak{h})$ where
$\mathfrak{h}$ is the Lie algebra of $H$ and $d=dim(\mathfrak{h})$ (we use the
adjoint action of $H$ in defining the $H$-equivariant structure).

  3. (3)

Any admissible stack $\mathcal{Y}$ of finite type can be represented as a
quotient $[X/\operatorname{GL}_{N}]$.

  4. (4)

The topological groupoid $X(K)/\operatorname{GL}_{N}(K)$ does not depend on a
choice of a presentation $\mathcal{Y}=[X/\operatorname{GL}_{N}]$.

######  Definition 2.5.

  1. (1)

For a line bundle $\mathcal{L}$ on an admissible stack of finite type over
$K$, $\mathcal{Y}=[X/\operatorname{GL}_{N}]$, and a character
$c:K^{*}\to\operatorname{\mathbb{C}}^{*}$, we denote by
$\mathcal{S}(\mathcal{Y}(K),\mathcal{L}_{c})$ (where
$z\in\operatorname{\mathbb{C}}$) the space of coinvariants

$\mathcal{S}(\mathcal{Y}(K),\mathcal{L}_{c}):=\mathcal{S}(X(K),\tilde{\mathcal{L}}_{c}\otimes|\omega_{X/\mathcal{Y}}|)_{\operatorname{GL}_{N}(K)},$

where $\tilde{\mathcal{L}}$ is the $\operatorname{GL}_{N}$-equivariant line
bundle on $X$ which represents $\mathcal{L}$.

  2. (2)

If $\mathcal{L}$ is a line bundle on an admissible stack $\mathcal{Y}$ which
is the union of increasing open substacks $\mathcal{Y}_{i}$ of finite type we
define
$\mathcal{S}(\mathcal{Y}(K),\mathcal{L}_{c})=\varinjlim\mathcal{S}(\mathcal{Y}_{i}(K),(\mathcal{L}|_{\mathcal{Y}_{i}})_{c})$.

######  Claim 2.6.

The space $\mathcal{S}(\mathcal{Y}(K),\mathcal{L}_{c})$ for an admissible
stack of finite type does not depend on a presentation of $\mathcal{Y}$ as a
quotient $[X/\operatorname{GL}_{N}]$. The space
$\mathcal{S}(\mathcal{Y}(K),\mathcal{L}_{c})$ for an admissible stack does not
depend on a choice of open admissible substacks of finite type
$\mathcal{Y}_{i}$ such that $\mathcal{Y}=\cup\mathcal{Y}_{i}$.

We define admissible stacks over $O$ in the same way as above, replacing $K$
by $O$.

######  Lemma 2.7.

Let $X$ be a scheme over $O$ equipped with an action of an algebraic group $H$
(defined over $O$), and let ${\mathcal{M}}$ be an $H$-equivariant line bundle
on $X$. Then the natural trivialization of $|{\mathcal{M}}|$ over $X(O)$ is
compatible with the $H(O)$-action.

###### Proof.

Let $s_{i}:{\mathcal{O}}_{U_{i}}\rTo{\sim}{\mathcal{M}}|_{U_{i}}$ be
trivializations of ${\mathcal{M}}$ over an open covering $(U_{i})$ of $X$.
Consider the covering of $H\times X$ by the open subsets

$V_{ij}=\\{(h,x)\in X\times H\ |x\in U_{i},hx\in U_{j}\\}.$

Over each $V_{ij}$ we have an equality

$s_{j}(hx)=f_{ij}(h,x)\cdot s_{i}(x),$

for some $f_{ij}\in{\mathcal{O}}^{*}(V_{ij})$ (we use the equivariant
structure to view both sides as sections of $p_{2}^{*}{\mathcal{M}}$). Hence,
taking absolute values we get the equality

$|s_{j}(hx)|=|s_{i}(x)|$

on $V_{ij}(O)$, which implies our assertion. ∎

######  Definition 2.8.

Let $\mathcal{Y}=[X/\operatorname{GL}_{N}]$ be an admissible stack of finite
type defined over $O$.

  1. (1)

We set
$\mathcal{S}(\mathcal{Y}(O)):=\mathcal{S}(X(O))_{\operatorname{GL}_{N}(O)}$.

  2. (2)

If $\mathcal{Y}$ is the union of increasing open substacks $\mathcal{Y}_{i}$
of finite type we define
$\mathcal{S}(\mathcal{Y}(O))=\varinjlim\mathcal{S}(\mathcal{Y}_{i}(O))$.

######  Remark 2.9.

By Lemma 2.7, we have
$\mathcal{S}(X(O))_{\operatorname{GL}_{N}(O)}=\mathcal{S}(X(O),|\omega_{X/{\mathcal{Y}}}|)_{\operatorname{GL}_{N}(O)}$.
The space $\mathcal{S}(\mathcal{Y}(O))$ does not depend on a representation of
$\mathcal{Y}$ as a quotient $[X/\operatorname{GL}_{N}]$.

If ${\mathcal{Y}}=[X/\operatorname{GL}_{N}]$ is an admissible stack of finite
type over $O$, ${\mathcal{L}}$ a line bundle on ${\mathcal{Y}}$, represented
by a $\operatorname{GL}_{N}$-equivariant line bundle
$\widetilde{{\mathcal{L}}}$ on $X$, then by Lemma 2.7, the identification

$\SS(X(O),|\widetilde{{\mathcal{L}}}|^{z}\otimes|\omega_{X/{\mathcal{Y}}}|)\simeq\SS(X(O))$

is compatible with the $\operatorname{GL}_{N}(O)$-actions. Hence, we get a
natural map

$r^{\mathcal{Y}}_{*}:\SS({\mathcal{Y}}(O))=\SS(X(O))_{\operatorname{GL}_{N}(O)}\to\SS(X(K),|\widetilde{{\mathcal{L}}}|^{z}\otimes|\omega_{X/{\mathcal{Y}}}|)_{\operatorname{GL}_{N}(K)}=\SS({\mathcal{Y}}_{K}(K),|{\mathcal{L}}|^{z}),$

where ${\mathcal{Y}}_{K}$ is corresponding stack
$[X_{K}/\operatorname{GL}_{N}]$ over $K$.

More generally, if $\mathcal{Y}$ is an admissible stack over $O$,
$\mathcal{Y}_{K}$ the corresponding admissible stack over $K$, then for any
line bundle $\mathcal{L}$ on $\mathcal{Y}$, we get a natural map

$r^{\mathcal{Y}}_{*}:\mathcal{S}(\mathcal{Y}(O))\to\mathcal{S}(\mathcal{Y}_{K}(K),|\mathcal{L}|^{z})$
(2.3)

defined as the limit of similar maps for
$\mathcal{Y}_{i}=[X/\operatorname{GL}_{N}]$,

######  Definition 2.10.

For a stack $\mathcal{Y}$ over $O$ we denote by $r^{\mathcal{Y}}$ the natural
map $\mathcal{Y}(O)\to\mathcal{Y}(K)$ of topological groupoids.

######  Proposition 2.11.

The image of the map $r^{\mathcal{Y}}_{*}$ (see (2.3)) consists of densities
supported on the open subgroupoid $r^{\mathcal{Y}}(\mathcal{Y}(O))$.

###### Proof.

Let $\mathcal{Y}_{i}=[X_{i}/\operatorname{GL}_{N_{i}}]$. For each $j\geq i$,
let us consider the cartesian square where $f_{i,j}$ is the composition of the
projection $X_{i}\to\mathcal{Y}_{i}$ with the open embedding
$\mathcal{Y}_{i}\to\mathcal{Y}_{j}$. Note that $X_{i,j}$ is a $GL_{N}$-torsor
over $X_{i}$.

Step 1. We claim that for every point $x\in X_{i}(K)$ in the preimage of
$r^{\mathcal{Y}}({\mathcal{Y}}({\mathcal{O}}))\subset{\mathcal{Y}}(K)$, there
exists $j\geq i$ and a point $\widetilde{x}\in X_{i,j}(K)$ over $x$ such that
$\widetilde{f}_{i,j}(\widetilde{x})\in X_{j}(O)$. Indeed, by definition there
exists $j\geq i$ such that the image of $x$ in $\mathcal{Y}_{j}(K)$ comes from
a point in $X_{j}(O)$. Thus, if $\widetilde{x}\in X_{i,j}(K)$ is any point
over $x$ then the $\operatorname{GL}_{N_{j}}(K)$-orbit of
$\widetilde{f}_{i,j}(x^{\prime})$ contains a point in $X_{j}(O)$. Hence,
changing $\widetilde{x}$ to $g\widetilde{x}$ for an appropriate
$g\in\operatorname{GL}_{N_{j}}(K)$, we can achieve that
$\widetilde{f}_{i,j}(\widetilde{x})\in X_{j}(O)$.

Step 2. Next, we claim that for any compact open subset $A$ in the preimage of
$r^{\mathcal{Y}}({\mathcal{Y}}({\mathcal{O}}))$ in $X_{i}(K)$, there exists
$j\geq i$ and a compact open $\widetilde{A}\subset X_{i,j}(K)$ such that
$p_{i,j}(\widetilde{A})=A$ and $\widetilde{f}_{i,j}(\widetilde{A})\subset
X_{j}(O)$. Indeed, by Step 1, for any point $x\in A$ we can find $j\geq i$ and
a point $\widetilde{x}\in X_{i,j}(K)$ over $x$ such that
$\widetilde{f}_{i,j}(\widetilde{x})\in X_{j}(O)$. Furthermore, there exists an
open compact neighborhood $U$ of $\widetilde{x}$ in $X_{i,j}(K)$ such that
$\widetilde{f}_{i,j}(U)\subset X_{j}(O)$. By compactness of $A$, finitely many
open sets of the form $p_{i,j}(U)$ cover $A$, which implies our claim.

Step 3. Given an element
$\phi\in\mathcal{S}(X_{i}(K),|\mathcal{L}\otimes\omega_{X_{i}/\mathcal{Y}}|)$,
supported on a compact open subset $A\subset X_{i}(K)$ contained in the
preimage of $r^{\mathcal{Y}}({\mathcal{Y}}({\mathcal{O}}))$, we can find
$j\geq i$ and $\widetilde{A}\subset X_{i,j}(K)$ as in Step 2. Since the
projection $\widetilde{A}\to A$ is smooth and surjective, as in the proof of
[15, Prop. 6.2], we can find
$\widetilde{\phi}\in\mathcal{S}(X_{i,j},|\mathcal{L}\otimes\omega_{X_{i,j}/\mathcal{Y}}|)$
supported on $\widetilde{A}$, such that $p_{i,j,!}(\widetilde{\phi})=\phi$
(note that
$\omega_{X_{i,j}/\mathcal{Y}}\simeq\omega_{X_{i}/\mathcal{Y}}\otimes\omega_{X_{i,j}/X_{i}}$).
Now
$\widetilde{f}_{i,j,!}(\widetilde{\phi})\in\mathcal{S}(X_{j},|\mathcal{L}\otimes\omega_{X_{j}/\mathcal{Y}}|)$
maps to the same element in $\mathcal{S}(\mathcal{Y}(K),|\mathcal{L}|)$ as
$\phi$. Since $\widetilde{f}_{i,j,!}(\widetilde{\phi})$ is supported on
$\widetilde{f}_{i,j}(\widetilde{A})\subset X_{j}(O)$, our assertion follows. ∎

For a smooth representable morphism $f:{\mathcal{X}}\to{\mathcal{Y}}$ of
admissible stacks over $K$, a line bundle ${\mathcal{L}}$ over
${\mathcal{Y}}$, and a character $c:K^{*}\to\operatorname{\mathbb{C}}^{*}$, we
define the push-forward maps

$f_{!}:\SS({\mathcal{X}}(K),(f^{*}{\mathcal{L}})_{c}\otimes|\omega_{{\mathcal{X}}/{\mathcal{Y}}}|)\to\SS({\mathcal{Y}}(K),{\mathcal{L}}_{c})$

using the corresponding maps (2.2) for varieties (see also [15, Sec. 6.8]).

Let ${\mathcal{Y}}$ be an admissible stack of finite type over $O$,
${\mathcal{L}}$ a line bundle over ${\mathcal{Y}}$. For each $n>0$, we have
the groupoid ${\mathcal{Y}}(O/{\mathfrak{m}}^{n})$ of
$O/{\mathfrak{m}}^{n}$-points of ${\mathcal{Y}}$, and the corresponding space
$\SS({\mathcal{Y}}(O/{\mathfrak{m}}^{n}))$ of finitely supported functions.
Furthermore, for we have a natural map

$r_{n,|{\mathcal{L}}|^{z}}:\SS({\mathcal{Y}}(O/{\mathfrak{m}}^{n}))\to\SS({\mathcal{Y}}(O))\rTo{r^{{\mathcal{Y}}}_{*}}\SS({\mathcal{Y}}(K),|{\mathcal{L}}|^{z})$

induced by the map (2.1). As in the case of varieties, we see that
$r^{{\mathcal{Y}}}_{*}(\SS({\mathcal{Y}}(O)))$ is the union of the subspaces
$r_{n,|{\mathcal{L}}|^{z}}(\SS({\mathcal{Y}}(O/{\mathfrak{m}}^{n})))$.

######  Proposition 2.12.

Let $f:{\mathcal{X}}\to{\mathcal{Y}}$ be a representable smooth morphism of
admissible stacks over $O$, and let
$f_{O/{\mathfrak{m}}^{n}}:{\mathcal{X}}(O/{\mathfrak{m}}^{n})\to{\mathcal{Y}}(O/{\mathfrak{m}}^{n})$
denote the corresponding functor between groupoids of
$O/{\mathfrak{m}}^{n}$-points. Then for any line bundle ${\mathcal{L}}$ over
${\mathcal{Y}}$, and any $\phi\in\SS({\mathcal{X}}(O/{\mathfrak{m}}^{n}))$, we
have

$f_{!}r_{n,|f^{*}{\mathcal{L}}|^{z}\otimes|\omega_{{\mathcal{X}}/{\mathcal{Y}}}|}(\phi)=\frac{1}{|O/{\mathfrak{m}}^{n}|^{\dim{\mathcal{X}}-\dim{\mathcal{Y}}}}\cdot
r_{n,|{\mathcal{L}}|^{z}}f_{O/{\mathfrak{m}}^{n},*}\phi.$

Here we use the natural push-forward map $f_{O/{\mathfrak{m}}^{n},*}$ for
finitely supported functions on groupoids.

###### Proof.

We can assume that ${\mathcal{X}}=X/G$, ${\mathcal{Y}}=Y/G$, where
$G=\operatorname{GL}_{N}$, and the morphism ${\mathcal{X}}\to{\mathcal{Y}}$ is
induced by a smooth $G$-equivariant map $f:X\to Y$. The commutative diagrams
show that it is enough to prove our assertion with
${\mathcal{X}}\to{\mathcal{Y}}$ replaced by $X\to Y$. We can also assume
${\mathcal{L}}$ to be trivial.

Let us start with $\phi\in\SS(X(O/{\mathfrak{m}}^{n}))$ then
$f_{!}r_{n,|\omega_{X/Y}|}(\phi)$ is supported on $Y(O)$ and we need to
compute its value at $y\in Y(O)$. Let $X_{y}\subset X$ be the fiber over $y$
(which is a smooth variety over $O$). Then

$f_{!}r_{n,|\omega_{X/Y}|}(\phi)(y)=\int
r_{n,|\omega_{X_{y}}|}(\phi|_{X_{y}(O/{\mathfrak{m}}^{n})}).$

Applying Lemma 2.2, we can rewrite the integral in terms of the summation of
values of $\phi$ over $X_{y}(O/{\mathfrak{m}}^{n})$, which gives the result. ∎

##  3\. Local Hecke algebras

In this section we discuss local Hecke algebras associated with a group $G$
and a $2$-dimensional local field $K(\\!(t)\\!)$, as well the related algebras
over $O$ and $O/{\mathfrak{m}}^{N}$.

First, in Sec. 3.1, 3.2 and 3.3 we discuss Hecke algebras associated with
$K(\\!(t)\\!)$ in the context of representations of $K(\\!(t)\\!)$ in pro-
vector spaces. Then in Sec. 3.4, we consider analogous algebras over $O$ and
$O/{\mathfrak{m}}^{N}$. In Sec. 3.5 we construct homomorphisms connecting the
three kind of algebras. Finally, in Sec. 3.6, we use these homomorphisms to
define the small Hecke algebra over $O/{\mathfrak{m}}^{N}$, and we formulate
our main theorem about its commutativity (Theorem 3.10).

Note that all constructions of this section are purely local and do not
involve $G$-bundles on curves.

###  3.1. Representations on pro-vector spaces and the local Hecke algebra
over $K$

Following [14], [15], we work in the framework of groups in
${\mathbb{S}}et={\operatorname{Ind}}({\operatorname{Pro}}({\operatorname{Ind}}({\operatorname{Pro}}(Set_{0}))))$
(where $Set_{0}$ is the category of finite sets) and their representations in
pro-vector spaces, i.e., objects of
${\mathbb{V}}ect={\operatorname{Pro}}(Vect)={\operatorname{Pro}}({\operatorname{Ind}}(Vect_{0}))$
(where $Vect_{0}$ is the category of finite-dimensional
$\operatorname{\mathbb{C}}$-vector spaces). Following [14], we write pro-
objects of a category ${\mathcal{C}}$ as $``\varprojlim"C_{j}$, where
$C_{j}\in{\mathcal{C}}$, to distinguish them from the projective limits taken
in the category ${\mathcal{C}}$.

As before, $G$ is a split connected reductive group over
$\operatorname{\mathbb{Z}}$.

Let $K$ be a local non-archimedean field. Then there is a natural group in
${\mathbb{S}}et$ which we denote as ${\mathbb{G}}=G(K(\\!(t)\\!))$. Namely,
$K(\\!(t)\\!)$ can be viewed as the ind-object $(t^{-n}K[\\![t]\\!])$, while
$K[\\![t]\\!]$ is the pro-object $(K[t]/(t^{n}))$, where each $K[t]/t^{n}$ is
an object of ${\bf S}et={\operatorname{Ind}}({\operatorname{Pro}}(Set_{0}))$.

Note that ${\mathbb{G}}$ comes from a group-indscheme $G(\\!(t)\\!)$ defined
over $\operatorname{\mathbb{Z}}$ (see [14, Sec. 2.12]). We also fix a central
extension $\hat{G}$ of $G(\\!(t)\\!)$ by ${\mathbb{G}}_{m}$ in the category of
group-indschemes, equipped with a splitting over $G[\\![t]\\!]$, and denote by
$\hat{{\mathbb{G}}}$ the corresponding extension of ${\mathbb{G}}$ by $K^{*}$
(see [14, Sec. 2.14]).

We consider the subgroup ${\mathbb{H}}:=G(K[\\![t]\\!])$ of ${\mathbb{G}}$
(also in ${\mathbb{S}}et$), and the natural congruenz subgroups
${\mathbb{G}}^{i}\subset{\mathbb{H}}$, $i\geq 0$ (see [14, Sec. 2/12]). Note
that ${\mathbb{H}}$ is a group object of ${\operatorname{Pro}}({\bf S}et)$, in
fact, it is given by the projective system $(G(K[t]/(t^{n})))$ of groups in
${\bf S}et$. Note that ${\mathbb{H}}$ is a thick  subgroup of ${\mathbb{G}}$
in the sense of [14, 2.12], and we have a splitting of the central extension
$\hat{{\mathbb{G}}}$ over ${\mathbb{H}}$.

Let ${\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})$ denote the category of
$\hat{{\mathbb{G}}}$-representations in pro-vector spaces at level $c$, where
$c:K^{*}\to\operatorname{\mathbb{C}}^{*}$ is a character (see [14, 2.14]). The
quotient ${\mathbb{G}}/{\mathbb{H}}\in{\operatorname{Ind}}({\bf S}et)$ is ind-
compact, i.e., a direct system of compact objects of ${\bf S}et$ (see [14,
3.4]). In this case there is a natural induction functor

$i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}:{\operatorname{Rep}}({\mathbb{H}})\to{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}}),$

forming an adjoint pair
$(r_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}},i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}})$
with the corresponding restriction functors (see [14, Prop. 3.5]).

We are interested in the induced representation
$i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}})$, where
$\operatorname{\mathbb{C}}$ is the trivial representation of ${\mathbb{H}}$.

######  Definition 3.1.

We define the local Hecke algebra at the level $c$ by

${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}:={\operatorname{End}}_{{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})}(i_{\mathbb{H}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}}))^{op}$

Using the adjunction of the restriction and the induction functors, we can
rewrite this definition as

${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}=\operatorname{Hom}_{{\mathbb{H}}}(i_{\mathbb{H}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}}),\operatorname{\mathbb{C}}).$

More concretely, let us consider the affine Grassmannian
$\operatorname{Gr}_{G}=G(\\!(t)\\!)/G[\\![t]\\!]$ (defined over
$\operatorname{\mathbb{Z}}$), and represent it as the union of finite-
dimensional proper schemes $\overline{\operatorname{Gr}}_{\lambda}$ (where
$\lambda$ runs over dominant coweights). The central extension $\hat{G}\to
G(\\!(t)\\!)$ induces a ${\mathbb{G}}_{m}$-torsor $\hat{G}/G[\\![t]\\!]$ over
$\operatorname{Gr}_{G}$ (equipped with a $G[\\![t]\\!]$-equivariant
structure), which we denote by ${\mathcal{L}}$. Taking the push-out with
respect to the character $c:K^{*}\to\operatorname{\mathbb{C}}^{*}$ we get a
complex line bundle ${\mathcal{L}}_{c}$ on $\operatorname{Gr}_{G}(K)$, and we
can identify $i_{\mathbb{H}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}})$
with the projective limit

$i_{\mathbb{H}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}})=``\varprojlim"\SS(\overline{\operatorname{Gr}}_{\lambda}(K),{\mathcal{L}}_{c}).$

Hence, as a vector space, ${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$
can be identified with the inductive limit,

${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}=\varinjlim\operatorname{Hom}_{{\mathbb{H}}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(K),{\mathcal{L}}_{c}),\operatorname{\mathbb{C}}).$

###  3.2. Endomorphisms of the functor of coinvariants

By [14, Prop. 2.5], we have the functor of coinvariants,

${\operatorname{Coinv}}_{{\mathbb{H}}}:{\operatorname{Rep}}({\mathbb{H}})\to{\mathbb{V}}ect,$

left adjoint to the functor
${\operatorname{triv}}_{{\mathbb{H}}}:{\mathbb{V}}ect\to{\operatorname{Rep}}({\mathbb{H}})$
of the trivial representations. We want to identify the endomorphisms of the
composed functor

${\operatorname{Coinv}}_{{\mathbb{H}}}r^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}:{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})\to{\mathbb{V}}ect$

with the opposite algebra to
${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$.

First, we observe that the functor
${\operatorname{Coinv}}_{{\mathbb{H}}}r^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}$
is left adjoint to the composition

$i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}}:{\mathbb{V}}ect\to{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}}).$

Hence, we have a natural identification of algebras

${\operatorname{End}}({\operatorname{Coinv}}_{{\mathbb{H}}}r^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}})\simeq{\operatorname{End}}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}.$

Now we consider the natural evaluation map

${\operatorname{End}}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}\to{\operatorname{End}}_{{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(\operatorname{\mathbb{C}}))^{op}={\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}.$
(3.1)

######  Proposition 3.2.

The map (3.1) is an isomorphism.

###### Proof.

We will construct a map in the opposite direction. Let us set
$F:=i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}}$.
First, we claim that every
$h\in{\operatorname{End}}_{{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})}(F(\operatorname{\mathbb{C}}))$
induces an endomorphism $h_{V}$ of $F(V)$ for every (usual) vector space $V$
that is uniquely characterized by the property that it is compatible with $h$
via any linear map $\operatorname{\mathbb{C}}\to V$.

To construct $h_{V}$ we choose a basis in $V$, so
$V=\bigoplus_{i}\operatorname{\mathbb{C}}e_{i}$. Let us consider the
corresponding embedding
$V\hookrightarrow\prod_{i}\operatorname{\mathbb{C}}e_{i}$. Since $F$ is a
right adjoint functor, it commutes with products and is left exact, hence, we
get the induced embedding

$F(V)\hookrightarrow\prod_{i}F(\operatorname{\mathbb{C}}e_{i}).$

Now the endomorphism $h$ induces an endomorphism $\prod_{i}h$ of
$\prod_{i}F(\operatorname{\mathbb{C}}e_{i})$. We claim that it preserves the
subobject $F(V)$. The pro-vector space underlying $F(V)$ is given by

$F(V)=``\varprojlim"(\SS_{\lambda}\otimes
V)=``\varprojlim"(\bigoplus_{i}\SS_{\lambda}\otimes\operatorname{\mathbb{C}}e_{i}).$

where we set
$\SS_{\lambda}:=\SS(\overline{\operatorname{Gr}}_{\lambda}(K),{\mathcal{L}}_{c})$.
On the other hand, as a pro-vector space,
$\prod_{i}F(\operatorname{\mathbb{C}}e_{i})$ is given by

$\prod_{i}F(\operatorname{\mathbb{C}}e_{i})=``\varprojlim"(\prod_{i}\SS_{\lambda}\otimes\operatorname{\mathbb{C}}e_{i}).$

Now by definition, we have

$\operatorname{Hom}_{{\mathbb{V}}ect}(``\varprojlim"\SS_{\lambda},``\varprojlim"\SS_{\lambda})=\varprojlim_{\lambda}\varinjlim_{\mu}\operatorname{Hom}(\SS_{\mu},\SS_{\lambda}).$

Thus, $h$ is given by a collection of linear maps
$h_{\lambda}:\SS_{\phi(\lambda)}\to\SS_{\lambda}$, for some function
$\phi:\Lambda\to\Lambda$, that are compatible in the following sense: for any
$\lambda^{\prime}<\lambda$ there exists a sufficiently large $\mu$ such that
the compositions

$\SS_{\mu}\to\SS_{\phi(\lambda)}\rTo{h_{\lambda}}\SS_{\lambda}\to\SS_{\lambda^{\prime}}\
\text{ and}$
$\SS_{\mu}\to\SS_{\phi(\lambda^{\prime})}\rTo{h_{\lambda^{\prime}}}\SS_{\lambda^{\prime}}$

are the same. Now the diagonal endomorphism $\prod_{i}h$ of
$\prod_{i}F(\operatorname{\mathbb{C}}e_{i})$ is given by the collection of
maps

$\prod_{i}h_{\lambda}:\prod_{i}\SS_{\phi(\lambda)}\otimes\operatorname{\mathbb{C}}e_{i}\to\prod_{i}\SS_{\lambda}\otimes\operatorname{\mathbb{C}}e_{i}.$

It is clear that these maps preserve the subspaces obtained by replacing
direct products by direct sums. The corresponding endomorphism of the pro-
vector space underlying $F(V)$ is given by

$h_{\lambda}\otimes\operatorname{id}:\SS_{\phi(\lambda)}\otimes
V\to\SS_{\lambda}\otimes V.$

Our argument shows that it is in fact compatible with the
$\hat{{\mathbb{G}}}$-action.

It is easy to check that the constructed $h_{V}$ is compatible with arbitrary
linear maps $V^{\prime}\to V$, and in particular, is uniquely determined.
Since $F$ commutes with projective limits, we can now construct
$h_{V}\in{\operatorname{End}}(F(V))$ for any pro-vector space $V$. One also
easily checks that the obtained map is inverse to (3.1). ∎

Next, we will give an explicit formula for the action of the Hecke algebra on
coinvariants of some $\hat{{\mathbb{G}}}$-representations. Recall that
$V\in{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})$ is called admissible if for
every congruenz-subgroup ${\mathbb{G}}^{i}\subset{\mathbb{H}}$, the pro-vector
space of coinvariants $V_{{\mathbb{G}}^{i}}$ is a usual vector space. The
action map $(g,v)\mapsto g^{-1}v$ of $\hat{{\mathbb{G}}}$ induces a collection
of weight-$c$ maps

$\alpha_{\lambda}:\operatorname{tot}_{\operatorname{Gr}_{\lambda}}({\mathcal{L}})\times
V_{{\mathbb{G}}_{i(\lambda)}}\to V_{{\mathbb{H}}},$ (3.2)

for some function $i(\lambda)$, where
$\operatorname{tot}_{\operatorname{Gr}_{\lambda}}({\mathcal{L}})$ is the total
space of the ${\mathbb{G}}_{m}$-bundle corresponding to ${\mathcal{L}}$.

######  Lemma 3.3.

For any admissible $V\in{\operatorname{Rep}}_{c}({\mathbb{G}})$, consider the
unit for the adjoint pair
$({\operatorname{Coinv}}_{\mathbb{H}}r^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}},i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})$,

$u_{V}:V\to
i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(V_{{\mathbb{H}}})\simeq``\varprojlim"\SS(\overline{\operatorname{Gr}}_{\lambda},{\mathcal{L}}_{c})\otimes
V_{{\mathbb{H}}}.$

Then for any $h\in{\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{c}$ and any
admissible $V\in{\operatorname{Rep}}_{c}({\mathbb{G}})$, the corresponding
endomorphism $h_{V}\in{\operatorname{End}}(V_{{\mathbb{H}}})$ is determined
from the commutative diagram where
$h_{0}\in\operatorname{Hom}_{{\mathbb{H}}}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(\operatorname{\mathbb{C}}),\operatorname{\mathbb{C}})$
corresponds to $h$.

More explicitly, if $h$ comes from
$h_{0}\in\operatorname{Hom}_{{\mathbb{H}}}(\SS(\overline{\operatorname{Gr}}_{\lambda},{\mathcal{L}}_{c}),\operatorname{\mathbb{C}})$,
then $h_{V}$ is induced by the composition

$V_{{\mathbb{G}}_{i(\lambda)}}\rTo{\alpha_{\lambda}^{*}}\SS(\overline{\operatorname{Gr}}_{\lambda},{\mathcal{L}}_{c})\otimes
V_{{\mathbb{H}}}\rTo{h_{0}\otimes\operatorname{id}}V_{H}.$

###### Proof.

Recall that $h$, viewed as an endomorphism of
$i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(\operatorname{\mathbb{C}})$ in
${\operatorname{Rep}}_{c}({\mathbb{G}})$, can be represented by a collection
of linear maps $h_{\lambda}:\SS_{\phi(\lambda)}\to\SS_{\lambda}$, where
$\SS_{\lambda}=\SS(\overline{\operatorname{Gr}}_{\lambda},{\mathcal{L}}_{c})$,
and for any $W\in Vect$, the corresponding endomorphism $h_{W}$ of
$i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(W)$ is given by the maps
$(h_{\lambda}\otimes\operatorname{id}_{W})$.

Recall that we use the isomorphism
${\operatorname{End}}({\operatorname{Coinv}}_{{\mathbb{H}}}r^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}})\simeq{\operatorname{End}}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}$
to define the action of ${\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{c}$ on
${\operatorname{Coinv}}_{{\mathbb{H}}}$. Hence, for any
$V\in{\operatorname{Rep}}_{c}({\mathbb{G}})$, the endomorphism
$h_{V}:V_{{\mathbb{H}}}\to V_{{\mathbb{H}}}$ corresponds by adjunction to the
composed map

$V\rTo{u_{V}}i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(V_{{\mathbb{H}}})\rTo{h_{V_{{\mathbb{H}}}}}i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(V_{{\mathbb{H}}}).$

This means that $h_{V}$ is the composition

$V_{{\mathbb{H}}}\rTo{{\operatorname{Coinv}}_{\mathbb{H}}(u_{V})}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(V_{{\mathbb{H}}}))_{\mathbb{H}}\rTo{(h_{V_{{\mathbb{H}}}})_{\mathbb{H}}}i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(V_{{\mathbb{H}}})_{\mathbb{H}}\rTo{\epsilon_{V_{{\mathbb{H}}}}}V_{{\mathbb{H}}},$

where $\epsilon_{W}$ is the counit of adjunction.

It is easy to see that for any $W\in Vect$, the composition

$i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(W)\to(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(W))_{\mathbb{H}}\rTo{(h_{W})_{\mathbb{H}}}i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}(W)_{\mathbb{H}}\rTo{\epsilon_{W}}W$

is given by $h_{0}\otimes\operatorname{id}_{W}$. Now the assertion follows
immediately from the commutative diagram ∎

###  3.3. Critical level and integration

Assuming that $[G,G]$ is simply connected, and let us consider the central
extension $\hat{G}_{crit}\to G(\\!(t)\\!)$ at the critical level. It
corresponds to the invariant form $-B/2$ on ${\mathfrak{g}}$, where $B$ is the
Killing form, and is characterized by the fact that the corresponding
$G[\\![t]\\!]$-equivariant line bundle ${\mathcal{L}}_{crit}$ on the affine
Grassmannian $\operatorname{Gr}_{G}$ satisfies

${\mathcal{L}}_{crit}|_{\operatorname{Gr}_{\lambda}}\simeq\omega_{\operatorname{Gr}_{\lambda}}\otimes
L_{\lambda},$ (3.3)

where $L_{\lambda}$ is a $1$-dimensional vector space depending
multiplicatively on $\lambda$ (see [2], [4, Thm. 2.4], [5, Thm. 5.1]). By
definition, the action of $G(\\!(t)\\!)$ lifts to an action of
$\hat{G}_{crit}$ on ${\mathcal{L}}_{crit}$ such that ${\mathbb{G}}_{m}$ acts
by the identity character.

Recall that the variety $\overline{\operatorname{Gr}}_{\lambda}$ is Gorenstein
and has rational singularities (see [12], [4, Thm. 2.2]). This implies that
the isomorphism (3.3) extends to a similar isomorphism on
$\overline{\operatorname{Gr}}_{\lambda}$ (see [4, Thm. 2.5]).

We denote by
${\operatorname{Rep}}_{crit}({\mathbb{G}}):={\operatorname{Rep}}_{|\cdot|}(\hat{{\mathbb{G}}}_{crit})$
the category of representations of the central extension
$\hat{{\mathbb{G}}}=\hat{G}_{crit}(K)$ in pro-vector spaces such that $K^{*}$
acts via $|\cdot|:K^{*}\to\operatorname{\mathbb{C}}^{*}$, and by
${\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{crit}:={\mathcal{H}}(\hat{G}_{crit},{\mathbb{H}})_{|\cdot|}$
the corresponding local Hecke algebra.

######  Definition 3.4.

For $\lambda\in\Lambda_{+}$, we denote by

$h^{\lambda}\in\operatorname{Hom}_{{\mathbb{H}}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(K),|{\mathcal{L}}_{crit}|),\operatorname{\mathbb{C}})\otimes
L_{\lambda}\subset{\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{crit}\otimes
L_{\lambda}$

the element given by the absolutely convergent integral

$\mu\mapsto\int_{\operatorname{Gr}_{\lambda}(K)}\mu,$

where we view elements of
$\SS(\operatorname{Gr}_{\lambda}(K),|{\mathcal{L}}_{crit}|)$ as smooth
$L_{\lambda}$-valued measures on $\operatorname{Gr}_{\lambda}(K)$.

The absolute convergence in the above definition follows from the fact that
$\overline{\operatorname{Gr}}_{\lambda}$ has rational singularities (see [1,
Sec. 3.4]). The next theorem is proved in [4].

######  Theorem 3.5.

The elements $(h^{\lambda})$ commute in
${\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{crit}$.

###  3.4. Local Hecke algebras over $O$ and $O/{\mathfrak{m}}^{N}$

The analogs of the constructions and results of Sections 3.1 and 3.2 also hold
if we replace $K$ with $O$ or with $O/{\mathfrak{m}}^{N}$.

Namely, we consider the group ${\mathbb{G}}_{O}:=G(O(\\!(t)\\!))$ in
${\mathbb{S}}et$ and its subgroup ${\mathbb{H}}_{O}:=G(O[\\![t]\\!])$, as well
as for the group ${\mathbb{G}}_{N}:=G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$ with
the subgroup ${\mathbb{H}}_{N}:=G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$. (Note
that we do not consider central extensions of ${\mathbb{G}}_{O}$ and $G_{N}$,
just the usual categories of representations.)

Similarly to Definition 3.1, we define local Hecke algebras

${\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O}):={\operatorname{End}}_{{\operatorname{Rep}}({\mathbb{G}}_{O})}(i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}(\operatorname{\mathbb{C}}))^{op},\
\
{\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N}):={\operatorname{End}}_{{\operatorname{Rep}}({\mathbb{G}}_{N})}(i_{{\mathbb{H}}_{N}}^{{\mathbb{G}}_{N}}(\operatorname{\mathbb{C}}))^{op}.$

We still have the identifications

${\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})=\operatorname{Hom}_{{\mathbb{H}}_{O}}(i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}(\operatorname{\mathbb{C}}),\operatorname{\mathbb{C}})=\varinjlim\operatorname{Hom}_{{\mathbb{H}}_{O}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(O)),\operatorname{\mathbb{C}}),$
${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})=\operatorname{Hom}_{{\mathbb{H}}_{N}}(i_{{\mathbb{H}}_{N}}^{{\mathbb{G}}_{N}}(\operatorname{\mathbb{C}}),\operatorname{\mathbb{C}})=\varinjlim\operatorname{Hom}_{{\mathbb{H}}_{N}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})),\operatorname{\mathbb{C}}).$

Next, we will show how to identify
${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$ with the algebra
${\mathcal{H}}_{G,O/{\mathfrak{m}}^{N}}$ of
$G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$-biinvariant distributions with compact
support on $G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$. Indeed, the group
$G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$ acts transitively on the fibers of the
map
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))\to\operatorname{Gr}(O/{\mathfrak{m}}^{N})$.
Hence, the orbits of $G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$ on
$\operatorname{Gr}(O/{\mathfrak{m}}^{N})$ are in bijection with the double
$G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$-cosets on
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$. Since the former orbits form a basis in
${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$ and the latter double
cosets form a basis in ${\mathcal{H}}_{G,O/{\mathfrak{m}}^{N}}$, we get an
identification

$\nu:{\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})\rTo{\sim}{\mathcal{H}}_{G,O/{\mathfrak{m}}^{N}}.$

More explicitly, given a $G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$-invariant
functional $\delta$ on
$\SS(\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N}))$, we
construct a $G(O/{\mathfrak{m}}^{N}[\\![t]\\!])$-biinvariant distribution on
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$ as follows: starting with a smooth
function on $G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))$ we restrict it to
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))_{\leq\lambda}$, the preimage of
$\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})$, then integrate
along the fibers of the map
$G(O/{\mathfrak{m}}^{N}(\\!(t)\\!))_{\leq\lambda}\to\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})$,
and finally apply $\delta$ to the resulting function.

######  Proposition 3.6.

The map $\nu$ is an isomorphism of algebras.

###### Proof.

To understand multiplication on
${\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$ let us look again at the
isomorphism

${\operatorname{End}}_{{\operatorname{Rep}}({\mathbb{G}}_{N})}(i_{{\mathbb{H}}_{N}}^{{\mathbb{G}}_{N}}(\operatorname{\mathbb{C}}))\rTo{\sim}\varinjlim\operatorname{Hom}_{{\mathbb{H}}_{N}}(\SS_{\lambda,N},\operatorname{\mathbb{C}}),$

where
$\SS_{\lambda,N}=\SS(\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N}))$
(these are finite dimensional ${\mathbb{H}}_{N}$-representations). We have
$i_{{\mathbb{H}}_{N}}^{{\mathbb{G}}_{N}}(\operatorname{\mathbb{C}})=``\varprojlim"\SS_{\lambda,N}$,
and an endomorphism $h$ of this pro-vector space is given by a compatible
collection

$h_{\lambda}:\SS_{\phi(\lambda),N}\to\SS_{\lambda,N}.$

The fact that $h$ commutes with the ${\mathbb{G}}_{N}$-action allows to
recover $(h_{\lambda})$ from
$h_{0}:\SS_{\phi(0),N}\to\operatorname{\mathbb{C}}$. Namely, for each
$\lambda,\mu$, the action map $(g,\phi)\mapsto(g^{-1}\phi)(x)=\phi(gx)$,
induces a map

$\alpha:\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})\times\SS_{\psi(\lambda,\mu),N}\to(\SS_{\mu,N})_{{\mathbb{H}}_{N}},$

Hence, for each $\lambda$, the composition

$\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})\times\SS_{\psi(\lambda,\phi(0)),N}\to(\SS_{\phi(0),N})_{{\mathbb{H}}_{N}}\rTo{h_{0}}\operatorname{\mathbb{C}}$

can be viewed as a map

$\SS_{\psi(\lambda,\phi(0)),N}\to\SS_{\lambda}$

which induces $h_{\lambda}$.

Now, for $h,h^{\prime}\in{\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N})$,
the map $(h^{\prime}h)_{0}$ is given as the composition

$\SS_{\lambda^{\prime}}\rTo{h_{\lambda}}\SS_{\lambda}\rTo{h^{\prime}_{0}}\operatorname{\mathbb{C}}.$

As we have seen above, $h_{\lambda}(\phi)(g)=h_{0}(g^{-1}\phi)$, where
$g^{-1}\phi(x)=\phi(gx)$. Thus, $(h^{\prime}h)_{0}(\phi)$ is obtained by
applying $h^{\prime}_{0}\otimes h_{0}$ to
$\phi(gx)\in\SS({\mathbb{G}}_{N,\lambda^{\prime}})\otimes\SS_{\lambda,N}$,
where ${\mathbb{G}}_{N,\lambda^{\prime}}\subset{\mathbb{G}}_{N}$ is the
preimage of
$\overline{\operatorname{Gr}}_{\lambda^{\prime}}(O/{\mathfrak{m}}^{N})$. The
latter definition is compatible with the usual convolution of distributions on
${\mathbb{G}}_{N}$. ∎

###  3.5. Homomorphisms between local Hecke algebras

As in Sec. 3.1 and 3.2, we continue to work with any central extension
$\hat{G}$ and a level $c:K^{*}\to\operatorname{\mathbb{C}}^{*}$.

First, we want to construct a natural homomorphism of algebras

$\nu_{K,O}:{\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}\to{\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O}).$

######  Lemma 3.7.

One has a natural isomorphism of functors
${\operatorname{Rep}}({\mathbb{H}})\to{\operatorname{Rep}}({\mathbb{G}}_{O})$,

$r^{\hat{{\mathbb{G}}}}_{{\mathbb{G}}_{O}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}\rTo{\sim}i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{H}}_{O}}^{\mathbb{H}}.$
(3.4)

Furthermore, the following diagram of functors
${\operatorname{Rep}}({\mathbb{H}})\to{\operatorname{Rep}}({\mathbb{H}}_{O})$
is commutative

$\begin{diagram}$ (3.5)

where the left vertical arrow is induced by (3.4).

###### Proof.

The isomorphism (3.4) is induced by the identifications

$\overline{\operatorname{Gr}}_{\lambda}(O)\rTo{\sim}\overline{\operatorname{Gr}}_{\lambda}(K)$

and by the trivialization of ${\mathcal{L}}_{c}$ over $\operatorname{Gr}(O)$.
The commutativity of the diagram follows from the fact that the adjunction
maps
$r_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}(V)\to
V$ and
$r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}(W)\to
W$ are both given by the evaluation at $1$. ∎

Now we define the homomorphism $\nu_{K,O}$ as the composition

$\displaystyle{\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}\simeq{\operatorname{End}}(i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}\to{\operatorname{End}}(r^{\hat{{\mathbb{G}}}}_{{\mathbb{G}}_{O}}i^{\hat{{\mathbb{G}}}}_{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}\simeq{\operatorname{End}}(i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}$
$\displaystyle\simeq{\operatorname{End}}(i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}})^{op}\simeq{\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O}),$

where we use the isomorphism (3.4), as well as the isomorphism (3.1) and a
similar isomorphism for ${\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})$.

Note that the isomorphism of ${\mathbb{G}}_{O}$-representations
$r^{\hat{{\mathbb{G}}}}_{{\mathbb{G}}_{O}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}(\operatorname{\mathbb{C}})\rTo{\sim}i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}(\operatorname{\mathbb{C}})$
comes from the identifications
$\SS(\overline{\operatorname{Gr}}_{\lambda}(K),|{\mathcal{L}}_{c}|)\simeq\SS(\overline{\operatorname{Gr}}_{\lambda}(O))$,
and $\nu_{K,O}$ is simply the induced map

$\varinjlim\operatorname{Hom}_{{\mathbb{H}}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(K),|{\mathcal{L}}_{c}|),\operatorname{\mathbb{C}})\to\varinjlim\operatorname{Hom}_{{\mathbb{H}}_{O}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(O)),\operatorname{\mathbb{C}}).$

The homomorphism $\nu_{K,O}$ is compatible with the action on the functors of
coinvariants as follows. Note that we have a natural morphism of functors from
${\operatorname{Rep}}({\mathbb{H}})$ to ${\mathbb{V}}ect$,

${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}\to{\operatorname{Coinv}}_{{\mathbb{H}}}$
(3.6)

obtained using adjunction from the natural isomorphism
$r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}}\rTo{\sim}{\operatorname{triv}}_{{\mathbb{H}}_{O}}$.
Namely, the map

$\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}}V,W)\to\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}},W)$

corresponding to (3.6) is given as the composition

$\displaystyle\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}}V,W)\simeq\operatorname{Hom}(V,{\operatorname{triv}}_{{\mathbb{H}}}W)\rTo{r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}}\operatorname{Hom}(r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}V,r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}}W)\simeq$
$\displaystyle\operatorname{Hom}(r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}V,{\operatorname{triv}}_{{\mathbb{H}}_{O}}W)\simeq\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}V,W).$

######  Lemma 3.8.

Suppose we have $V\in{\operatorname{Rep}}({\mathbb{G}})$,
$V_{O}\in{\operatorname{Rep}}({\mathbb{G}}_{O})$, and a morphism $f:V_{O}\to
V$ compatible with the action of ${\mathbb{G}}_{O}$. Then for any
$h\in{\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$, the action of $h$ on
${\operatorname{Coinv}}_{{\mathbb{H}}}(V)$ is compatible with the action of
$\nu_{K,O}(h)$ on ${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}(V_{O})$, i.e.,
the following diagram is commutative where the vertical maps are induced by
$f$ and by the morphism (3.6).

###### Proof.

Step 1. First, we claim that for any map
$f:{\operatorname{Coinv}}_{{\mathbb{H}}}(V)\to W$, where
$V\in{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})$, the morphism
$f^{\prime}:V\to
i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}}W$ in
${\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})$ corresponding to $f$ by
adjunction, viewed as a morphism in ${\operatorname{Rep}}({\mathbb{G}}_{O})$,
coincides with the morphism obtained by adjunction from the composed morphism

${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}(V)\to{\operatorname{Coinv}}_{{\mathbb{H}}}(V)\to
W,$

where the first arrow is given by the canonical morphism (3.6). Indeed, this
is equivalent to checking the commutativity of the diagram of functors

$\begin{diagram}$ (3.7)

where $\gamma$ is induced by the isomorphism (3.4), $\beta$ is induced by
(3.6), and the remaining two arrows are counits of adjunction. We claim that
this follows from the commutativity of the diagram (3.5). Indeed,
$\operatorname{can}\circ\gamma$ is equal to the composition

${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}}\rTo{\gamma}{\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}}\to{\operatorname{Coinv}}_{{\mathbb{H}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}}\to\operatorname{Id}_{{\mathbb{V}}ect}.$

The composition of the first two arrows is induced (by post-composing with
${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}$ and pre-composing with
${\operatorname{triv}}_{{\mathbb{H}}}$) by the map

$r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}\to
r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}\to
r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}},$

which by commutativity of (3.5), is equal to the map

$r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}\simeq
r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}r_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}\to
r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}$

induced by the adjunction. Hence, $\operatorname{can}\circ\gamma$ is equal to
the composition

${\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}}\simeq{\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}r_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}}\to{\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{H}}}{\operatorname{triv}}_{{\mathbb{H}}}\simeq{\operatorname{Coinv}}_{{\mathbb{H}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}}\to\operatorname{id}_{{\mathbb{V}}ect},$

induced by adjunctions. Now the fact that this is equal to
$\operatorname{can}\circ\beta$ follows from the commutative diagram in which
commutativity of the right square follows from the definition of the morphism
(3.4).

Step 2. Step 1 implies that for any
$V\in{\operatorname{Rep}}_{c}(\hat{{\mathbb{G}}})$ and $W\in{\mathbb{V}}ect$,
the following square (in which the horizontal arrows are isomorphisms) is
commutative

$\begin{diagram}$ (3.8)

where $\alpha$ is given by the adjunction, $\beta$ is induced by (3.6), and
the bottom horizontal arrow is the composition of the map

$\alpha_{O}:\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}_{O}}(V),W)\rTo{\sim}\operatorname{Hom}_{{\mathbb{G}}_{O}}(V,i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}}W)$

given the adjunction with the map induced by the isomorphism
$\gamma:i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}}\rTo{\sim}r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}}$
(see (3.4)).

For
$h\in{\mathcal{H}}_{{\mathbb{G}},{\mathbb{H}}}={\operatorname{End}}(i_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}}{\operatorname{triv}}_{{\mathbb{H}}})^{op}$,
let $h^{\prime}$ be the corresponding element of
${\operatorname{End}}({\operatorname{Coinv}}_{{\mathbb{H}}}r_{{\mathbb{H}}}^{\hat{{\mathbb{G}}}})$.
By definition, for any
$f\in\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}}(V),W)$, we have

$h_{W}\circ\alpha(f)=\alpha(f\circ h^{\prime}_{V}).$ (3.9)

Similarly, for
$h_{O}\in{\mathcal{H}}_{{\mathbb{G}}_{O},{\mathbb{H}}_{O}}={\operatorname{End}}(i_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}}{\operatorname{triv}}_{{\mathbb{H}}_{O}})^{op}$
and the corresponding element
$h^{\prime}_{O}\in{\operatorname{End}}({\operatorname{Coinv}}_{{\mathbb{H}}_{O}}r_{{\mathbb{H}}_{O}}^{{\mathbb{G}}_{O}})$,
we have

$h_{O,W}\circ\alpha_{O}(f_{O})=\alpha_{O}(f_{O}\circ h^{\prime}_{O,V_{O}}),$
(3.10)

where
$f_{O}\in\operatorname{Hom}({\operatorname{Coinv}}_{{\mathbb{H}}_{O}}(V_{O}),W)$,
with $V_{O}\in{\operatorname{Rep}}({\mathbb{G}}_{O})$.

Now let $h_{O}=\nu_{K,O}(h)$. Note that we have an endomorphism of every
vertex of the square (3.8): for the two right vertices they are induced by
$h_{W}$ and $r^{\hat{{\mathbb{G}}}}_{{\mathbb{G}}_{O}}(h_{W})$, while for the
two left vertices they are induced by $h^{\prime}$ and $h^{\prime}_{O}$. We
want to check that the map $\beta$ is compatible with these endomorphism,
i.e.,

$\beta(f\circ h^{\prime}_{V})=\beta(f)\circ h^{\prime}_{O,V_{O}}.$

It is enough to check instead that three other edges in the square are
compatible with the endomorphisms. For the arrow
$r^{\hat{{\mathbb{G}}}}_{{\mathbb{G}}_{O}}$, this is clear. For $\alpha$ and
$\alpha_{O}$ the compatibility is given by (3.9) and (3.10). Finally, for
$\gamma$ this corresponds to the identity

$\gamma_{W}h_{O,W}=r_{{\mathbb{G}}_{O}}^{{\mathbb{G}}}(h_{W})\gamma_{W}$

which follows from the definition of $\nu_{K,O}$: ∎

Next, we will construct a homomorphism

$\nu_{O,O/{\mathfrak{m}}^{N}}:{\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})\to{\mathcal{H}}({\mathbb{G}}_{N},{\mathbb{H}}_{N}).$

As a map of vector spaces it is induced by the natural maps

$\operatorname{Hom}_{{\mathbb{H}}_{O}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(O)),\operatorname{\mathbb{C}})\to\operatorname{Hom}_{{\mathbb{H}}_{N}}(\SS(\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N})),\operatorname{\mathbb{C}})$

dual to the pull-back maps
$\SS(\overline{\operatorname{Gr}}_{\lambda}(O/{\mathfrak{m}}^{N}))\to\SS(\overline{\operatorname{Gr}}_{\lambda}(O))$.
We claim that it is a homomorphism of algebras.

For this, we use the interpretation of both algebras as endomorphisms of the
functor of coinvariants. Namely, given an element
$h\in{\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})$ it induces an
endomorphism $h_{V}\in{\operatorname{End}}(V_{{\mathbb{H}}_{O}})$ for any
$V\in{\mathbb{G}}_{O}$. Now given a representation $V_{N}$ of
${\mathbb{G}}_{N}$, we can view it as a representation of ${\mathbb{G}}_{O}$
via the surjective homomorphism ${\mathbb{G}}_{O}\to{\mathbb{G}}_{N}$. Then we
observe that $(V_{N})_{{\mathbb{H}}_{O}}=(V_{N})_{{\mathbb{H}}_{N}}$ since
${\mathbb{H}}_{O}$ surjects onto ${\mathbb{H}}_{N}$. Hence, we can view
$h_{V_{N}}$ as an endomorphism of $(V_{N})_{{\mathbb{H}}_{N}}$. Clearly, this
construction is compatible with the algebra structure. The fact that it
coincides with the map $\nu_{O,O/{\mathfrak{m}}^{N}}$ follows easily from the
analogs of Lemma 3.3 for ${\mathbb{G}}_{O}$\- and
${\mathbb{G}}_{N}$-representations.

###  3.6. Coordinate-free Hecke algebras and the small Hecke algebra over
$O/{\mathfrak{m}}^{N}$

All the notions related to the groups $G(\\!(t)\\!)$, $G[\\![t]\\!]$ over a
local field $K$, including the Hecke algebras
${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$ and their actions on the
space of ${\mathbb{H}}$-coinvariants, can be formulated starting with a local
complete $K$-algebra $A_{K}$, isomorphic to $K[\\![t]\\!]$ (then the field of
quotients of $A_{K}$ will be isomorphic to $K(\\!(t)\\!)$). We denote the
corresponding Hecke algebra as ${\mathcal{H}}_{\hat{G},A_{K},c}$ (which is
isomorphic to ${\mathcal{H}}(\hat{{\mathbb{G}}},{\mathbb{H}})_{c}$). For
example, we can take $A_{K}=\hat{{\mathcal{O}}}_{C,v}$, the completion of a
local ring of a $K$-point on smooth curve over $K$.

The situation is slightly different with the $O$-integral version. Here, we
need to start with an $O$-algebra $A_{O}$, together with a principal ideal
$I\subset A_{O}$ such that $A_{O}/I\simeq O$ and $A_{O}$ is $I$-adically
complete, such that there exists an isomorphism $A_{O}\simeq O[\\![t]\\!]$
sending $I$ to $tO[\\![t]\\!]$. Then since $I$ is free of rank $1$ as an
$A_{O}$-module, we can define powers $I^{-n}$ for $n>0$, and consider the ring
$\varinjlim I^{-n}$ as a replacement of $O(\\!(t)\\!)$ (to which it is
isomorphic). This allows to define the corresponding Hecke algebra
${\mathcal{H}}_{G,A_{O},I}$ which is isomorphic to
${\mathcal{H}}({\mathbb{G}}_{O},{\mathbb{H}}_{O})$. For example, we can take
$A_{O}=\hat{{\mathcal{O}}}_{C_{O},v_{O}}$, the completion of a smooth curve
$C_{O}$ over $O$ along an $O$-point
$v_{O}:\operatorname{Spec}(O)\hookrightarrow C_{O}$, where the ideal $I\subset
A_{O}$ is the ideal of $v_{O}(\operatorname{Spec}(O))$.

Finally, for the $O/{\mathfrak{m}}^{N}$-version, we start with a local
complete $O/{\mathfrak{m}}^{N}$-algebra $A_{O/{\mathfrak{m}}^{N}}$, isomorphic
to $O/{\mathfrak{m}}^{N}[\\![t]\\!]$. Note that for such an algebra we can
consider the complete ring of quotients $QA_{O/{\mathfrak{m}}^{N}}$ (by
inverting every non-zero divisor), which is easily seen to be isomorphic to
$O/{\mathfrak{m}}^{N}(\\!(t)\\!)$. Then we can define the corresponding Hecke
algebra ${\mathcal{H}}_{G,A_{O/{\mathfrak{m}}^{N}}}$. For example, if $C_{N}$
is a smooth curve over $O/{\mathfrak{m}}^{N}$ and
$\overline{v}\in\overline{C}(k)$ is a $k$-point of the reduction
$\overline{C}=(C_{N})_{k}$, then we can take
$A_{O/{\mathfrak{m}}^{N}}=\hat{{\mathcal{O}}}_{C_{N},\overline{v}}$. Note that
if $v_{N}\in C_{N}(O/{\mathfrak{m}}^{N})$ is a lifting of $\overline{v}$ then
the completion of $C_{O/{\mathfrak{m}}^{N}}$ along $v_{N}$ gives the same
algebra.

Now all the constructions of the previous sections can be adjusted so that
they would work with above definitions. For example, starting with the data
$(A_{O},I)$ as above, we can define $A_{K}$ as the completion of $K\otimes
A_{O}$ and set
$A_{O/{\mathfrak{m}}^{N}}=A_{O}\otimes_{O}O/{\mathfrak{m}}^{N}$. Then we will
have natural homomorphisms

$\nu_{K,O/{\mathfrak{m}}^{N}}:{\mathcal{H}}_{G,A_{K},crit}\rTo{\nu_{K,O}}{\mathcal{H}}_{G,A_{O},I}\rTo{\nu_{O,O/{\mathfrak{m}}^{N}}}{\mathcal{H}}_{G,A_{O/{\mathfrak{m}}^{N}}},$
(3.11)

where
${\mathcal{H}}_{G,A_{K},crit}:={\mathcal{H}}_{\hat{G}_{crit},A_{K},|\cdot|}$,
which are compatible with the action on spaces of coinvariants.

######  Definition 3.9.

Let $A_{O/{\mathfrak{m}}^{N}}$ be an $O/{\mathfrak{m}}^{N}$-algebra as above.
We define the small Hecke algebra
${\mathcal{H}}^{sm}_{G,A_{O/{\mathfrak{m}}^{N}}}$ as the subalgebra of
${\mathcal{H}}_{G,A_{O/{\mathfrak{m}}^{N}}}$ generated by the elements
$\nu_{K,O/{\mathfrak{m}}^{N}}(h^{\lambda})$, where $\lambda\in\Lambda^{+}$,
and we consider all pairs $(A_{O},I)$ as above equipped with an isomorphism
$A_{O}\otimes_{O}O/{\mathfrak{m}}^{N}\simeq A_{O/{\mathfrak{m}}^{N}}$, and use
the corresponding homomorphism (3.11).

######  Theorem 3.10.

Assume that

  * •

$G$ is a split reductive group over $\operatorname{\mathbb{Z}}$, such that
$[G,G]$ is simple and simply connected;

  * •

the characteristic $p$ of $k=O/{\mathfrak{m}}$ satisfies
$p>\max(a_{{\mathfrak{g}}^{\prime}},4(h_{{\mathfrak{g}}^{\prime}}-1))$, where
${\mathfrak{g}}^{\prime}=[{\mathfrak{g}},{\mathfrak{g}}]$.

Then the small Hecke algebra
${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$ is commutative.

We will give a proof of this theorem in Section 5.3 using the action of the
Hecke algebras on the spaces associated with ${\operatorname{Bun}}_{G}$. A
purely local proof of commutation of some of the elements in
${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$ for
$G=\operatorname{GL}_{2}$ was given in [6].

##  4\. Some results on $G$-bundles

In this section we prove some auxiliary results about $G$-bundles on curves.
First, we recall some facts on principal $\mathfrak{s}l_{2}$ subalgebra in
Sec. 4.1, working with a group $G$ over any field $k$ whose characteristic is
either zero or sufficiently large. Then in Sec. 4.2, still working over any
such field, we introduce and study nice $G$-bundles on smooth proper curves.
In Sec. 4.3 we specialize to the case of a finite field and give a
construction of such $G$-bundles, while in Sec. 4.4 we consider $G$-bundles
over $O/{\mathfrak{m}}^{N}$ whose reduction over $k=O/{\mathfrak{m}}$ is nice.
Finally in Sec. 4.5 we consider curves over $O$ and $K$ and discuss generic
triviality of $G$-bundles on them.

###  4.1. Principal $\mathfrak{s}l_{2}$

Let $G$ be a connected split simple group over a field $k$, and let
${\mathfrak{g}}$ denote its Lie algebra. Let $p$ be the characteristic of $k$.
We will recall here some results about the principal $\mathfrak{s}l_{2}$
subalgebra in the case $p$ is either zero or sufficiently large (depending on
the type of ${\mathfrak{g}}$).

Let $r$ be the rank, $h$ the Coxeter number of ${\mathfrak{g}}$, and let us
define positive integers $(a_{i})_{i=1,\ldots,r}$ by

$2\rho^{\vee}=\sum_{\alpha\in R_{+}}\alpha^{\vee}=\sum
a_{i}\alpha_{i}^{\vee},$

where $R_{+}$ is the set of positive roots, $(\alpha^{\vee})$ are coroots,
$(\alpha_{i}^{\vee})$ are simple coroots. We set

$a_{{\mathfrak{g}}}:=\max_{i}a_{i}.$

For example, for type $D_{\ell}$ we have
$a_{{\mathfrak{g}}}=(\ell-2)(\ell+1)$.

We use the standard basis $(H_{i}=H_{\alpha_{i}})_{1\leq i\leq
r},(X_{\pm\alpha})_{\alpha\in R_{+}}$ of ${\mathfrak{g}}$.

######  Lemma 4.1.

Assume $p=0$ or $p>\max(a_{{\mathfrak{g}}},4(h-1))$.

(i) Set

$h^{0}=\sum_{\alpha\in R_{+}}H_{\alpha}=\sum_{i=1}^{r}a_{i}H_{i},\ \
e=\sum_{i=1}^{r}X_{\alpha_{i}},\ \ f=\sum_{i=1}^{r}a_{i}X_{-\alpha_{i}}.$

Then $(e,h^{0},f)$ is a principal $\mathfrak{s}l_{2}$-triple, i.e., $e$ and
$h^{0}$ are regular, and we have a decomposition

${\mathfrak{g}}=\bigoplus_{i=1}^{r}V_{2d_{i}}$ (4.1)

of $\mathfrak{s}l_{2}$-modules, where $d_{1},\ldots,d_{r}$ are the exponents
of ${\mathfrak{g}}$, and $V_{m}$ is the $m$th symmetric power of the standard
representation of $\mathfrak{s}l_{2}$.

(ii) There exists a homomorphism (defined over $k$)
$i:{\operatorname{SL}}_{2}\to G$ inducing the above
$\mathfrak{s}l_{2}$-triple.

###### Proof.

(i) The fact that $(e,h^{0},f)$ is an $\mathfrak{s}l_{2}$-triple is [7, Sec.
7.2, Lem. 2]. It is well known that the element $e$ is regular (see e.g., [8,
Prop. 5.1.3]).

For each $\alpha\in R_{+}$, the element $\operatorname{ad}(h^{0})$ acts on
${\mathfrak{g}}_{\pm\alpha}$ by $\pm\langle\alpha,2\rho^{\vee}\rangle$. Note
that if $\alpha=\sum n_{i}\alpha_{i}$ then

$\langle\alpha,2\rho^{\vee}\rangle=2(\sum n_{i})$

which is positive integer $\leq 2(h-1)$. In particular, all these numbers are
nonzero in $k$, which implies that $h^{0}$ is regular.

We have $(\operatorname{ad}e)^{2h-1}=(\operatorname{ad}f)^{2h-1}=0$. By [8,
Thm. 5.4.8], this implies that ${\mathfrak{g}}$ is a direct sum of standard
irreducible $\mathfrak{s}l_{2}$-modules $V_{i}$, with $i\leq 2h-2$. Now we
observe that since $p>4(h-1)$, the eigenvalues of $\operatorname{ad}(h^{0})$
on ${\mathfrak{g}}_{\pm\alpha}$ are all distinct (and distinct from $0$).
Hence, the character of ${\mathfrak{g}}$ as $\mathfrak{s}l_{2}$-module is
determined by the numbers of $\alpha=\sum n_{i}\alpha_{i}\in R_{+}$ with given
$\sum n_{i}$, so this character is the same as in characteristic zero. Thus
the decomposition (4.1) follows from the same decomposition in characteristic
zero.

(ii) As is explained in [8, Prop. 5.5.5] the adjoint action of
$\mathfrak{s}l_{2}$ on ${\mathfrak{g}}$ integrates to an action of the group
${\operatorname{SL}}_{2}$ by automorphisms of ${\mathfrak{g}}$. Hence, we get
a homomorphism
${\operatorname{SL}}_{2}\to\operatorname{Aut}({\mathfrak{g}})^{0}$. Do to our
assumptions on characteristic, we have
$G/Z_{G}\simeq\operatorname{Aut}({\mathfrak{g}})^{0}$ (see [8, Sec. 1.13]).
Since ${\operatorname{SL}}_{2}$ does not have any central extensions, there
exists a lifting $i:{\operatorname{SL}}_{2}\to G$. ∎

It is important for us that there are no multiplicities in the
$\mathfrak{s}l_{2}$-decomposition (4.1) (since all exponents $d_{i}$ are
distinct). In the case when $G$ is of type $D_{\ell}$ where $\ell$ is even,
the $\mathfrak{s}l_{2}$-representation $V_{2\ell-2}$ shows up with
multiplicity $2$ in (4.1). In this case the standard representation $W$ of
$\operatorname{SO}(2\ell)$ can be viewed as projective representation
$G\to\operatorname{PGL}(W)$, for which we have an
$\mathfrak{s}l_{2}$-decomposition

$W=V_{2\ell-2}\oplus V_{0}.$

Note that the center of $G$ acts trivially on $W^{\prime}:=S^{2}W/k$, so we
can consider the corresponding representation

$\rho:G\to\operatorname{GL}(W^{\prime}).$

The corresponding representation $d\rho$ of the Lie algebra ${\mathfrak{g}}$
is faithful.

######  Lemma 4.2.

Assume ${\mathfrak{g}}$ is of type $D_{\ell}$ with even $\ell$ and that either
$p=0$ or $p>\max(a_{{\mathfrak{g}}},4(h-1))$. Let $W$ be the standard
representation of ${\mathfrak{g}}$ (of dimension $2\ell$). Then we have
$\mathfrak{s}l_{2}$-decompositions, with respect to the
$\mathfrak{s}l_{2}$-triple of Lemma 4.1(i),

$W=V_{2\ell-2}\oplus V_{0},$ $W^{\prime}\simeq
V_{2\ell-2}\oplus\bigoplus_{i=0}^{\ell-1}V_{4i}.$

###### Proof.

As in Lemma 4.1, this reduces to the corresponding statement in characteristic
zero. The decomposition of $W$ is well known, and the decomposition of
$W^{\prime}$ follows from

$S^{2}V_{2\ell-2}=V_{4\ell-4}\oplus V_{4\ell-8}\oplus\ldots V_{0}.$

∎

######  Lemma 4.3.

Assume as before that either $p=0$ or $p>\max(a_{{\mathfrak{g}}},4(h-1))$.
Consider the algebraic subgroup $C(i)\subset G$ of $g\in G$ such that
$\operatorname{Ad}(g)$ centralizes $i(\mathfrak{s}l_{2})\subset{\mathfrak{g}}$
defined in Lemma 4.1. Then $C(i)$ coincides with center $Z_{G}$ of $G$.

###### Proof.

It is enough to prove that the centralizer of $i({\operatorname{SL}}_{2})$ in
$G/Z_{G}$ is trivial, so we can assume that $G$ is of adjoint type. We use the
fact that the centralizer of $e$ is contained in the unipotent radical of the
standard Borel subgroup. Hence, it is enough to prove that no elements in
${\mathfrak{u}}=\oplus_{\alpha\in R_{+}}{\mathfrak{g}}_{\alpha}$ centralize
$i(\mathfrak{s}l_{2})$. But this follows from the fact that $h^{0}$ acts on
${\mathfrak{u}}$ with nonzero eigenvalues. ∎

###  4.2. Nice $G$-bundles

Let $G$ be a connected split reductive group over a field $k$, such that
$[G,G]$ is simple. We also assume that the characteristic of $k$ is either $0$
or $p>\max(a_{{\mathfrak{g}}^{\prime}},4(h_{{\mathfrak{g}}^{\prime}}-1))$,
where ${\mathfrak{g}}^{\prime}=[{\mathfrak{g}},{\mathfrak{g}}]$. We denote by
$Z_{G}\subset G$ the center of $G$, and by $Z^{0}_{G}\subset Z_{G}$ its
neutral component (which is a split torus). Note that we have a decomposition
of the Lie algebra
${\mathfrak{g}}=[{\mathfrak{g}},{\mathfrak{g}}]\oplus\mathfrak{z}$, where
$\mathfrak{z}$ is the center of ${\mathfrak{g}}$. Also, we have a surjective
homomorphism $Z_{[G,G]}\to Z_{G}/Z^{0}_{G}$, and if the characteristic $p$ of
$k$ is nonzero then the orders of both groups are relatively prime to $p$.

Applying the construction of Sec. 4.1 to the split simple group $[G,G]$, we
get a homomorphism $i:{\operatorname{SL}}_{2}\to[G,G]\subset G$, corresponding
to a principal $\mathfrak{s}l_{2}$ in
$[{\mathfrak{g}},{\mathfrak{g}}]\subset{\mathfrak{g}}$. Note that the
statement of Lemma 4.3 still holds in this situation (since we can apply the
said Lemma to $G/Z_{G}$).

In the case when $[G,G]$ is of type $D_{\ell}$ with even $\ell$, we denote by
$W^{\prime}$ the representation of $G/Z_{G}$ defined in Sec. 4.1.

Let $C_{0}$ be a smooth projective curve over $k$ such that
$H^{0}(C_{0},{\mathcal{O}})=k$. Note that for every $G$-bundle $P$ we have the
induced vector bundle ${\mathfrak{g}}_{P}$, so any automorphism $\phi$ of $P$
over an open subset $U\subset C_{0}$ induces an automorphism
$\phi_{\mathfrak{g}}$ of the vector bundle ${\mathfrak{g}}_{P}|_{U}$. In the
case when $G$ is of type $D_{\ell}$ with even $\ell$, for any $G$-bundle $P$
we have the induced bundle $W^{\prime}_{P}$, so an automorphism $\phi$ of
$P|_{U}$ induces an automorphism $\phi_{W^{\prime}}$ of $W^{\prime}_{P}|_{U}$.

######  Definition 4.4.

Let $P$ and $P^{\prime}$ be $G$-bundles on $C_{0}$ and let
$\phi:P|_{C_{0}-p}\to P^{\prime}_{C_{0}-p}$ be an isomorphism of $G$-bundles
over $C_{0}-p$. We say that $\phi$ has a pole of order $\leq n$ at $p$, if
this holds for the induced homomorphism of the adjoint vector bundles
$\phi_{{\mathfrak{g}}}:{\mathfrak{g}}_{P}|_{C_{0}-p}\to{\mathfrak{g}}_{P^{\prime}}|_{C_{0}-p}$
over $C_{0}-p$ (resp., for the homomorphism
$\phi_{W^{\prime}}:W^{\prime}_{P}|_{C_{0}-p}\to
W^{\prime}_{P^{\prime}}|_{C_{0}-p}$, if $G$ is of type $D_{\ell}$ with even
$\ell$).

######  Definition 4.5.

Let $p\in C_{0}(k)$ be a point. We say that a $G$-bundle $P$ on $C_{0}$ is
nice of level $n$ at $p$ if

  * •

$H^{0}(C_{0},{\mathfrak{g}}_{P}(np))=\mathfrak{z}(k)$;

  * •

every automorphism of $P|_{C_{0}-p}$ with a pole of order $\leq n$ at $p$, is
given by an element of the center $Z_{G}(k)$.

Note that since
${\mathfrak{g}}_{P}=[{\mathfrak{g}},{\mathfrak{g}}]_{P}\oplus\mathfrak{z}\otimes{\mathcal{O}}_{C_{0}}$,
the condition $h^{0}(C_{0},{\mathfrak{g}}_{P}(np))=0$ is a consequence of the
following two conditions:

$h^{0}(C_{0},[{\mathfrak{g}},{\mathfrak{g}}]_{P}(np))=0,\ \
H^{0}(C_{0},{\mathcal{O}}(np))=k$ (4.2)

(the second condition is only needed when $\mathfrak{z}\neq 0$).

######  Lemma 4.6.

For any point $p\in C_{0}(k)$, one has $Z_{G}(C_{0}-p)=Z_{G}(C_{0})=k$.

###### Proof.

Since $Z^{0}_{G}$ is a split torus, we have $H^{1}(k,Z^{0}_{G})=1$, so the
sequence

$1\to Z^{0}_{G}(k)\to Z_{G}(k)\to(Z_{G}/Z^{0}_{G})(k)\to 1$

is exact. Given a regular map $f:C-p\to Z_{G}$, let us consider the induced
map $\overline{f}:C_{0}-p\to Z_{G}/Z^{0}_{G}$. Since $C_{0}-p$ is connected,
and $Z_{G}/Z^{0}_{G}$ is finite, the image of $\overline{f}$ is a closed
point. Furthermore, since $H^{0}(C_{0},{\mathcal{O}})=k$, $\overline{f}$
factors through the projection $C-p\to\operatorname{Spec}(k)$. The
corresponding $k$-point of $Z_{G}/Z^{0}_{G}$ can be lifted to $Z_{G}(k)$.
Thus, modifying $f$, we reduce to the case when $f$ factors through
$Z^{0}_{G}$. Now the assertion follows from the fact that any invertible
function on $C_{0}-p$ is constant. ∎

######  Lemma 4.7.

Fix $n\geq 0$ and a point $p\in C_{0}(k)$. Set $m=2d_{r}+2d_{r-1}$ (resp.,
$m=4(2\ell-3)$ if $G$ is of type $D_{\ell}$ with even $\ell$), and let $V$ an
${\operatorname{SL}}_{2}$-bundle on $C_{0}$ satisfying
$H^{0}(C_{0},S^{i}V(np))=0$ for $1\leq i\leq m$. If $\mathfrak{z}\neq 0$,
assume in addition that such that $H^{0}(C_{0},{\mathcal{O}}(np))=k$. Let $P$
be the $G$-bundle associated with $V$ via the homomorphism
$i:{\operatorname{SL}}_{2}\to G$. Then $P$ is nice of level $n$ at $p$.

###### Proof.

First, taking into account the decomposition of ${\mathfrak{g}}$ as
$\mathfrak{s}l_{2}$-representation, we see that our assumptions imply the
conditions (4.2). Thus, we only have to check that every automorphism of
$P|_{C_{0}-p}$ with a pole of order $\leq n$ at $p$ comes from an element of
the center $Z_{G}(k)$.

Next, we claim that it is enough to consider the case when $Z_{G}$ is trivial.
Indeed, let $P/Z_{G}$ be the induced $G_{ad}$-bundle, where $G_{ad}=G/Z_{G}$.
Suppose we know that $P/Z_{G}$ is nice of level $n$. Then for any automorphism
$\phi$ of $P|_{C_{0}-p}$ with a pole of order $\leq n$ at $p$, the induced
automorphism of $P/Z_{G}|_{C_{0}-p}$ is trivial. Hence, $\phi$ is given by an
element $f\in Z_{G}(C_{0}-p)$, and the assertion follows from Lemma 4.6.

Thus, we can assume that $G$ is simple and $Z_{G}$ is trivial. Assume first
that $G$ is not of type $D_{\ell}$ with even $\ell$. Then we have a
decomposition

${\mathfrak{g}}_{P}=\bigoplus_{i=1}^{r}S^{2d_{i}}V.$ (4.3)

Hence, for an automorphism $\phi$ of $P$ (defined over some open $U\subset
C_{0}$), the endomorphism $\phi_{{\mathfrak{g}}}$ of ${\mathfrak{g}}_{P}$ is
an element of $\bigoplus_{i,j}H^{0}(C,S^{2d_{i}}V\otimes S^{2d_{j}}V(np))$.
For $i\neq j$, using the decomposition

$S^{2d_{i}}V\otimes S^{2d_{j}}V\simeq S^{2d_{i}+2d_{j}}V\oplus
S^{2d_{i}+2d_{j}-2}V\oplus\ldots,$

we get $H^{0}(C,S^{2d_{i}}V\otimes S^{2d_{j}}V(np))=0$, while for $i=j$, we
get that $H^{0}(C,S^{2d_{i}}V\otimes S^{2d_{i}}V(np))$ is $1$-dimensional.
Thus, we deduce that $\phi_{{\mathfrak{g}}}$ preserves the decomposition (4.3)
and acts by a scalar on each summand.

Let us denote by ${\mathcal{A}}\subset\underline{\operatorname{Aut}}_{G}(P)$
the subsheaf of the sheaf of automorphisms of $P$ as a $G$-bundle, consisting
of automorphisms such that induced automorphism of ${\mathfrak{g}}_{P}$
preserves the decomposition (4.3) and acts by scalar on each summand. Then
Lemma 4.3 implies that ${\mathcal{A}}$ is trivial, and hence so is $\phi$.

In the case when $G$ is of type $D_{\ell}$ with even $\ell$ , the argument is
similar, but instead of $\phi_{\mathfrak{g}}$ we consider endomorphisms
$\phi_{W^{\prime}}$ of $W^{\prime}_{P}$. ∎

###  4.3. Construction of nice bundles over a curve over finite field

Now we will construct nice ${\operatorname{SL}}_{2}$-bundles over curves over
a finite field.

######  Proposition 4.8.

(i) Fix integers $n\geq 0$ and $d>0$. Assume the characteristic of $k$ is
$>d/2$. Let $C_{0}$ be a geometrically irreducible smooth projective curve
over $k$ of genus $g\geq n+2$, $p\in C_{0}(k)$ a point such that
$h^{0}({\mathcal{O}}(np))=1$, $L$ a degree $0$ line bundle on $C_{0}$ such
that $H^{0}(L^{\pm i}(np))=0$ for $1\leq i\leq d$. Then there exists an
extension

$0\to L\to V\to L^{-1}\to 0$ (4.4)

such that $H^{0}(C_{0},S^{i}V(np))=0$ for $1\leq i\leq d$.

(ii) Let $k$ be a finite field of characteristic $>2$. For any $n\geq 0$ and
$d\geq 1$ there exists data as in (i) over $k$.

###### Proof.

(i) Since $g\geq n+2$, we have $\deg(\omega_{C_{0}}\otimes L^{-2}(-np))\geq
g$, so $h^{1}(L^{2}(np))=h^{0}(\omega_{C_{0}}\otimes L^{-2}(-np))\neq 0$. The
natural map $H^{1}(C_{0},L^{2})\to H^{1}(C_{0},L^{2}(np))$ is surjective and
we choose an element $e\in H^{1}(C_{0},L^{2})$ with a nonzero image. Then we
consider the extension (4.4) with the extension class $e$. Let us consider the
decreasing filtration of $S^{i}V$ by the subbundles $F_{p}=L^{p}\cdot
S^{i-p}V\subset S^{i}V$, where $0\leq p\leq i$ (and we set $F_{i+1}=0$). We
have

$F_{p}/F_{p+1}\simeq L^{p}\otimes(L^{-1})^{i-p}=L^{2p-i}.$

It is easy to check that the extension

$0\to L^{2p+2-i}\to F_{p}/F_{p+2}\to L^{2p-i}\to 0$

has the class $(p-i)\cdot e\in H^{1}(C_{0},L^{2})$.

In the case when $i$ is odd, we get a filtration of $S^{i}V(np)$ with the
successive quotients of the form $L^{\pm j}(np)$, where $1\leq j\leq i$. By
our assumptions this implies the vanishing of $H^{0}(C_{0},S^{i}V(np))$. In
the case when $i$ is even, the only successive quotient not of the above form
is $F_{i/2}/F_{i/2+1}\simeq{\mathcal{O}}$. Thus, it is enough to prove that

$H^{0}(C_{0},F_{i/2}/F_{i/2+2}(np))=0.$

Consider the exact sequence

$0\to L^{2}(np)\to F_{i/2}/F_{i/2+2}(np)\to{\mathcal{O}}(np)\to 0.$

Note that the corresponding extension class is $-i/2\cdot e$, where $i/2\neq
0$ by the assumption on the characteristic. It is enough to prove the
injectivity of the corresponding coboundary map
$H^{0}(C_{0},{\mathcal{O}}(np))\to H^{1}(C_{0},L^{2}(np))$, which is given by
the cup product by $-i/2\cdot e$. By our assumption,
$H^{0}(C_{0},{\mathcal{O}}(np))$ is generated by $1$, and by the choice of
$e$, its image under the above coboundary map is nonzero. This proves the
required injectivity.

(ii) Let $C_{0}$ be a hypergeometric curve of genus $g\geq n+d+2$, and let
$p,p_{1},p_{2}\in C_{0}(k)$ be distinct points, which are not ramification
points of the double covering $\pi:C_{0}\to{\mathbb{P}}^{1}$. Let us set
$L={\mathcal{O}}(p_{1}-p_{2})$. Then we claim that the assumptions of (i) are
satisfied. Indeed, for every $i$, $1\leq i\leq d$, we have
$h^{0}({\mathcal{O}}(np+ip_{1}))=h^{0}({\mathcal{O}}(np+ip_{2}))=1$ (since
every element of
$H^{0}({\mathcal{O}}(n\pi^{-1}(\pi(p))+i\pi^{-1}(\pi(p_{1}))))$ is a pull-back
from ${\mathbb{P}}^{1}$). This immediately implies that
$h^{0}({\mathcal{O}}(np))=1$. Also, if there exists an effective divisor $D$
such that $L^{i}(np)\simeq{\mathcal{O}}(D)$ (resp.,
$L^{-i}(np)\simeq{\mathcal{O}}(D)$) then we would get
${\mathcal{O}}(np+ip_{1})\simeq{\mathcal{O}}(ip_{2}+D)$ (resp.,
${\mathcal{O}}(np+ip_{2})\simeq{\mathcal{O}}(ip_{1}+D)$) which is a
contradiction.

Thus, it is enough to find a monic separable polynomial $f$ of degree $\geq
n+d+4$ such that $f(0)$ and $f(1)$ are squares in $k^{*}$. Indeed, then we can
define $C_{0}$ as the completion of

$y^{2}=f(x),$

and take $p,p_{1},p_{2}$ to be some points over $x=0,1,\infty$.

To find such $f$, let us pick three distinct irreducible monic polynomials
$f_{1}$, $f_{2}$ and $f_{3}$ of degrees $\geq n+d+4$. Then it is easy to see
that one of the polynomials

$f_{1},\ f_{2},\ f_{3},\ f_{1}f_{2},\ f_{1}f_{3},\ f_{2}f_{3},\
f_{1}f_{2}f_{3}$

can be taken as $f$: one should consider the images of these elements under
the map

$f\mapsto(f(0)\mod(k^{*})^{2},f(1)\mod(k^{*})^{2})\in k^{*}/(k^{*})^{2}\times
k^{*}/(k^{*})^{2}\simeq\operatorname{\mathbb{Z}}_{2}\times\operatorname{\mathbb{Z}}_{2}$

and observe that one of them will go to zero. ∎

######  Corollary 4.9.

Let $G$ be a connected split reductive group over a finite field $k$, such
that $[G,G]$ is simple. Assume that the characteristic of $k$ satisfies
$p>\max(a_{{\mathfrak{g}}^{\prime}},4(h_{{\mathfrak{g}}^{\prime}}-1))$, where
${\mathfrak{g}}^{\prime}=[{\mathfrak{g}},{\mathfrak{g}}]$. Then for any $n\geq
0$, there exists a geometrically irreducible smooth projective curve $C_{0}$
over $k$, a point $p\in C_{0}(k)$, and a $G$-bundle $P$ on $C_{0}$ such that
$P$ is nice of level $n$ at $p$.

###### Proof.

This follows from Lemma 4.7 and Proposition 4.8. ∎

###  4.4. $G$-bundles with nice reduction

Now assume that $G$ is a connected split reductive group over
$\operatorname{\mathbb{Z}}$, such that $[G,G]$ is simple, and
$k=O/{\mathfrak{m}}$ for the ring of integers $O$ in a local field. As before,
we assume that the characteristic of $k$ satisfies
$p>\max(a_{{\mathfrak{g}}},4(h-1))$.

Let $C$ be a smooth proper curve over $O/{\mathfrak{m}}^{N}$, $P$ a $G$-bundle
over $C$. We use an obvious analog of Definition 4.4 to define the order of a
pole of an automorphism of $P$ along an $O/{\mathfrak{m}}^{N}$-point of $C$.

######  Lemma 4.10.

Let $C$ be a smooth proper curve over $O/{\mathfrak{m}}^{N}$, $p\in
C(O/{\mathfrak{m}}^{N})$, and let $C_{0}$ be the corresponding curve over
$k=O/{\mathfrak{m}}$, $p_{0}=p\mod{\mathfrak{m}}$. Assume that $P$ is a
$G$-bundle on $C$ such that $P|_{C_{0}}$ is nice of level $n$ at $p_{0}$. Then
any automorphism of $P|_{C-p}$ with a pole of order $\leq n$ at $p$ comes from
an element of $Z_{G}(O/{\mathfrak{m}}^{N})$.

###### Proof.

We will prove this by induction by considering restrictions of $P$ to
$C_{n}=C\times\operatorname{Spec}(O/{\mathfrak{m}}^{n+1})$.

Let $d$ denote the order of $Z_{G}/Z^{0}_{G}$. Then the group $Z_{G}$ is
smooth over $\operatorname{\mathbb{Z}}[d^{-1}]$. Since $d$ is invertible in
$O/{\mathfrak{m}}$, we deduce that each map $Z_{G}(O/{\mathfrak{m}}^{n+1})\to
Z_{G}(O/{\mathfrak{m}}^{n})$ is surjective.

Let $\phi$ be an automorphism of $P|_{C_{n}-p}$ with a pole of order $\leq n$
at $p$, such that the induced automorphism of $P|_{C_{n-1}-p}$ comes from an
element of $Z_{G}(O/{\mathfrak{m}}^{n})$. Lifting it to an element of
$Z_{G}(O/{\mathfrak{m}}^{n+1})$ and modifying $\phi$ we can assume that $\phi$
is identity modulo ${\mathfrak{m}}^{n}$. Hence, $\phi$ corresponds to a
section of
$H^{0}(C_{0},{\mathfrak{g}}_{P}(np)|_{C_{0}})\otimes{\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1}$.
It remains to use the equality
$H^{0}(C_{0},{\mathfrak{g}}_{P}(np)|_{P_{0}})=\mathfrak{z}(k)$. ∎

###  4.5. Generic triviality and $O$-structures

Let $C$ be a smooth complete irreducible curve over a local non-archimedian
field $K$. Let $G$ be a connected split reductive group over
$\operatorname{\mathbb{Z}}$. We denote by
$\underline{{\operatorname{Bun}}}_{G}$ the stack of $G$-bundles on $C$, and
set ${\operatorname{Bun}}_{G}:=\underline{{\operatorname{Bun}}}_{G}(K)$. It is
known that the stack $\underline{{\operatorname{Bun}}}_{G}$ is admissible (see
[15, Sec. 7.1]).

Let us consider the following condition on $G$:

($\star$) $G$ is connected split reductive, $[G,G]$ is simply connected, and
there exists a principal homomorphism $i:{\operatorname{SL}}_{2}\to G$,
defined over $k=O/{\mathfrak{m}}$.

Here by a principal homomorphism ${\operatorname{SL}}_{2}\to G$ we mean a
homomorphism inducing a principal ${\mathfrak{s}l}_{2}$-subalgebra in
${\mathfrak{g}}$. Let $B\subset G$ be a Borel subgroup containing a split
torus $T$.

Note that by Lemma 4.1, condition ($\star$) is satisfied if $[G,G]$ is simply
connected and the characteristic of $k$ is sufficiently large.

######  Lemma 4.11.

Assume that $C$ has a smooth model $C_{O}$ over $O$, and let
${\operatorname{Bun}}^{O}_{G}\subset{\operatorname{Bun}}_{G}$ denote the
subgroupoid of $G$-bundles on $C$ that extend to $G$-bundles over $C_{O}$. Let
also ${\operatorname{Bun}}^{gt}_{G}\subset{\operatorname{Bun}}_{G}$ denote the
subgroupoid of $G$-bundles that are trivial at the general point of $C$.

(i) We have an inclusion
${\operatorname{Bun}}^{gt}_{G}\subset{\operatorname{Bun}}^{O}_{G}$.

(ii) For every $n\geq 0$, set
$C_{n}=C_{O}\times_{\operatorname{Spec}(O)}\operatorname{Spec}(O/{\mathfrak{m}}^{n+1})$.
Assume $G$ satisfies condition ($\star$). Then every $G$-bundle over $C_{O}$
(resp., over $C_{n}$) admits a $B$-structure and is trivial at the general
point. In particular, we have
${\operatorname{Bun}}^{gt}_{G}={\operatorname{Bun}}^{O}_{G}$.

###### Proof.

(i) First, let us show that any $G$-bundle $P$ on $C$, trivial at the general
point, can be extended to a $G$-bundle on $C_{O}$. Let $t_{U}$ be a
trivialization of $P$ on an open subset $U=C\setminus D$, where $D\subset C$
is an effective divisor. We can extend $D$ to a divisor $D_{O}\subset C_{O}$.
Let $D_{k}\subset C_{k}$ be the corresponding divisor on the special fiber.
Then using our trivialization $t_{U}$ we can glue $P$ and the trivial bundle
on $C_{O}\setminus D_{O}$ into a $G$-bundle $P^{\prime}$ over $C_{O}\setminus
D_{k}$.

Let us realize $G$ as a closed subgroup in some $\operatorname{GL}_{N}$, and
let $V^{\prime}$ be the $\operatorname{GL}_{N}$-bundle over $C_{O}\setminus
D_{k}$ associated with $P^{\prime}$. Then it is well known that $V^{\prime}$
extends to a $\operatorname{GL}_{N}$-bundle $V_{O}$ over $C_{O}$ (using the
fact that reflexive coherent sheaves on $C_{O}$ are locally free). Let
$\pi:X_{O}\to C_{O}$ be the $\operatorname{GL}_{N}/G$-fibration classifying
reductions of $V_{\mathcal{O}}$ to a $G$-bundle. The $G$-bundle $P^{\prime}$
corresponds to a section $\sigma$ of $\pi$ over $C_{O}\setminus D_{k}$. Since
$\operatorname{GL}_{N}/G$ is affine, $\sigma$ extends regularly to a section
of $\pi$ on the entire $C_{O}$. This gives the required extension of $P$ to
$C_{O}$.

(ii) By (i), we need to show that if $G$ satisfies ($\star$) then every
$G$-bundle over $C_{O}$ (resp., $C_{n}$) admits a $B$-structure and is trivial
at the general point. This follows from the results of [16]. We will give a
somewhat different proof following closely the arguments from the work [3].

Since every $B$-bundle is trivial at the general point, it is enough to prove
that every $G$-bundle $P$ over $C_{O}$ (resp., $C_{n}$) admits a
$B$-structure. Note that a $B$-structure over $C_{O}$ (resp., $C_{n}$) is a
section of a $G/B$-bundle $P/B$ over $C_{O}$ (resp., over $C_{n}$). Hence, by
Grothendieck’s existence theorem (see [13, Cor. 8.4.7]), it is enough to
construct a compatible family of $B$-structures on the restrictions $P_{n}$ of
$P$ to $C_{n}$. We do this by first constructing a $B$-structure of special
type on $P_{0}$, and then showing that it extends to a required compatible
family.

The starting point is the fact that for $G$ satisfying condition $(\star)$ any
$G$-bundle on the curve $C_{0}$ over finite field $k$, is trivial at the
general point. Indeed, for $G$ simply connected and semisimple, this is a
theorem of Harder [17] on the vanishing of the Galois cohomology
$H^{1}(k(C_{0}),G)$. In general, let $G^{\prime}$ be the commutator subgroup
in $G$. Then the result follows from the vanishing of
$H^{1}(k(C_{0}),G^{\prime})$ and $H^{1}(k(C_{0}),G/G^{\prime})$ (note that
$G/G^{\prime}$ is a split torus).

Let $\Delta$ denote the set of simple roots with respect to $(T,B)$. Recall
that for every simple root $\alpha$, we can define the degree
$\deg_{\alpha}(F)\in\operatorname{\mathbb{Z}}$ of a $B$-bundle $F$, as the
degree of the line bundle associated with $F$ and the homomorphism $B\to
T\to{\mathbb{G}}_{m}$ given by $\alpha$. We claim that for every $N>0$ and
every $G$-bundle $P_{0}$ on $C_{0}$ there exists a $B$-bundle $F_{0}$ inducing
$P_{0}$ such that $\deg_{\alpha}(F_{0})\leq-N$. We prove this as in [3, Prop.
3]. First, we observe that if $P^{\prime}_{0}$ and $P_{0}$ are isomorphic over
an open subset then the assertions for $P_{0}$ and for $P^{\prime}_{0}$ are
equivalent. Indeed, we can choose an isomorphism of $P_{0}$ and
$P^{\prime}_{0}$ over $C_{0}\setminus S$, for a finite set of points $S$. Then
any $B$-structure on $P_{0}$ induces one on $P^{\prime}_{0}$ and the
difference between the corresponding degrees $\deg_{\alpha}$ is bounded by a
constant depending only on the isomorphism $P_{0}|_{C_{0}\setminus S}\simeq
P^{\prime}_{0}|_{C_{0}\setminus S}$. Since any $P_{0}$ is trivial at the
general point, it is enough to construct a required $B$-structure for the
trivial $G$-bundle on $C_{0}$. Furthermore, we claim that it is enough to do
this for the trivial $G$-bundle on ${\mathbb{P}}^{1}$. Indeed, we can choose a
finite morphism $f:C_{0}\to{\mathbb{P}}^{1}$ over $k$ and take the pull-backs
of a $B$-structure under $f$. Finally, using the existence of a principal
${\operatorname{SL}}_{2}$ in $G$, we reduce to the case
$G={\operatorname{SL}}_{2}$ (see [3, Sec. 5]). In this case we use the
$B$-structure corresponding to the subbundle
${\mathcal{O}}(-N)\subset{\mathcal{O}}^{2}$ generated by
$(x_{0}^{N},x_{1}^{N})$, where $(x_{0},x_{1})$ is the standard basis of
$H^{0}({\mathbb{P}}^{1},{\mathcal{O}}(1))$.

Next, starting with a $B$-structure $F_{0}$ on $P_{0}$, with
$\deg_{\alpha}(F_{0})$ sufficiently small (in fact we need them $<2-2g$), we
claim that it extends to a collection of compatible $B$-structures $F_{n}$ on
$P_{n}$, for all $n\geq 0$. Indeed, we can think of such $B$-structures as
sections $\sigma_{n}:C_{n}\to P_{n}/B$. We just need to check that each
$\sigma_{n}$ extends to a section $\sigma_{n+1}$. As explained in [3, Prop.
1], the obstruction to such an extension lies in $H^{1}(C_{0},V)$, where $V$
is the vector bundle on $C_{0}$ associated with the $B$-bundle $F_{0}$ and
with the $B$-representation ${\mathfrak{g}}/{\mathfrak{b}}$. Now the
assumption that $\deg_{\alpha}(F_{0})$ are sufficiently small for
$\alpha\in\Delta$ implies that $H^{1}(C_{0},V)=0$. ∎

##  5\. Hecke operators on ${\operatorname{Bun}}_{G}$

In this section we will use Hecke operators on ${\operatorname{Bun}}_{G}$ to
prove commutativity of the small Hecke algebra over $O/{\mathfrak{m}}^{N}$.

First, in Sec. 5.1 we connect standard Hecke operators for $G$-bundles on a
curve $C$ over $K$ with the action of the local Hecke algebra on
${\mathbb{H}}$-coinvariants of ${\mathbb{G}}$-representations described in
Sec. 3.2. We also establish commutativity of a natural global Hecke algebra
$H(C)$ associated with $C$ (see Theorem 5.3): the case of two Hecke operators
at the same point goes back to [4], while the case of two Hecke operators at
different points is new (but not difficult).

Then in Sec. 5.2, we consider actions of local Hecke algebras on $G$-bundles
for curves over $O$ and $O/{\mathfrak{m}}^{N}$. Note that we do not have
explicit descriptions of the Hecke operators over $O$ or over
$O/{\mathfrak{m}}^{N}$ associated with non-minuscule weights.

Finally, in Sec. 5.3, we give a global proof of Theorem 3.10, using the
connection with global Hecke operators over $K$ and nice $G$-bundles.

###  5.1. Hecke operators on the Schwartz space of half-densities on
${\operatorname{Bun}}_{G}$

We refer to [5] for more details on some constructions and assertions sketched
below.

As before, $G$ is a connected split reductive group over
$\operatorname{\mathbb{Z}}$, $C$ is a smooth complete irreducible curve over
$K$.

We write by $\omega$ instead of $\omega_{{\operatorname{Bun}}_{G}}$ and fix a
square root $\omega^{1/2}$ of $\omega$ and a square root $\omega_{C}^{1/2}$ of
$\omega_{C}$ (which we assume to exist over $K$).

Let us consider the space

$\mathcal{W}:=\mathcal{S}({\operatorname{Bun}}_{G},|\omega^{1/2}|).$

For each $i\geq 0$, let us consider the $G[t]/(t^{i})$-torsor
${\operatorname{Bun}}^{i,v,t}_{G}\to{\operatorname{Bun}}_{G}$, defined, over
$O$, classifying $G$-bundles together with a trivialization on the $i$-th
infinitesimal neighborhood of $v$ given by the ideal $(t^{i})$ (we will often
write ${\operatorname{Bun}}^{i,v}_{G}={\operatorname{Bun}}^{i,v,t}_{G}$ for
brevity). For any open substack of finite type
${\mathcal{Y}}\subset{\operatorname{Bun}}_{G}$, we have the corrresponding
torsor ${\mathcal{Y}}^{i,v}\to{\mathcal{Y}}$, which is a smooth scheme of
finite type for $i\gg 0$. We denote by ${\mathcal{Y}}^{\infty,v}$ the
corresponding object of ${\operatorname{Pro}}(Sch^{ft})$, where $Sch^{ft}$ is
the category of schemes of finite type. Let us denote by
${\operatorname{Bun}}_{G}^{\infty,v}$ the object
$({\mathcal{Y}}^{\infty,v})_{{\mathcal{Y}}}$ of
${\operatorname{Ind}}({\operatorname{Pro}}(Sch^{ft}))$.

Abusing the notation, we denote by $\omega_{{\operatorname{Bun}}_{G}}^{1/2}$
the pull-back of the square root of the canonical bundle on
${\operatorname{Bun}}_{G}$ to ${\operatorname{Bun}}_{G}^{\infty,v}$. We have a
natural action of $G(\\!(t)\\!)$, viewed as a group in
${\operatorname{Ind}}({\operatorname{Pro}}(Sch^{ft}))$, on
${\operatorname{Bun}}_{G}^{\infty,v}$ (see [15, Sec. 7]). This action does not
change a $G$-bundle away from $v$, and multiplies the transition function on
the formal punctured neighborhood of $v$ by an element of $G(\\!(t)\\!)$.

Furthermore, we have a compatible action of the central extension
$\hat{G}_{crit}$ at the critical level on the line bundle
$\omega_{{\operatorname{Bun}}_{G}}^{1/2}$ over
${\operatorname{Bun}}_{G}^{\infty,v}$, such that the center ${\mathbb{G}}_{m}$
acts with weight $1$. Note that the action of $G(\\!(t)\\!)$ on
${\operatorname{Bun}}_{G}^{\infty,v}$ fits into a commutative diagram, where
the horizontal maps are given by $(g,x)\mapsto g^{-1}x$,

$\begin{diagram}$ (5.1)

and the above statement about the weight-$1$ action of $\hat{G}_{crit}$ on
$\omega_{{\operatorname{Bun}}_{G}}$ corresponds to an isomorphism

$\beta^{*}\omega_{{\operatorname{Bun}}_{G}}^{1/2}\simeq
p^{*}{\mathcal{L}}_{crit}^{-1}\otimes
q^{*}\omega_{{\operatorname{Bun}}_{G}}^{1/2},$ (5.2)

where $p$ and $q$ are the projections onto the factors $\operatorname{Gr}_{G}$
and ${\operatorname{Bun}}_{G}^{\infty,v}$ (the appearance of
${\mathcal{L}}_{crit}^{-1}$ is due to the fact that we use the action of
$g^{-1}$). The latter isomorphism descends to an isomorphism of line bundles
on the Hecke correspondence that we will use later (see (5.3)).

Following [15] we set
$W^{i}:=\varinjlim_{{\mathcal{Y}}}\SS({\mathcal{Y}}^{i,v}(K),|\omega_{{\operatorname{Bun}}_{G}}|^{1/2})$.
Due to the above action of $\hat{G}_{crit}$ on the line bundle
$\omega_{{\operatorname{Bun}}_{G}}^{1/2}$, the pro-vector space

$W:=``\varprojlim_{i}"W^{i}$

acquires a structure of an object of
${\operatorname{Rep}}_{crit}({\mathbb{G}})$, such that
$W_{{\mathbb{G}}^{i}}\simeq W^{i}$. In particular,

${\mathcal{W}}:=W^{0}=\SS({\operatorname{Bun}}_{G}(K),|\omega|^{1/2})\simeq
W_{{\mathbb{H}}}.$

Thus, we get an action of the local Hecke algebra
${\mathcal{H}}({\mathcal{G}},{\mathcal{H}})_{crit}$ on
$\SS({\operatorname{Bun}}_{G}(K),|\omega|^{1/2})$.

Note that we can also define this action without a choice of a formal
parameter $t$ at $v$, by replacing $K[\\![t]\\!]$ with the algebra
$\hat{{\mathcal{O}}}_{C,v}$ (see Sec. 3.6).

The action of the elements
$h^{\lambda}\in{\mathcal{H}}({\mathcal{G}},{\mathcal{H}})_{crit}$ on
$\SS({\operatorname{Bun}}_{G}(K),|\omega|^{1/2})$, for $\lambda\in\Lambda^{+}$
and $v\in C(K)$, can be described in terms of the Hecke correspondences.

Let us denote by $Z_{\lambda,v}$ the Hecke correspondence which is the stack
of triples $(\mathcal{F},\mathcal{G},j)$ where
$\mathcal{F},\mathcal{G}\in{\operatorname{Bun}}_{G}$ and
$j:\mathcal{F}_{C\setminus v}\to\mathcal{G}_{C\setminus v}$ is an isomorphism
which is in the position $\mu\leq\lambda$ at $v$. The natural projections
$p_{1},p_{2}:Z_{\lambda,v}\to{\operatorname{Bun}}_{G}$ are representable
proper maps. If $\lambda$ is a minuscule coweight then these projections are
also smooth.

The map
$\beta:\operatorname{Gr}_{G}\times{\operatorname{Bun}}_{G}^{\infty,v}\to{\operatorname{Bun}}_{G}$
(see (5.1)) gives a collection of maps

$Z^{i}_{\lambda,v}:=\overline{\operatorname{Gr}}_{\lambda}\times{\operatorname{Bun}}_{G}^{i,v}\rTo{\beta_{\lambda}}{\operatorname{Bun}}_{G},$

with $i=i(\lambda)$, The maps $\beta_{\lambda}$ induce the action maps (3.2)
for $W\in{\operatorname{Rep}}_{crit}({\mathbb{G}})$. Furthermore,
$\beta_{\lambda}$ factors as a composition

$\beta_{\lambda}:Z^{i}_{\lambda,v}\rTo{\pi_{\lambda}}Z_{\lambda,v}\rTo{p_{2}}{\operatorname{Bun}}_{G},$

where $\pi_{\lambda}$ is a ${\mathbb{H}}/{\mathbb{G}}^{i}$-torsor. Also, the
natural projection
$Z^{i}_{\lambda,v}\to{\operatorname{Bun}}_{G}^{i,v}\to{\operatorname{Bun}}_{G}$
coincides with $p_{1}\pi_{\lambda}$.

The isomorphism (5.2) descends to an isomorphism of line bundles on
$Z_{\lambda,v}$,

$p_{2}^{*}\omega^{1/2}\simeq p_{1}^{*}\omega^{1/2}\otimes
p^{*}{\mathcal{L}}_{crit}^{-1},$

where $p$ is the projection to the quotient stack
$[\operatorname{Gr}_{G}/G[\\![t]\\!]]$. Using the isomorphism (3.3), we can
rewrite this as

$p_{1}^{*}\omega^{1/2}\rTo{\sim}p_{2}^{*}\omega^{1/2}\otimes\omega_{p_{2}}\otimes
L_{\lambda,v},$ (5.3)

where $L_{\lambda,v}$ is a $1$-dimensional space depending on $\lambda$ and
$v$ (in fact $L_{\lambda,v}=\omega_{C}|_{v}^{-<\lambda,\rho^{\vee}>}$, where
$\rho^{\vee}$ is the sum of positive coroots).

Thus, in the case when $\lambda$ is minuscule we have a well defined
composition

$\displaystyle
T_{v}^{\lambda}:\SS({\operatorname{Bun}}_{G}(K),|\omega^{1/2}|)\rTo{p_{1}^{\ast}}\SS(Z_{\lambda,v}(K),|p_{1}^{\ast}(\omega^{1/2})|)\simeq\SS(Z_{\lambda,v}(K),|p_{2}^{\ast}(\omega^{1/2})\otimes\omega_{p_{2}}|)\otimes|L_{v}|$
$\displaystyle\rTo{p_{2,!}}\SS({\operatorname{Bun}}_{G}(K),|\omega^{1/2}|)\otimes|L_{\lambda,v}|.$

Trivializing $L_{\lambda,v}$, we can view $T_{v}^{\lambda}$ as an operator on
$\SS({\operatorname{Bun}}_{G}(K),|\omega^{1/2}|)$ defined up to rescaling.

In the case when $\lambda$ is not necessarily minuscule, we can still define
the Hecke operators using a resolution of $Z_{\lambda,v}$. Namely, the variety
$\overline{\operatorname{Gr}}_{\lambda}$ admits a
$G(K[\\![t]\\!])$-equivariant smooth resolution

$\rho:\widetilde{\operatorname{Gr}}_{\lambda}\to\overline{\operatorname{Gr}}_{\lambda}$

such that
$\rho^{*}\omega_{\overline{\operatorname{Gr}}_{\lambda}}\simeq\omega_{\widetilde{\operatorname{Gr}}_{\lambda}}(-E)$
for some effective divisor $E$ (see [12]). Let us set

$\widetilde{Z}_{\lambda,v}=\widetilde{\operatorname{Gr}}_{\lambda}\times_{G[\\![t]\\!]/G_{i})}{\operatorname{Bun}}_{G}^{i,v},\
\ D:=E\times_{G[\\![t]\\!]/G_{i}}{\operatorname{Bun}}_{G}^{i,v}.$

We have a natural projection $\rho:\widetilde{Z}_{\lambda,v}\to
Z_{\lambda,v}$, which is a simultaneous resolution of singularities of the
fibers of $p_{2}$.

Furthermore, locally on a smooth covering $B$ of ${\operatorname{Bun}}_{G}$,
the map $p_{2}\pi$ has form $\widetilde{\operatorname{Gr}}_{\lambda}\times
B\to B$, so that the divisor $D$ corresponds to $E\times B$. Thus, as was
explained in Sec. 2.1, we have a well defined push-forward map

$(p_{2})_{!}:\SS(Z_{\lambda,v}(K),|p_{2}^{*}\omega^{1/2}\otimes\omega_{p_{2}}|)\to\SS({\operatorname{Bun}}_{G}(K),|\omega^{1/2}|).$

Hence, using the isomorphism (5.3), we can still define the operator
$T_{v}^{\lambda}$ on $\SS({\operatorname{Bun}}_{G}(K),|\omega^{1/2}|)$ as the
composition $(p_{2})_{!}p_{1}^{*}$. It is easy to see that this definition
does not depend on a choice of resolution of
$\overline{\operatorname{Gr}}_{\lambda}$.

######  Lemma 5.1.

Under the identification ${\mathcal{W}}=W_{\mathbb{H}}$, the action of the
element $h^{\lambda}\in{\mathcal{H}}({\mathbb{G}},{\mathbb{H}})_{crit}$ on
$W_{\mathbb{H}}$ coincides (up to rescaling) with the operator
$T_{v}^{\lambda}$ defined above.

###### Proof.

This follows from Lemma 3.3 and from the relation between the Hecke
correspondence and $G(\\!(t)\\!)$-action on
${\operatorname{Bun}}_{G}^{\infty,v}$ described above. ∎

######  Definition 5.2.

We denote by $H(C)\subset{\operatorname{End}}(\mathcal{W})$ the subalgebra
generated by operators $T_{v}^{\lambda}$ for $v\in
C(K),\lambda\in\Lambda^{+}$.

######  Theorem 5.3.

The algebra $H(C)$ is commutative.

###### Proof.

For a fixed point $v\in C$, the operators $T_{v}^{\lambda}$ and $T_{v}^{\mu}$
commute by Theorem 3.5. Let us now show that the operators $T_{v}^{\lambda}$
and $T_{v^{\prime}}^{\mu}$ for distinct points $v,v^{\prime}\in C(K)$ commute.
Since we want to prove the equality of some convergent integrals, we can work
over open parts of our correspondences. The composition $T_{v}^{\lambda}\circ
T_{v^{\prime}}^{\mu}$ is given by the composed correspondence

$Z(v^{\prime},v):=Z_{\mu,v^{\prime}}\times_{{\operatorname{Bun}}_{G}}Z_{\lambda,v}$

equipped with projections $p_{1},p_{2}$ to ${\operatorname{Bun}}_{G}$, and by
the isomorphism

$\alpha_{v^{\prime},v}:p_{1}^{*}\omega^{1/2}\rTo{\sim}p_{2}^{*}\omega^{1/2}\otimes\omega_{p_{2}}\otimes
L_{\lambda,v}\otimes L_{\mu,v^{\prime}}$

induced by 5.3. The composition $T_{v^{\prime}}^{\mu}\circ T_{v}^{\lambda}$ is
given by the correspondence $Z(v,v^{\prime})$ which is naturally identified
with $Z(v^{\prime},v)$ and some isomorphism $\alpha_{v,v^{\prime}}$ between
the same line bundles. Note that the composed operators depend only on the
absolute value of the isomorphisms $\alpha_{v,v^{\prime}}$ and
$\alpha_{v^{\prime},v}$. We claim that in fact

$\alpha_{v^{\prime},v}=\pm\alpha_{v,v^{\prime}},$

which implies the required commutation.

Indeed, to prove this we can work over an algebraically closed field. A priori
we have $\alpha_{v^{\prime},v}=f_{v^{\prime},v}\cdot\alpha_{v,v^{\prime}}$ for
some invertible function $f_{v,v^{\prime}}$ on
$Z(v,v^{\prime})=Z(v^{\prime},v)$. Note that by construction, we have

$f_{v,v^{\prime}}\cdot f_{v^{\prime},v}=1.$ (5.4)

Let us work over a fixed connected component of ${\operatorname{Bun}}_{G}$ via
$p_{2}$. Then all global functions on this component are constant, hence, all
global functions on the corresponding component of $Z(v,v^{\prime})$ are
constant, so the restriction of $f_{v,v^{\prime}}$ is constant. Now let us
vary the points $v$ and $v^{\prime}$, so that $f_{v,v^{\prime}}$ will become
an invertible function on the complement of the diagonal in $C\times C$. The
relation (5.4) shows that in fact, $f_{v,v^{\prime}}$ is regular on $C\times
C$, hence constant. Now the same relation shows that $f_{v,v^{\prime}}=\pm 1$.
∎

###  5.2. Hecke operators over $K$, over $O$, and over $O/{\mathfrak{m}}^{n}$

Now, assume $C$ is a smooth proper curve over $O$. Then the stack
${\operatorname{Bun}}_{G}$ is also defined over $O$, and we can consider the
spaces

$\mathcal{W}_{O}:=\mathcal{S}({\operatorname{Bun}}_{G}(O),|\omega^{1/2}|)=\mathcal{S}({\operatorname{Bun}}_{G}(O)),\
\
\mathcal{W}_{N}:=\mathcal{S}({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N})),$

where in the second case we consider finitely supported functions on the
isomorphism classes of $G$-bundles on
$C_{n}:=C\times_{\operatorname{Spec}(O)}\operatorname{Spec}(O/{\mathfrak{m}}^{n})$.
We have natural maps constructed in Sec. 2.2,

$\mathcal{W}_{N}\to\mathcal{W}_{O}\to\mathcal{W},$ (5.5)

where the image of map $\mathcal{W}_{O}\to\mathcal{W}$ consists of densities
supported on ${\operatorname{Bun}}^{O}_{G}$ (see (Prop. 2.11).

Given a point $v\in C(O)=C_{K}(K)$, we have homomorphisms of local Hecke
algebras

${\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C_{K},v},crit}\rTo{\nu_{K,O}}{\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C,v},I_{v}}\rTo{\nu_{O,O/{\mathfrak{m}}^{N}}}{\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C_{n},\overline{v}}},$
(5.6)

where $\overline{v}$ is the reduction of $v$ modulo ${\mathfrak{m}}$. We also
have three algebra actions:

  * •

${\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C_{K},v},crit}$-action on
$\mathcal{W}$;

  * •

${\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C,v},I_{v}}$-action on
$\mathcal{W}_{O}$;

  * •

${\mathcal{H}}_{G,\hat{{\mathcal{O}}}_{C_{n},\overline{v}}}$-action on
$\mathcal{W}_{N}$.

We claim that the maps (5.5) are compatible with these Hecke actions via
homomorphisms (5.6). Indeed, for $\nu_{K,O}$ this follows immediately from
Lemma 3.8. For $\nu_{O,O/{\mathfrak{m}}^{N}}$ the compatibility of the Hecke
actions on coinvariants was explained in Sec. 3.5, as part of the definition.

######  Definition 5.4.

For $\lambda\in\Lambda^{+}$, let us set

$h^{\lambda}_{O}:=\nu_{K,O}(h^{\lambda}),\ \
h^{\lambda}_{N}:=\nu_{O,O/{\mathfrak{m}}^{N}}\nu_{K,O}(h^{\lambda}).$

We denote by $T_{v,O}^{\lambda}$ (resp.,
$T_{v,O/{\mathfrak{m}}^{N}}^{\lambda}$) the operator on ${\mathcal{W}}_{O}$
(resp., ${\mathcal{W}}_{N}$) given by the action of $h^{\lambda}_{O}$ (resp.,
$h^{\lambda}_{N}$) associated with $v\in C(O)$.

######  Lemma 5.5.

Assume $\lambda$ is minuscule. Then for any point $v\in C(O)$, the operator
$T_{v,O/{\mathfrak{m}}^{N}}^{\lambda}$ coincides (up to rescaling) with the
operators on $\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{n}))$ coming from
the Hecke correspondence $Z_{\lambda,v}$.

###### Proof.

This follows from Proposition 2.12. ∎

###  5.3. Commuting Hecke operators

Now assume that $G$ is a split reductive group over
$\operatorname{\mathbb{Z}}$, such that $[G,G]$ is simple and simply connected.
We also assume that the characteristic $p$ of $k=O/{\mathfrak{m}}$ satisfies
$p>\max(a_{{\mathfrak{g}}^{\prime}},4(h_{{\mathfrak{g}}^{\prime}}-1))$, where
${\mathfrak{g}}^{\prime}=[{\mathfrak{g}},{\mathfrak{g}}]$.

Let $C_{O/{\mathfrak{m}}^{N}}$ be a smooth proper curve over
$O/{\mathfrak{m}}^{N}$, $C_{0}$ the corresponding curve over
$k=O/{\mathfrak{m}}$. For a $G$-bundle $P$ over $C_{O/{\mathfrak{m}}^{N}}$,
and a point $p\in C_{O/{\mathfrak{m}}^{N}}(O/{\mathfrak{m}}^{N})$, we have a
map

$h_{p}:\operatorname{Gr}_{G}(O/{\mathfrak{m}}^{N})\to{\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}):x\mapsto
P(x)$

associating with a point of $\operatorname{Gr}_{G}$ the corresponding Hecke
transform $P(x)$ at $p$. Note that for every point
$x\in\operatorname{Gr}_{G}$, we have an isomorphism $\alpha_{x}:P\to P(x)$ on
$C_{N}-p$. Let us denote by
$\operatorname{Gr}_{n}\subset\operatorname{Gr}_{G}$ the subscheme of $x$ such
that $\alpha_{x}$ and $\alpha_{x}^{-1}$ have poles of order $\leq n$ at $p$
(in the sense of Definition 4.4).

######  Lemma 5.6.

Let $P$ be a $G$-bundle over $C_{O/{\mathfrak{m}}^{N}}$, such that the
corresponding $G$-bundle over $C_{0}$ is nice of level $2n$ at
$p\mod{\mathfrak{m}}$. Then

  1. (1)

the map
$h_{p}|_{\operatorname{Gr}_{n}}:\operatorname{Gr}_{n}(O/{\mathfrak{m}}^{N})\to{\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N})$
is injective;

  2. (2)

for every $x\in\operatorname{Gr}_{n}(O/{\mathfrak{m}}^{N})$, one has
$\operatorname{Aut}(P(x))=Z_{G}(O/{\mathfrak{m}}^{N})$.

###### Proof.

(1) For every pair $x,x^{\prime}\in\operatorname{Gr}_{n}$, the automorphism
$\alpha_{x}^{-1}\alpha_{x^{\prime}}$ of $P|_{C_{N}-p}$ has a pole of order
$\leq 2n$ at $p$. Then by Lemma 4.10, there exists an element $z\in
Z_{G}(O/{\mathfrak{m}}^{N})$ such that $\alpha_{x}^{-1}\alpha_{x^{\prime}}=z$.
Hence, $\alpha_{x^{\prime}}=z\alpha_{x}=\alpha_{zx}=\alpha_{x}$, where we use
the fact that $Z_{G}(O/{\mathfrak{m}}^{N})\subset
Z_{G}(O/{\mathfrak{m}}^{N}[\\![t]\\!])$ acts trivially on
$\operatorname{Gr}_{G}$. Hence, $x^{\prime}=x$.

(2) Let $\phi$ be an automorphism of $P(x)$. Then
$\alpha_{x}^{-1}\phi\alpha_{x}$ is an automorphism of $P|_{C-p}$ with a pole
of order $\leq 2n$ at $p$. Hence, $\alpha_{x}^{-1}\phi\alpha_{x}=z\in
Z_{G}(O/{\mathfrak{m}}^{N})$, which implies that
$\phi=\alpha_{x}z\alpha_{x}^{-1}=z$. ∎

Now let $C_{O}$ be a smooth proper curve over $O$, $C$ (resp., $C_{0}$) the
corresponding curve over $K$ (resp., $k$). Let $v_{1},v_{2}\in C(K)=C_{O}(O)$
be points such that

$v_{1}\mod{\mathfrak{m}}=v_{2}\mod{\mathfrak{m}}=v_{0}\in C(k),$

and let $T_{1}$ and $T_{2}$ be the Hecke operators on
$\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$ associated with $v_{1}$
and $v_{2}$ and some dominant coweights $\lambda_{1},\lambda_{2}$.

By Lemma 4.1, our assumptions on $G$ and on the characteristic of $k$ imply
that the assumption ($\star$) from Sec. 4.5 is satisfied. Hence, by Lemma
4.11(ii), every $G$-bundle over $C_{O}$ admits a $B$-structure, and so is
Zariski locally trivial.

######  Lemma 5.7.

There exists $n$ depending only on the coweights $\lambda_{1},\lambda_{2}$,
such that if there exists a $G$-bundle $P_{0}$ on $C_{k}$, which is nice of
level $n$ at $v_{0}$, then the operators $T_{1}$ and $T_{2}$ on
$\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$ commute.

###### Proof.

Let us denote by ${\mathcal{U}}\subset{\operatorname{Bun}}_{G}$ the open
substack of $G$-bundles $P$ with no infinitesimal automorphisms (i.e., such
that ${\mathfrak{g}}_{P}$ has no global sections).

Step 1. The map
$r_{n}:\SS({\mathcal{U}}(O/{\mathfrak{m}}^{N}))\to\SS({\operatorname{Bun}}_{G}(K),|\omega|^{1/2})$
is injective. Indeed, this map factors through
$\SS({\mathcal{U}}(K),|\omega|^{1/2})$. Now we observe that for an open
substack $X/\operatorname{GL}_{m}\subset{\mathcal{U}}$ of finite type, all
points of $X$ have finite stablizers, hence, the orbits are closed. Thus, we
have well defined functionals on $\SS({\mathcal{U}}(K),|\omega|^{1/2})$ given
by integration over orbits, which recovers the values of a function in
$\SS({\mathcal{U}}(O/{\mathfrak{m}}^{N}))$.

Step 2. Let $P_{0}$ be a $G$-bundle on $C_{0}$, which is nice of level $n\gg
0$ at $v_{0}$. Let us consider the subspace
$V_{P_{0}}\subset\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$
consisting of functions supported on $G$-bundles over $O/{\mathfrak{m}}^{N}$
that reduce to $P_{0}$. Then we claim that

$T_{1}T_{2}\phi=T_{2}T_{1}\phi$

for any $\phi\in V_{P_{0}}$. Indeed, for any $G$-bundle on $C_{0}$ in the
support of $T_{1}T_{2}\delta_{P_{0}}$ and $T_{2}T_{1}\delta_{P_{0}}$ belongs
to ${\mathcal{U}}(k)$. Hence $T_{1}T_{2}\phi$ and $T_{2}T_{1}\phi$ belong to
$\SS({\mathcal{U}}(O/{\mathfrak{m}}^{N}))$. Now the assertion follows from
Step 1 and the fact that the corresponding Hecke operators $\widetilde{T}_{1}$
and $\widetilde{T}_{2}$ on $\SS({\operatorname{Bun}}_{G}(K),|\omega|^{1/2})$
commute (by Theorem 5.3).

Step 3. Let ${\mathcal{H}}_{\neq v_{0}}$ denote the (non-commutative)
subalgebra of Hecke operators on
$\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$ transforming $G$-bundles
at effective divisors away from $v_{0}$. Then $T_{1}$ and $T_{2}$ commute with
${\mathcal{H}}_{\neq v_{0}}$ and ${\mathcal{H}}_{\neq
v_{0}}V_{P_{0}}=\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$. Indeed,
for any $G$-bundle $P$ over $C_{O/{\mathfrak{m}}^{n}}$ there exists a Zariski
open neighborhood of $v_{0}$ such that $P|_{C_{0}}$ is isomorphic to $P_{0}$.
This implies that $\delta_{P}\in{\mathcal{H}}_{\neq v_{0}}V_{P_{0}}$.

Finally, we conclude that $T_{1}$ and $T_{2}$ commute on ${\mathcal{H}}_{\neq
v_{0}}V_{P_{0}}=\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$. ∎

###### Proof of Theorem 3.10.

We need to prove that for any pair of ideals $I,I^{\prime}\subset
O[\\![t]\\!]$, complementary to $O$, and a pair of coweights
$\lambda,\lambda^{\prime}\in\Lambda^{+}$, the corresponding elements
$h,h^{\prime}$ in the small Hecke algebra
${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$ commute.

By Corollary 4.9, we can choose a pointed curve $(C_{0},v_{0})$ over $k$ and a
nice $G$-bundle $P_{0}$ over $C_{0}$ of sufficiently large level with respect
to $v_{0}$. Next, we choose a lifting of $(C_{0},P_{0})$ to a similar data
$(C_{N},p_{N},P_{N})$ over $O/{\mathfrak{m}}^{N}$, and then to $(C,P)$ over
$O$. Let $v,v^{\prime}\in C(O)$ be $O$-points reducing to $v_{0}$,
corresponding to the ideals $I,I^{\prime}$. Then by Lemma 5.7, the operators
$h$ and $h^{\prime}$ on $\SS({\operatorname{Bun}}_{G}(O/{\mathfrak{m}}^{N}))$
commute. Finally, we use Lemma 5.6 to deduce that they commute in the local
Hecke algebra ${\mathcal{H}}^{sm}_{G,O/{\mathfrak{m}}^{N}[\\![t]\\!]}$. ∎

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