#  $S=T$ for Shimura Varieties and $p$-adic Shtukas

Zhiyou Wu

###### Abstract.

We prove the $S=T$ conjecture proposed by Liang Xiao and Xinwen Zhu in [1],
making use of Scholze’s theory of diamonds and v-stacks and Fargues-Scholze’s
geometric Satake equivalence. We deduce the Eichler-Shimura relation for
Shimura varieties of Hodge type.

###### Contents

  1. 1 Introduction
  2. 2 Basic Definitions
    1. 2.1 $p$-adic Shtukas and Hecke stacks
    2. 2.2 Witt vector Shtukas and Hecke stacks
  3. 3 Excursion operators
    1. 3.1 Geometric Satake
    2. 3.2 Hecke Correspondence
    3. 3.3 Creation correspondence
    4. 3.4 Annihilation correspondence
    5. 3.5 Partial Frobenius
    6. 3.6 Excursion Operators
  4. 4 S=T for Shtukas
  5. 5 S=T for Shimura varieties
  6. A Comparison between algebraic stacks and v-stacks
    1. A.1 Comparison of cohomology
    2. A.2 Comparison of cohomological correspondences
    3. A.3 Truncated Witt vector Shtukas

##  1\. Introduction

Liang Xiao and Xinwen Zhu have raised a question on S=T for Shimura varieties
in [1], which states that excursion operators on the compactly supported
cohomology is the same as Hecke operators, and is an analogue for Vincent
Lafforgue’s S=T theorem in [2]. This is a fundamental question which can be
interpreted as a local-global compatibility in a categorical formulation of
Langlands correspondence, see [3]. Moreover, it has consequences on Eichler-
Shimura relation for compactly supported cohomology of Shimura varieties of
Hodge type.

Strictly speaking, there are two versions of S=T in mixed characteristics
(while there is only one in equal characteristic). The first is S=T for mixed
characteristic Shtukas, and the second is the one for Shimura varieties. In
[1], Xiao-Zhu have proved S=T for Witt vector Shtukas, and zero dimensional
Shimura varieties. However, even their version for Shtukas is strictly weaker
than Lafforgue’s S=T theorem in equal characteristics, and is again zero
dimensional in nature. More precisely, they have proved that the excursion
operator associated to a representation of the Langlands dual group acts on
the cohomology of zero dimensional Shtukas by the function associated to the
representation through classical Satake. While Lafforgue’s S=T theorem states
that the Hecke correspondence associated to a representation $V$ is the same
as the excursion operator associated to $V$ on any Shtukas.

The reason Xiao-Zhu only prove their results for zero dimensional objects is
the same for both Shimura varieties and Shtukas. Namely, they work with Witt
vector affine Grassmannian and Shtukas, which is over a characteristic p base,
so they can only talk about the special fiber of Shimura varieties and
Shtukas. However, Hecke operators at $p$ are only naturally visible (as etale
correspondence) in characteristic 0, or at least prime to $p$. The usual
treatment is to take the closure of the Hecke correspondence in an integral
model and reduce to characteristic p, this usually creates a lot of
degeneration, as the classical example $T_{p}\equiv Frob+V$ for modular curves
shows. This problem only goes away in zero dimension, where the Hecke
operators ”flatten” to characteristic $p$. Since Xiao-Zhu can only work with
the special fiber, they can not even define Hecke operators for general Witt
vector Shtukas. Indeed, they only state their conjecture and results for
cohomology, which is nothing but passing to zero dimension, where they have
the obvious notion of Hecke operators, namely the action of a function.

There is a natural way to proceed. Namely the theory of diamonds and v-stacks
developed by Scholze allows us to talk about mixed characteristic Shtukas over
generic fibers, so we can define Hecke operators for Shtukas. Further, we have
integral models of Shtukas using this framework, whose special fiber is
essentially the Witt vector Shtukas considered by Xiao-Zhu, and the proof in
this article is in some sense a nearby cycle construction relating cohomology
in characteristic $0$ to cohomology in characteristic $p$.

More precisely, the excursion operators on Shtukas are defined by the
composition of creation, partial Frobenius and annihilation correspondences,
each of which are essentially pullback of cohomological correspondences on
Hecke stacks. The key to the construction of these correspondences on the
Hecke Stacks is the geometric Satake established by Fargues and Scholze in
[4], which can be viewed as a globalization of the Witt vector geometric
Satake established by Xinwen Zhu. We can perform this construction on the
integral models of Shtukas, whose restriction to the special fiber are the
excursion operators for Witt vector Shtuaks constructed by Xiao-Zhu.

On the other hand, we can prove that the excursion operators on the generic
fiber is the same as the Hecke operators. We follow the same strategy as
Vincent Lafforgue’s proof in [2], which is basically to move legs to reduce to
the zero dimensional case that has already been established by Xiao-Zhu.

Now we have the $S=T$ for $p$-adic Shtukas, and we want to deduce the
corresponding $S=T$ for Shimura varieties. Recall that excursion operators for
the special fiber of Shimura varieties are defined as the pullback of the
excursion operators for Witt vector Shtukas through a crystalline period map
from Shimura varieties to Shtukas, see [1]. We generalise this construction to
a period map from integral models of Shimura varieties to (integral) $p$-adic
Shtukas. The special fiber of this map is essentially the old crystalline
period map, while the generic fiber is closely related to (a quotient of)
Hodge-Tate period map in [5] or [6]. The construction of this map makes
essential use of the classification of $p$-divisible groups over integral
perfectoid rings in terms of Breuil-Kisin-Fargues modules in appendix of
lecture 17 in [7]. Now the pullback of the excursion operators on $p$-adic
Shtukas along this period map produces the excursion operators on Shimura
varieties, and the $S=T$ for Shtukas pullback to the $S=T$ for Shimura
varieties.

The integral period map also appears in the recent work [8] of Georgios Pappas
and Michael Rapoport. Indeed, they constructed it more generally for Shimura
varieties with parohoric reduction, while we work only with the good reduction
case. Moreover, they construct it on the full adic space associated to the
integral model of Shimura varieties, while we work only with the good
reduction locus.

The Eichler-Shimura relation for Shimura varieties of Hodge type has also
recently been established by Si Ying Lee in [9]. We provide an extensive
remark on the comparison between the two approaches to congruence relations,
which we believe to help clarify the ideas underlying this paper.

The strategy of proof in [9] is similar to that of Faltings-Chai ([10]) ,
Torsten Wedhorn ([11]) and many others. The key to all approaches to
congruence relations is to detect the degeneration of $p$-isogenies between
abelian varieties from characteristic $0$ to characteristic $p$ using linear
algebra data coming from cohomology theory, and the difference lies in which
cohomology theory to use. The approach in [10] is to use étale cohomology to
detect the effect of specialization of isogenies. For ordinary reduction, this
is easy, since the specialization just adds a canonical filtration on the Tate
module, and the effect on isogenies is simply to preserve this filtration.
Hence their proof makes crucial use of the ordinary abelian varieties, which
restricts the applicability of the method, namely one has to add a condition
on the density of ordinary isogeny locus.

To get rid of this condition, it is necessary to look at the specialization of
general abelian varieties and their isogenies, and detect them using linear
algebra data. This is exactly what $p$-adic Hodge theory could help us. In the
ordinary case, $p$-adic Hodge theory is hidden since the comparison already
takes places at the rational level, namely the canonical filtration. In
general, the period rings manifest themselves. However, it is hard to find a
direct linear algebra description of the specialization as in the ordinary
case. Indeed, it is well-known that the relation between the Hodge-Tate period
of $p$-divisible groups over $\mathcal{O}_{\mathbb{C}_{p}}$ and the Dieudonnie
module of its special fiber is mysterious. The way we proceed is exactly the
way Scholze describes the mysterious relation in [7] using intermediate
integral objects, namely $p$-adic Shtukas.

The lack of explicit linear algebra models in the previous approach is
remedied by the high power functorial geometric models, which puts us in the
framework of geometric Langlands. In particular, the degeneration is described
by the comparison theorems in $p$-adic Hodge, in the disguise of the Fargues
Fontaine curve (or closely related objects), and the lack of explicit
identification is remedied by the functoriality of Fargues Fontaine curve,
which significantly mechanizes the situation by putting us in the framework of
diamonds and v-stacks, so we can mimic what has been done in geometric
Langlands. More precisely, the degeneration is reflected by the behaviour of
excursion operators when the legs of Shtukas collide, somehow the excursion
operators ”flatten” the situation by allowing extra freedom on adding legs and
intermediate modifications, where the magic ultimately boils down to the
fusion product in geometric Satake.

We learned from Peter Scholze after initial progress on this project that
Xinwen Zhu also has a proof of $S=T$ conjecture for some time, which in
principle should be similar to the proof presented here.

We now briefly describe the content of this paper. In section 2, we provide
definitions for all the spaces that we will use. In section 3, we construct
the excursion operators for Shtukas using geometric Satake. In section 4, we
prove the $S=T$ for Shtukas. In the last section, we construct the integral
period map from Shimura varieties to Shtukas, and prove the $S=T$ for Shimura
varieties. We collect some results on the comparison between cohomology of
algebraic stacks and v-stacks, which is used in the article to compare the
construction given here with the construction in [1].

### Convention

We fix a prime $p$ throughout. We will work with the theory of v-stacks and
diamonds as developed in [12], by definition they are stacks over the category
of characteristic $p$ perfectoid spaces with v-topology. Unlike the usual
notation, a prestack is meant to be a groupoid valued functor from the
category of characteristic $p$ perfectoid spaces, without separatedness
condition.

We follow the notation in [12] to denote $X^{\diamond}$ the diamond associated
to an adic space $X$ over $\mathbb{Z}_{p}$, which parameterizes untilts
$S^{\sharp}$ of $S$ together with a map $S^{\sharp}\rightarrow X$. When
$X=\text{Spa}(A,A^{+})$, we denote as usual
$X^{\diamond}=\text{Spd}(A,A^{+})$. When $A^{+}$ is clear from the situation,
we abbreviate the notation by writing $\text{Spd}(A)=\text{Spd}(A,A^{+})$. For
example, when $K$ is a non-archimedean field, we write
$\text{Spd}(K)=\text{Spd}(K,\mathcal{O}_{K})$.

Let $X$ be an algebraic stack or scheme locally of finite type over a discrete
valuation ring $\mathcal{O}$ with perfect residue field, as in appendix A.1,
we write $X^{\diamond}$ (resp. $X^{\diamond\diamond}$) for the stackification
of the prestack sending a characteristic $p$ perfectoid space
$\text{Spa}(R,R^{+})$ to untilts $\text{Spa}(R^{\sharp},R^{\sharp,+})$ over
$\mathcal{O}$ together with a map $\text{Spec}(R^{\sharp,+})\rightarrow X$
(resp. $\text{Spec}(R^{\sharp})\rightarrow X$) of schemes over $\mathcal{O}$.
When $X$ is a scheme, this deviates from the notation in [12] section 27,
where they denote $X^{\diamond}$ for our $X^{\diamond\diamond}$.

For $E$ an extension of $\mathbb{Q}_{p}$, $Spd(E)$ is the diamond
parameterizing characteristic 0 untilts with a morphism to
$\text{Spa}(E,\mathcal{O}_{E})$, and $Spd(\mathcal{O}_{E})$ is the v-sheaf
parameterizing all untilts with a morphism to
$\text{Spa}(\mathcal{O}_{E},\mathcal{O}_{E})$.

Following the notation of [7], we denote

$\mathcal{Y}_{[0,\infty)}(S):=\text{Spa}(W(R^{+}),W(R^{+}))\setminus
V([\omega])$

for $S=\text{Spa}(R,R^{+})$ a characteristic $p$ affinoid perfectoid space
with topological nilpotent units $\omega$, and the general $S$ by gluing. We
have a natural identification

$\mathcal{Y}_{[0,\infty)}(S)^{\diamond}=S\times Spd(\mathbb{Z}_{p})$

Then an untilt of $S$ is equivalent to a section of
$\mathcal{Y}_{[0,\infty)}(S)^{\diamond}\rightarrow S$, giving rise to an
effective Cartier divisor on $\mathcal{Y}_{[0,\infty)}(S)$.

We denote

$\mathcal{X}_{\text{FF},S}:=\mathcal{Y}_{(0,\infty)}(S)/Frob_{S}^{\mathbb{Z}}$

the relative Fargues-Fontaine curve with respect to $S$, where

$\mathcal{Y}_{(0,\infty)}(S):=\text{Spa}(W(R^{+}),W(R^{+}))\setminus
V(p[\omega])$

for $S=\text{Spa}(R,R^{+})$. We note that $\mathcal{Y}_{(0,\infty)}(S)$ is an
open subspace of $\mathcal{Y}_{[0,\infty)}(S)$, the generic fiber of
$\mathcal{Y}_{[0,\infty)}(S)$ over $\mathbb{Z}_{p}$, whose analytification can
be identified with $S\times\text{Spd}(\mathbb{Q}_{p})$.

Now let $S^{\sharp}=\text{Spa}(R^{\sharp},R^{\sharp,+})$ be an untilt of $S$,
we denote

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}}:=W(R^{+})[\frac{1}{[\omega]}]^{\wedge}_{\xi},$

i.e. the $\xi$-adic completion of $W(R^{+})[\frac{1}{[\omega]}]$, where
$\xi\in W(R^{+})$ is a generator of the kernel of $W(R^{+})\rightarrow
R^{\sharp,+}$, and $\omega\in R^{+}$ is a pseudouniformizer. Note that

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}}=\mathbb{B}_{\text{dR}}^{+}(R^{\sharp})$

if $R^{\sharp}$ is of characteristic $0$, and

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}}=W(R)$

if $S^{\sharp}=S$, where $\mathbb{B}_{\text{dR}}^{+}(R^{\sharp})$ is the
$\xi$-adic completion of $W(R^{+})[\frac{1}{p}]$ with $\xi\in W(R^{+})$ being
a generator of the kernel of the canonical map
$W(R^{+})[\frac{1}{p}]\rightarrow R^{\sharp}$.

When we have finite many untilts
$S_{i}^{\sharp}=\text{Spa}(R^{\sharp}_{i},R^{\sharp,+}_{i})$ of $S$ (possibly
not all distinct), we denote

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),\sum
S_{i}^{\sharp}}:=W(R^{+})[\frac{1}{[\omega]}]^{\wedge}_{\xi},$

where $\xi=\prod\xi_{i}$ with $\xi_{i}$ a generator of the kernel of
$W(R^{+})\rightarrow R_{i}^{\sharp,+}$.

We will use frequently the formal properties of cohomological correspondences.
Let $X,Y$ and $Z$ be small v-stacks which forms a diagram

${Z}$${X}$${Y.}$$\scriptstyle{q}$$\scriptstyle{p}$

Let $A\in D_{\text{ét}}(X,\Lambda)$ and $B\in D_{\text{ét}}(Y,\Lambda)$, we
denote

$\mathscr{C}:(X,A)\longrightarrow(Y,B)$

a cohomological correspondence supported on $Z$ to be a map

$\mathscr{C}:p^{*}A\longrightarrow q^{!}B.$

Let us given a commutative diagram

${\mathcal{Z}}$${\mathcal{X}}$${\mathcal{Y}}$${Z}$${X}$${Y.}$$\scriptstyle{f}$$\scriptstyle{q^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{q}$$\scriptstyle{p}$

of small $v$-stacks. Let $\mathscr{C}:(X,A)\rightarrow(Y,B)$ be a
cohomological correspondence supported on $Z$, then we can pullback the
$\mathscr{C}$ along the diagram to obtain a cohomological correspondence

$f^{*}\mathscr{C}:(\mathcal{X},h^{*}A)\longrightarrow(\mathcal{Y},g^{*}B)$

supported on $\mathcal{Z}$, if either the right square of the diagram is
Cartesian or $f,g$ and $h$ are cohomologically smooth, see [1] appendix A.2.

On the other hand, let $\mathscr{C}:(\mathcal{X},A)\rightarrow(\mathcal{Y},B)$
be a cohomological correspondence supported on $\mathcal{Z}$, then we can
!-pushforward $\mathscr{C}$ to obtain a cohomological correspondence

$Rf_{!}\mathscr{C}:(X,Rh_{!}A)\longrightarrow(Y,Rg_{!}B)$

supported on $Z$, if $p^{\prime}$ and $p$ are proper , see [1] appendix A.2.

Moreover, we can compose cohomological correspondence, and pullback and
!-pushforward commute with composition.

### Acknowledgments

I would like to thank Peter Scholze for his constant help and encouragement on
this project. I am particularly grateful for the early access of his
fundamental work [4] with Fargues. I would like also to thank Xinwen Zhu for
comments on the initial draft and sharing ideas with me. I am grateful to Max
Planck Institute for Mathematics in Bonn for its hospitality and financial
support.

##  2\. Basic Definitions

Let $G$ be a reductive group over $\mathbb{Q}_{p}$, together with a reductive
model $\mathcal{G}$ over $\mathbb{Z}_{p}$. Then $G$ is unramified, so split
over $L:=W(\overline{\mathbb{F}_{p}})[\frac{1}{p}]$. Let $\mu$ be a conjugacy
class of cocharacters $\mathbb{G}_{m}\rightarrow
G_{\overline{\mathbb{Q}_{p}}}$, whose field of definition is $E$, which is
contained in $L$.

###  2.1. $p$-adic Shtukas and Hecke stacks

######  Definition 2.1.

The moduli space of Shtukas associated to $\mathcal{G}$ and $\mu$ is the
prestack

$Sht_{\mu}$

over $Spd(\mathcal{O}_{L})$ whose value at a characteristic $p$ perfectoid
space $S$ is the groupoid of the data of

$\bullet$ a $\mathcal{G}$-torsor $\mathcal{P}$ over
$\mathcal{Y}_{[0,\infty)}(S)$,

$\bullet$ an (not necessarily of characteristic 0) untilt
$S^{\sharp}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$ of $S$ defined over
$\mathcal{O}_{L}$, and

$\bullet$ an isomorphism

$\varphi_{\mathcal{P}}:\mathcal{P}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus
S^{\sharp}}\cong\text{Frob}_{S}^{*}\mathcal{P}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus
S^{\sharp}}.$

that is meromorphic along the Cartier divisor
$S^{\sharp}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$, and bounded by $\mu$
in the sense that for any geometric rank 1 point $\mathbb{C}$ of $S$, and
trivializations of $\text{Frob}_{\mathbb{C}}^{*}\mathcal{P}$ and $\mathcal{P}$
over
$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(\mathbb{C}),\mathbb{C}^{\sharp}}[\frac{1}{\xi}]$,
the ”punctured formal neighborhood of $\mathbb{C}^{\sharp}$”,
$\varphi_{\mathcal{P}}$ lies in
$\mathcal{P}(\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(\mathbb{C}),\mathbb{C}^{\sharp}})\mu(t)\mathcal{P}(\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(\mathbb{C}),\mathbb{C}^{\sharp}})$,
where
$t\in\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(\mathbb{C}),\mathbb{C}^{\sharp}}$
is a uniformizer. Note that this is a pointwise condition on $S$.

The isomorphism is defined to be an isomorphism of the torsors that preserves
$\varphi_{\mathcal{P}}$.

######  Remark 2.2.

Although we work with the base $\mathcal{O}_{L}$, the definition of Shtukas
really depends on the group $\mathcal{G}$ over $\mathbb{Z}_{p}$ rather than
its base change to $\mathcal{O}_{L}$, as otherwise the Frobenius pullback
$\text{Frob}_{S}^{*}\mathcal{P}$ does not make sense (it will be a torsor with
respect to a Frobenius twist of the structure group over $\mathcal{O}_{L}$).

######  Remark 2.3.

The definition given here is close to the classical function field
counterpart, which is slightly different from the ones in [7] in that we do
not fix a level structure, nor do we fix the type of the Shtuka in $B(G)$. The
treatment in [7] is to emphasize the way Shtukas generalize Rapoport-Zink
spaces, whereas we want to mimic the treatment of Vincent Lafforgue.

We will also make use of Shtukas with several legs.

######  Definition 2.4.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. Let $\\{I_{1},\cdots,I_{k}\\}$ be a
partition of $I$. The moduli space of Shtukas associated to $G$,
$\mu_{\bullet}$ and $\\{I_{1},\cdots,I_{k}\\}$ is the prestack

$Sht_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$

over $Spd(\mathcal{O}_{L})^{I}$ whose value at a characteristic $p$ perfectoid
space $S$ is the groupoid of the data of

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k}$ over
$\mathcal{Y}_{[0,\infty)}(S)$,

$\bullet$ (not necessarily of characteristic 0) untilts
$S^{\sharp}_{i}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$ of $S$ defined over
$\mathcal{O}_{L}$ indexed by $i\in I$, and

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}}\cong\mathcal{P}_{i+1}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}}$

with $\mathcal{P}_{k+1}=\text{Frob}_{S}^{*}\mathcal{P}_{1}$, which are
meromorphic along the Cartier divisor $\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$, and
bounded by $\underset{S^{\sharp}_{j}=S^{\sharp}}{\sum}\mu_{j}$ at every
geometric point of $S^{\sharp}$.

When the partition is trivial, we drop the upper script to simplify the
notation, i.e.

$Sht_{\mu_{\bullet}}:=Sht_{\mu_{\bullet}}^{(I)},$

which parametrizes

$\mathcal{P}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{i\in
I}{\cup}S^{\sharp}_{i}}\cong\text{Frob}_{S}^{*}\mathcal{P}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{i\in
I}{\cup}S^{\sharp}_{i}}.$

The following variant defines a correspondence between Shtukas, and will be
very important for our purpose.

######  Definition 2.5.

Let $\mu,\mu^{\prime}$ and $\nu$ be conjugacy classes of cocharacters
$\mathbb{G}_{m}\rightarrow G_{\overline{\mathbb{Q}_{p}}}$, with $E$ and $F$
the field of definition of $\mu$ and $\mu^{\prime}$ respectively. We define

$Sht^{\nu}_{\mu|\mu^{\prime}}$

to be the prestack with value at a characteristic $p$ perfectoid space $S$
being the groupoid of the data of

$\bullet$ $\mathcal{P}\in Sht_{\mu}(S)$ and $\mathcal{P}^{\prime}\in
Sht_{\mu^{\prime}}(S)$,

$\bullet$ a modification bounded by $\nu$ (in the sense similar to the above)

$\gamma:\mathcal{P}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}|_{\mathcal{Y}_{(0,\infty)}(S)}$

at the charctersictic $p$ untilt $S\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$
as Shtukas, i.e. the following diagram is commutative

${\mathcal{P}}$${\text{Frob}_{S}^{*}\mathcal{P}}$${\mathcal{P}^{\prime}}$${\text{Frob}_{S}^{*}\mathcal{P}^{\prime}}$$\scriptstyle{\varphi_{\mathcal{P}}}$$\scriptstyle{\gamma}$$\scriptstyle{\text{Frob}_{S}^{*}\gamma}$$\scriptstyle{\varphi_{\mathcal{P}^{\prime}}}$

with the notation that dashed arrows denote generically defined modifications.

We denote $Sht_{\mu|\mu^{\prime}}$ the union of $Sht^{\nu}_{\mu|\mu^{\prime}}$
for all $\nu$.

Similarly, we have the version for Shtukas with several legs.

######  Definition 2.6.

Let $I,J$ be finite sets, and $\mu_{\bullet},\mu^{\prime}_{\bullet}$ be
collections of dominant cocharacters of $G$ indexed by $I$ and $J$
respectively. Let $\\{I_{1},\cdots,I_{k}\\}$ and $\\{J_{1},\cdots,J_{l}\\}$ be
partitions of $I$ and $J$. Let $\nu$ be a dominant cocharacter of $G$. Then we
define

$Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$

to be the prestack with value at a characteristic $p$ perfectoid space $S$
being the groupoid of the data of

$\bullet$
$\\{\mathcal{P}_{1},\cdots,\mathcal{P}_{k},\varphi_{1},\cdots,\varphi_{k}\\}\in
Sht^{(I_{1},\cdots,I_{k})}_{\mu_{\bullet}}(S)$ and
$\\{\mathcal{P}^{\prime},\cdots,\mathcal{P}_{l}^{\prime},\varphi_{1}^{\prime},\cdots,\varphi_{l}^{\prime}\\}\in
Sht^{(J_{1},\cdots,J_{l})}_{\mu^{\prime}_{\bullet}}(S)$,

$\bullet$ a modification bounded by $\nu$ (in the sense similar to the above)

$\gamma:\mathcal{P}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}|_{\mathcal{Y}_{(0,\infty)}(S)}$

at the charctersictic $p$ untilt $S\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$
as Shtukas, i.e. the following diagram is commutative

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\mathcal{P}_{1}^{\prime}}$${\mathcal{P}_{2}^{\prime}}$${\cdots\cdots}$${\mathcal{P}^{\prime}_{l}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}^{\prime}}$$\scriptstyle{\gamma}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$$\scriptstyle{\text{Frob}_{S}^{*}\gamma}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\varphi^{\prime}_{l-1}}$$\scriptstyle{\varphi_{l}^{\prime}}$

with the usual notation that dashed arrows denote generically defined
modifications.

When the partitions of $I$ and $J$ are trivial, we will simplify the notation
by writing

$Sht^{\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}:=Sht^{(I)|(J),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}},$

which parametrizes

${\mathcal{P}}$${\text{Frob}_{S}^{*}\mathcal{P}}$${\mathcal{P}^{\prime}}$${\text{Frob}_{S}^{*}\mathcal{P}^{\prime}.}$$\scriptstyle{\varphi_{\mathcal{P}}}$$\scriptstyle{\gamma}$$\scriptstyle{\text{Frob}_{S}^{*}\gamma}$$\scriptstyle{\varphi_{\mathcal{P}^{\prime}}}$

######  Remark 2.7.

Let $E_{i}$ be the reflex field of $\mu_{i}$, then
$Sht_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$ is actually defined over
$\prod\text{Spd}(\mathcal{O}_{E_{i}})$, and similarly for
$Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$.
The same refinement works for all the spaces defined in this section.

######  Notation 2.8.

Throughout the paper, we use the following notation to denote the restriction
to the generic fiber,

$Sht_{\mu_{\bullet},\eta}^{(I_{1},\cdots,I_{k})}:=Sht_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}|_{Spd(L)^{I}},$
$Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu^{\prime}_{\bullet},\eta}:=Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu^{\prime}_{\bullet}}|_{Spd(L)^{I}\times
Spd(L)^{J}},$

and in particular

$Sht^{\nu}_{\mu|\mu^{\prime},\eta}:=Sht^{\nu}_{\mu|\mu^{\prime}}|_{Spd(L)\times
Spd(L)}.$

Note that in the last case, $\mu$ has to equal to $\mu^{\prime}$.

Similarly, we have the restriction to the special fiber

$Sht_{\mu_{\bullet},s}^{(I_{1},\cdots,I_{k})}:=Sht_{\mu_{\bullet}}|_{Spd(\overline{\mathbb{F}_{p}})^{I}},$
$Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu^{\prime}_{\bullet},s}:=Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu^{\prime}_{\bullet}}|_{Spd(\overline{\mathbb{F}_{p}})^{I}\times
Spd(\overline{\mathbb{F}_{p}})^{J}}.$

We now introduce the Hecke stacks.

######  Definition 2.9.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. Let $\\{I_{1},\cdots,I_{k}\\}$ be a
partition of $I$. We define

$Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$

to be the prestack over $Spd(\mathcal{O}_{L})^{I}$ whose value at $S$ is the
groupoid of

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k+1}$ on
$\mathcal{Y}_{[0,\infty)}(S)$

$\bullet$ untilts $S^{\sharp}_{i}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$
of $S$ defined over $\mathcal{O}_{L}$ indexed by $i\in I$, and

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1},}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}}\cong\mathcal{P}_{i+1}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}}$

that are meromorphic along the Cartier divisor $\underset{j\in
I_{i}}{\cup}S^{\sharp}_{j}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$, and
bounded by $\underset{S^{\sharp}_{j}=S^{\sharp}}{\sum}\mu_{j}$ at every
geometric point of $S^{\sharp}$.

When the partition is trivial, we make the following simplification

$Hecke_{\mu_{\bullet}}:=Hecke_{\mu_{\bullet}}^{(I)},$

which parametrizes

$\varphi:\mathcal{P}_{1}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{i\in
I}{\cup}S^{\sharp}_{i}}\cong\mathcal{P}_{2}|_{\mathcal{Y}_{[0,\infty)}(S)\setminus\underset{i\in
I}{\cup}S^{\sharp}_{i}}.$

Lastly, we denote

$Hecke^{(I_{1},\cdots,I_{k})}:=\underset{\mu_{\bullet}}{\bigcup}Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}.$

######  Remark 2.10.

We have remarked that $Hecke^{(I_{1},\cdots,I_{k})}_{\mu_{\bullet}}$ is really
defined over the product of the defining fields of $\mu_{\bullet}$. The base
can be further refined if we do not specify the boundedness condition. Indeed,
$Hecke^{(I_{1},\cdots,I_{k})}$ can be naturally defined over
$\text{Spd}(\mathbb{Z}_{p})^{I}$.

Similarly, we have the the correspondence version.

######  Definition 2.11.

Let $I,J$ be finite sets, and $\mu_{\bullet},\mu^{\prime}_{\bullet}$ be
collections of dominant cocharacters of $G$ indexed by $I$ and $J$
respectively. Let $\nu$ be a dominant cocharacter of $G$. Then we define

$Hecke^{\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$

to be the prestack with value at a characteristic $p$ perfectoid space $S$
being the groupoid of the data of

$\bullet$ $\\{\mathcal{P}_{1},\mathcal{P}_{2},\varphi\\}\in
Hecke_{\mu_{\bullet}}(S)$ and
$\\{\mathcal{P}^{\prime}_{1},\mathcal{P}_{2}^{\prime},\varphi^{\prime}\\}\in
Hecke_{\mu^{\prime}_{\bullet}}(S)$,

$\bullet$ modifications bounded by $\nu$ (in the sense similar to the above)

$\gamma_{1}:\mathcal{P}_{1}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}_{1}|_{\mathcal{Y}_{(0,\infty)}(S)}$
$\gamma_{2}:\mathcal{P}_{2}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}_{2}|_{\mathcal{Y}_{(0,\infty)}(S)}$

at the charctersictic $p$ untilt $S\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$
such that the following diagram is commutative

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\mathcal{P}^{\prime}_{1}}$${\mathcal{P}^{\prime}_{2}}$$\scriptstyle{\varphi}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\varphi^{\prime}}$

with the usual notation that dashed arrows denote generically defined
modifications.

We will occasionally use the following more general version.

######  Definition 2.12.

Let $I,J$ be finite sets, and $\mu_{\bullet},\mu^{\prime}_{\bullet}$ be
collections of dominant cocharacters of $G$ indexed by $I$ and $J$
respectively. Let $\\{I_{1},\cdots,I_{k}\\}$ and $\\{J_{1},\cdots,J_{l}\\}$ be
partitions of $I$ and $J$. Let $\nu$ be a dominant cocharacter of $G$. Then we
define

$Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$

to be the prestack with value at a characteristic $p$ perfectoid space $S$
being the groupoid of the data of

$\bullet$
$\\{\mathcal{P}_{1},\cdots,\mathcal{P}_{k+1},\varphi_{1},\cdots\varphi_{k}\\}\in
Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}(S)$ and
$\\{\mathcal{P}^{\prime}_{1},\cdots,\mathcal{P}_{l+1}^{\prime},\varphi^{\prime},\cdots\varphi_{l}^{\prime}\\}\in
Hecke_{\mu^{\prime}_{\bullet}}^{(J_{1},\cdots,J_{l})}(S)$,

$\bullet$ modifications bounded by $\nu$ (in the sense similar to the above)

$\gamma_{1}:\mathcal{P}_{1}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}_{1}|_{\mathcal{Y}_{(0,\infty)}(S)}$
$\gamma_{2}:\mathcal{P}_{2}|_{\mathcal{Y}_{(0,\infty)}(S)}\cong\mathcal{P}^{\prime}_{2}|_{\mathcal{Y}_{(0,\infty)}(S)}$

at the charctersictic $p$ untilt $S\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$
such that the following diagram is commutative

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1}}$${\mathcal{P}_{1}^{\prime}}$${\mathcal{P}_{2}^{\prime}}$${\cdots\cdots}$${\mathcal{P}^{\prime}_{l}}$${\mathcal{P}^{\prime}_{l+1}}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\varphi_{l-1}^{\prime}}$$\scriptstyle{\varphi_{l}^{\prime}}$

with the usual notation that dashed arrows denote generically defined
modifications.

We will also make use of a local version of Hecke stacks, which parameterizes
modifications of torsors over the formal neighborhoods of Cartier divisors
corresponding to untilts.

######  Definition 2.13.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. Let $\\{I_{1},\cdots,I_{k}\\}$ be a
partition of $I$. We define

$Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}$

to be the prestack over $Spd(\mathcal{O}_{L})^{I}$ whose value at $S$ is the
groupoid of

$\bullet$ untilts $S^{\sharp}_{i}$ of $S$ defined over $\mathcal{O}_{L}$
indexed by $i\in I$,

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k+1}$ on
$Spec(\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),\sum S_{i}^{\sharp}})$,
and,

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1},}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),\sum
S_{k}^{\sharp}}[\frac{1}{\underset{j\in
I_{i}}{\prod}\xi_{j}}]}\cong\mathcal{P}_{i+1}|_{\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),\sum
S_{k}^{\sharp}}[\frac{1}{\underset{j\in I_{i}}{\prod}\xi_{j}}]}$

that are bounded by $\underset{S^{\sharp}_{j}=S^{\sharp}}{\sum}\mu_{j}$ at the
point $S^{\sharp}$, where again we have product over distinct
$S^{\sharp}_{j}$.

When the partition is trivial, we make the following simplification

$Hecke^{\text{loc}}_{\mu_{\bullet}}:=Hecke_{\mu_{\bullet}}^{{\text{loc}},(I)}.$

Lastly, we denote

$Hecke^{{\text{loc}},(I_{1},\cdots,I_{k})}:=\underset{\mu_{\bullet}}{\bigcup}Hecke_{\mu_{\bullet}}^{{\text{loc}},(I_{1},\cdots,I_{k})}.$

######  Remark 2.14.

The drawback of local Hecke stacks is that there is no obvious correspondence
between them, due to that the torsors are defined on different domains.
However, this problem disappears if we fix the legs at characteristic $p$
untilts. An important example is the Witt vector local Hecke stacks to be
defined below, where we have the correspondences.

There is a canonical map

$Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}\longrightarrow
Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}$

being the restriction from global $G$-torsors to local $G$-torsors. We will
use it to define the canonical perverse sheaves on
$Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$.

Let $L^{+}G$ be the group v-sheaf over $Spd(\mathcal{O}_{L})$ whose value at
$S$ is

$L^{+}G(S)=\\{(x,S^{\sharp})|S^{\sharp}\in
Spd(\mathcal{O}_{L})(S),x\in\mathcal{G}(\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}})\\},$

and $Gr_{\mathcal{G},Spd(\mathcal{O}_{L}),\leq\mu}$ be the Beilinson-Drinfeld
affine Grassmannian as defined in [7] 20.3.1, which classifies
$\mathcal{G}$-torsors on the formal neighborhood of $S^{\sharp}$ together with
a trivialization of the generic fiber that is bounded by $\mu$. By [7] 20.5.4,
$Gr_{\mathcal{G},Spd(\mathcal{O}_{L}),\leq\mu}\rightarrow
Spd(\mathcal{O}_{L})$ is representable in spatial diamonds, and so in
particular a small v-sheaf. By definition, we have

$Hecke_{\mu}^{{\text{loc}}}=L^{+}G\setminus
Gr_{\mathcal{G},Spd(\mathcal{O}_{L}),\leq\mu}$

where $L^{+}G$ acts on the trivialization, so $Hecke_{\mu}$ is a small
v-stack.

We have similar expression for the global Hecke stack $Hecke_{\mu}$, which can
be written as the quotient of $Gr_{\mathcal{G},Spd(\mathcal{O}_{L}),\leq\mu}$
by the group v-sheaf parameterizing automorphisms of trivial global
$G$-torsors on $\mathcal{Y}_{[0,\infty)}$. Moreover, for Hecke stacks with
multiple legs, it can be written again as quotients of twisted products of
affine Grassmannians.

If the characteristic of the untilt $S^{\sharp}$ is $p$, then $S^{\sharp}=S$,
and

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}}=W(R)$

if $S=Spa(R,R^{+})$. Then the fiber of the Beilinson-Drinfeld affine
Grassmannian at the characteristic $p$ untilt is the Witt vector affine
Grassmannian considered in [13] (we embed perfect schemes into the category of
v-sheaves as explained in lecture 18 of [7]). On the other hand, if
$S^{\sharp}=Spa(R^{\sharp},R^{\sharp,+})$ is a characteristic zero untilt, we
have

$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}}=\mathbb{B}^{+}_{\text{dR}}(R^{\sharp}),$

then the fiber of $Gr_{\mathcal{G},Spd(\mathcal{O}_{L}),\leq\mu}$ at
$S^{\sharp}$ is the $\mathbb{B}^{+}_{\text{dR}}$-affine Grassmannian as
defined in [7] lecture 19.

######  Notation 2.15.

We have the similar notation for the restriction of Hecke stack to the generic
fiber,

$Hecke_{\mu_{\bullet},\eta}^{(I_{1},\cdots,I_{k})}:=Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}|_{Spd(L)^{I}}$
$Hecke_{\mu_{\bullet},\eta}^{\text{loc},(I_{1},\cdots,I_{k})}:=Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}|_{Spd(L)^{I}}$
$Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime},\eta}:=Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}|_{Spd(L)^{I}\times
Spd(L)^{J}}.$

Similarly for the restriction to the special fiber,

$Hecke_{\mu_{\bullet},s}^{(I_{1},\cdots,I_{k})}:=Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}|_{Spd(\overline{\mathbb{F}_{p}})^{I}}$
$Hecke_{\mu_{\bullet},s}^{\text{loc},(I_{1},\cdots,I_{k})}:=Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}|_{Spd(\overline{\mathbb{F}_{p}})^{I}}$
$Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime},s}:=Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}|_{Spd(\overline{\mathbb{F}_{p}})^{I}\times
Spd(\overline{\mathbb{F}_{p}})^{J}}.$

We have the following alternative characterization of Shtukas over the generic
fiber.

######  Theorem 2.16.

$Sht_{\mu,\eta}$ is equivalent to the prestack over $Spd(L)$ sending a
characteristic $p$ perfectoid space $S$ over $L$ to the groupoid of

$\bullet$ $G$-torsors $\mathcal{E}_{0}$ and $\mathcal{E}$ over the Fargues-
Fontaine curve $\mathcal{X}_{\text{FF},S}$ such that $\mathcal{E}_{0}$ is
trivial at every geometric point of $S$, which corresponds to a pro-étale
$G(\mathbb{Q}_{p})$-torsor $\mathbb{L}_{\eta}$ on $S$ by [14],

$\bullet$ a characteristic 0 untilt
$S^{\sharp}\hookrightarrow\mathcal{Y}_{[0,\infty)}(S)$ of $S$ defined over
$L$,

$\bullet$ an isomorphism

$\alpha:\mathcal{E}_{0}|_{\mathcal{X}_{\text{FF},S}\setminus
S^{\sharp}}\cong\mathcal{E}|_{\mathcal{X}_{\text{FF},S}\setminus S^{\sharp}}$

that is meromorphic along $S^{\sharp}$ and bounded by $\mu$,

$\bullet$ a sub $\mathcal{G}(\mathbb{Z}_{p})$-torsor $\mathbb{L}$ inside
$\mathbb{L}_{\eta}$.

###### Proof.

The proof of [7] 23.3.1 goes without change. Indeed, in $loc.cit.$ they fix
the type of Shtuka in $\text{B}(G)$, which corresponds to rigidifying
$\mathcal{E}$. Dropping this extra condition gives exactly the data as in the
statement. ∎

We have a natural correspondence diagram

${Sht^{\nu}_{\mu|\mu,\eta}}$${Sht_{\mu,\eta}}$${Sht_{\mu,\eta}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$
(1)

where $p_{1}$ (resp. $p_{2}$) sends
$\\{\mathcal{P},\mathcal{P}^{\prime},\gamma,S^{\sharp}\\}$ to
$\\{\mathcal{P},S^{\sharp}\\}$ (resp.
$\\{\mathcal{P}^{\prime},S^{\sharp}\\}$).

######  Proposition 2.17.

The correspondence (1) is an étale correspondence, i.e. both $p_{1}$ and
$p_{2}$ are finite étale morphisms.

###### Proof.

With the characterization of theorem 2.16, we see that
$Sht_{\mu|\mu,\eta}^{\nu}(S)$ parametrizes

$\bullet$ $\\{\mathcal{E}_{0},\mathcal{E},\alpha,\mathbb{L},S^{\sharp}\\}\in
Sht_{\mu,\eta}(S)$,

$\bullet$
$\\{\mathcal{E}^{\prime}_{0},\mathcal{E}^{\prime},\alpha^{\prime},\mathbb{L}^{\prime},S^{\sharp}\\}\in
Sht_{\mu,\eta}(S)$,

$\bullet$ isomorphisms

$\beta_{0}:\mathcal{E}_{0}\cong\mathcal{E}^{\prime}_{0}$
$\beta:\mathcal{E}\cong\mathcal{E}^{\prime}$

such that $\alpha^{\prime}\circ\beta_{0}=\beta\circ\alpha$, and under the
isomorphism

$\mathbb{L}_{\eta}\cong\mathbb{L}^{\prime}_{\eta}$

of pro-étale $G(\mathbb{Q}_{p})$-torsors on $S$ induced by $\beta_{0}$, the
sub-$\mathcal{G}(\mathbb{Z}_{p})$-torsors $\mathbb{L}$ and
$\mathbb{L}^{\prime}$ have relative position bounded by $\nu$ at every rank 1
geometric point of $S$.

Now for a given $S$-point of $Sht_{\mu,\eta}$ specified by the data
$\\{\mathcal{E}_{0},\mathcal{E},\alpha,\mathbb{L},S^{\sharp}\\}\in
Sht_{\mu,\eta}(S)$, we see that the fiber of $p_{1}$ at this point is the data
of sub-$\mathcal{G}(\mathbb{Z}_{p})$-torsors $\mathbb{L}^{\prime}$ of
$\mathbb{L}_{\eta}$ which have relative position with respect to $\mathbb{L}$
bounded by $\nu$ at every geometric point of $S$. Since pro-étale locally
$\mathbb{L}_{\eta}$ is the trivial torsor $S\times G(\mathbb{Q}_{p})$, the
relevant sub-$\mathcal{G}(\mathbb{Z}_{p})$-torsors corresponds to a subset of
$G(\mathbb{Q}_{p})/\mathcal{G}(\mathbb{Z}_{p})$, which is visibly étale over
$S$. Thus we see that $p_{1}$ is étale after base change to a pro-étale cover,
hence it is finite étale by [12] 9.11.

The same argument shows that $p_{2}$ is finite étale. ∎

###  2.2. Witt vector Shtukas and Hecke stacks

We will need the Witt vector version of Shtukas and Hecke stacks as well.

######  Definition 2.18.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. The moduli space of Witt vector local
Shtukas associated to $G$ and $\mu_{\bullet}$ is the prestack

$Sht_{\mu_{\bullet}}^{W}$

over $Spd(\overline{\mathbb{F}_{p}})$ whose value at a characteristic $p$
affinoid perfectoid space $S=Spa(R,R^{+})$ over $\overline{\mathbb{F}_{p}}$ is
the groupoid of the data of

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k}$ over
$\text{Spec}(W(R^{+}))$, and

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{\text{Spec}(W(R^{+})[\frac{1}{p}])}\cong\mathcal{P}_{i+1}|_{\text{Spec}(W(R^{+})[\frac{1}{p}])}$

with $\mathcal{P}_{k+1}=\text{Frob}_{S}^{*}\mathcal{P}_{1}$, which are bounded
by $\mu_{i}$ at every geometric point of $S$.

We have the local version as well.

######  Definition 2.19.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. The moduli space of Witt vector local
Shtukas associated to $G$ and $\mu_{\bullet}$ is the prestack

$Sht_{\mu_{\bullet}}^{loc,W}$

over $Spd(\overline{\mathbb{F}_{p}})$ whose value at a characteristic $p$
perfectoid space $S$ over $\overline{\mathbb{F}_{p}}$ is the groupoid of the
data of

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k}$ over
$\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})$, and

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})\setminus
S}\cong\mathcal{P}_{i+1}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})\setminus
S}$

with $\mathcal{P}_{k+1}=\text{Frob}_{S}^{*}\mathcal{P}_{1}$, which are bounded
by $\mu_{i}$ at every geometric point of $S$.

The possibility of having such a definition come from that Frobenius fixes the
characteristic $p$ legs, which is not the case for the characteristic 0 ones.

######  Remark 2.20.

Note that there is no extra data on the partition of $I$ since we fix legs at
the characteristic $p$ untilt, so essentially the only possibility of the
partition is $(\\{1\\},\cdots,\\{k\\})$.

When $S=\text{Spa}(R,R^{+})$ is affinoid perfectoid,
$\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),S}=W(R)$ by definition,
whence the name.

Again, there is the correspondence version.

######  Definition 2.21.

Let $I,J$ be finite sets, and $\mu_{\bullet},\mu^{\prime}_{\bullet}$ be
collections of dominant cocharacters of $G$ indexed by $I$ and $J$
respectively. Let $\nu$ be a dominant cocharacter of $G$. Then we define

$Sht^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$

to be the prestack with value at a characteristic $p$ affinoid perfectoid
space $S=Spa(R,R^{+})$ over $\overline{\mathbb{F}_{p}}$ being the groupoid of
the data of

$\bullet$
$\\{\mathcal{P}_{1},\cdots,\mathcal{P}_{k},\varphi_{1},\cdots,\varphi_{k}\\}\in
Sht^{W}_{\mu_{\bullet}}(S)$ and
$\\{\mathcal{P}^{\prime},\cdots,\mathcal{P}_{l}^{\prime},\varphi_{1}^{\prime},\cdots,\varphi_{l}^{\prime}\\}\in
Sht^{W}_{\mu^{\prime}_{\bullet}}(S)$,

$\bullet$ a modification bounded by $\nu$

$\gamma:\mathcal{P}|_{W(R^{+})[\frac{1}{p}]}\cong\mathcal{P}^{\prime}|_{W(R^{+})[\frac{1}{p}]}$

as Shtukas, i.e. the following diagram is commutative

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\mathcal{P}_{1}^{\prime}}$${\mathcal{P}_{2}^{\prime}}$${\cdots\cdots}$${\mathcal{P}^{\prime}_{l}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}^{\prime}.}$$\scriptstyle{\gamma}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$$\scriptstyle{\text{Frob}_{S}^{*}\gamma}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\varphi^{\prime}_{l-1}}$$\scriptstyle{\varphi_{l}^{\prime}}$

We now define the Witt vector Hecke stacks.

######  Definition 2.22.

Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection of dominant
cocharacters of $G$ indexed by $I$. We define

$Hecke_{\mu_{\bullet}}^{W}$

to be the prestack whose value at $S=Spa(R,R^{+})$ over
$\overline{\mathbb{F}_{p}}$ is the groupoid of

$\bullet$ $\mathcal{G}$-torsors $\mathcal{P}_{1},\cdots,\mathcal{P}_{k+1}$ on
$Spec(W(R^{+}))$,

$\bullet$ modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1},}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

where the dotted arrows $\varphi_{i}$ are isomorphisms

$\varphi_{i}:\mathcal{P}_{i}|_{W(R^{+})[\frac{1}{p}]}\cong\mathcal{P}_{i+1}|_{W(R^{+})[\frac{1}{p}]}$

that are bounded by $\mu_{i}$ at every geometric point of $S$.

######  Definition 2.23.

Let $I,J$ be finite sets, and $\mu_{\bullet},\mu^{\prime}_{\bullet}$ be
collections of dominant cocharacters of $G$ indexed by $I$ and $J$
respectively. Let $\nu$ be a dominant cocharacter of $G$. Then we define

$Hecke^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$

to be the prestack with value at a characteristic $p$ affinoid perfectoid
space $S=Spa(R,R^{+})$ over $\overline{\mathbb{F}_{p}}$ being the groupoid of
the data of

$\bullet$
$\\{\mathcal{P}_{1},\cdots,\mathcal{P}_{k+1},\varphi_{1},\cdots\varphi_{k}\\}\in
Hecke_{\mu_{\bullet}}^{W}(S)$ and
$\\{\mathcal{P}^{\prime}_{1},\cdots,\mathcal{P}_{l+1}^{\prime},\varphi^{\prime},\cdots\varphi_{l}^{\prime}\\}\in
Hecke_{\mu^{\prime}_{\bullet}}^{W}(S)$,

$\bullet$ modifications bounded by $\nu$

$\gamma_{1}:\mathcal{P}_{1}|_{W(R^{+})[\frac{1}{p}]}\cong\mathcal{P}^{\prime}_{1}|_{W(R^{+})[\frac{1}{p}]}$
$\gamma_{2}:\mathcal{P}_{2}|_{W(R^{+})[\frac{1}{p}]}\cong\mathcal{P}^{\prime}_{2}|_{W(R^{+})[\frac{1}{p}]}$

such that the following diagram is commutative

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1}}$${\mathcal{P}_{1}^{\prime}}$${\mathcal{P}_{2}^{\prime}}$${\cdots\cdots}$${\mathcal{P}^{\prime}_{l}}$${\mathcal{P}^{\prime}_{l+1}.}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\varphi_{l-1}^{\prime}}$$\scriptstyle{\varphi_{l}^{\prime}}$

We observe that the Witt vector Shtukas and Hecke stacks are analytifications
of perfect stacks. Let $Hk_{\mu_{\bullet}}$ be the perfect Hecke stack as
defined in [1] definition 5.1.1, then it is easy to see that

$Hk_{\mu_{\bullet}}^{\diamond}=Hecke^{W}_{\mu_{\bullet}}$

and

$Hk_{\mu_{\bullet}}^{\diamond\diamond}=Hecke^{\text{loc},(\\{1\\},\cdots,\\{n\\})}_{\mu_{\bullet},s}$

with notation as in A.1. On the other hand, we have

$Sht_{\mu_{\bullet}}^{perf,\diamond}=Sht_{\mu_{\bullet}}^{W}$

and

$Sht_{\mu_{\bullet}}^{perf,\diamond\diamond}=Sht_{\mu_{\bullet}}^{\text{loc},W}$

where $Sht_{\mu_{\bullet}}^{perf}$ is the perfect Shtuka as defined in [1]
definition 5.2.1. Similarly for the correspondence version
$Hecke^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$ and
$Sht^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$.

As a sanitary check, all the prestacks defined above are small v-stacks.

######  Proposition 2.24.

$Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$,
$Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}$,
$Hecke^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$,
$Hecke^{W}_{\mu_{\bullet}}$,
$Hecke^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$,
$Sht_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$,
$Sht_{\mu_{\bullet}}^{\text{loc},W}$, $Sht_{\mu_{\bullet}}^{W}$,
$Sht^{\nu,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}$ and
$Sht^{(I_{1},\cdots,I_{k})|(J_{1},\cdots,J_{l}),\nu}_{\mu_{\bullet}|\mu^{\prime}_{\bullet}}$
are small v-stack.

###### Proof.

Being stacks follows from [7] proposition 19.5.3. The smallness follows from
that the data we are parameterizing are sets, see the proof of [4] proposition
<EMAIL_ADDRESS>for example.

∎

##  3\. Excursion operators

###  3.1. Geometric Satake

We follow the same notation as in the previous section. Let us first recall
the geometric Satake proved by Fargues and Scholze.

######  Theorem 3.1.

(Fargues-Scholze) Let $I$ be a finite set, and $\mu_{\bullet}$ be a collection
of dominant cocharacters of $G$ indexed by $I$. Let $\\{I_{1},\cdots,I_{k}\\}$
be a partition of $I$. There is a natural functor

$U\longrightarrow\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}$

from the category of representations of $\hat{G}^{I}$, the product of the dual
group of $G$ (with $\overline{\mathbb{Q}_{l}}$-coefficient), to the category
$D_{\text{ét}}^{b}(Hecke^{(I_{1},\cdots,I_{k})},\overline{\mathbb{Q}_{l}})$.
It has the following properties.

1) If $U$ is irreducible with highest weight $\mu_{\bullet}$, then
$\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}$ is supported on
$Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$.

2) Let $\zeta:I\rightarrow J$ be a map of finite sets, $(J_{1},\cdots,J_{k})$
be a partition of $J$, and $I_{i}=\zeta^{-1}J_{i}$, there exists a natural
closed immersion

$i:Hecke^{(I_{1},\cdots,I_{k})}\times_{Spd(\mathcal{O}_{L})^{I},\zeta}Spd(\mathcal{O}_{L})^{J}\hookrightarrow
Hecke^{(J_{1},\cdots,J_{k})},$

where we abuse the notation by writing
$\zeta:Spd(\mathcal{O}_{L})^{J}\rightarrow Spd(\mathcal{O}_{L})^{I}$ the
permutation map corresponding to $\zeta$, i.e. sending $(x_{j})_{j\in J}$ to
$(x_{\zeta(i)})_{i\in I}$. Let

$p:Hecke^{(I_{1},\cdots,I_{k})}\times_{Spd(\mathcal{O}_{L})^{I},\zeta}Spd(\mathcal{O}_{L})^{J}\rightarrow
Hecke^{(I_{1},\cdots,I_{k})}$

be the projection on the first factor, then there exists a canonical
isomorphism

$i_{*}p^{*}\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}\cong\mathscr{S}_{\Delta_{\zeta}^{*}U}^{(J_{1},\cdots,J_{k})}$

where $\Delta_{\zeta}:\hat{G}^{J}\rightarrow\hat{G}^{I}$ is the map sending
$(g_{j})_{j\in J}$ to $(g_{\zeta(i)})_{i\in I}$.

3) Let

$\pi^{(I_{1},\cdots,I_{k})}:Hecke^{(I_{1},\cdots,I_{k})}\longrightarrow
Hecke^{(I)}$

be the convolution map, i.e. it takes

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

to

${\mathcal{P}_{1}}$${\mathcal{P}_{k+1},}$$\scriptstyle{\varphi_{k}\cdots\varphi_{1}}$

then there is a canonical isomorphism

$(\pi^{(I_{1},\cdots,I_{k})})_{!}\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}\cong\mathscr{S}_{U}^{(I)}.$

4) Let $(J_{1},\cdots,J_{l})$ be a partition of $I$ such that
$J_{j}=I_{i_{j}}\cup\cdots I_{i_{j+1}-1}$ for some partition
$1=i_{1}<i_{2}<\cdots<i_{l+1}=k+1$ of $k+1$. Let

$\kappa^{(I_{1},\cdots,I_{k})}_{(J_{1},\cdots,J_{l})}:Hecke^{(I_{1},\cdots,I_{k})}\longrightarrow\underset{j}{\prod}Hecke^{(I_{i_{j}},\cdots,I_{i_{j+1}-1})}$

be the forgetting map that sends

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\cdots\cdots}$${\mathcal{P}_{k}}$${\mathcal{P}_{k+1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{k-1}}$$\scriptstyle{\varphi_{k}}$

to

$\underset{j}{\prod}(\mathcal{P}_{i_{j}},\cdots,\mathcal{P}_{i_{j+1}},\varphi_{i_{j}},\cdots,\varphi_{i_{j+1}-1}),$

then we have a canonical isomorphism

$(\kappa^{(I_{1},\cdots,I_{k})}_{(J_{1},\cdots,J_{l})})^{*}\underset{j}{\boxtimes}\mathscr{S}_{U_{j}}^{(I_{i_{j}},\cdots,I_{i_{j+1}-1})}\cong\mathscr{S}_{\boxtimes
U_{j}}^{(I_{1},\cdots,I_{k})}.$

where $U_{j}$ is a representation of $\hat{G}^{J_{j}}$.

5) Let $U$ be irreduccible with hightest weight $\mu_{\bullet}$, then the
pullback of $\mathscr{S}_{U}^{(\\{1\\},\cdots,\\{n\\})}$ restricted on
$Hecke_{\mu_{\bullet},s}^{(\\{1\\},\cdots,\\{n\\})}$ through the canonical map

$Hecke^{W}_{\mu_{\bullet}}\longrightarrow
Hecke_{\mu_{\bullet},s}^{(\\{1\\},\cdots,\\{n\\})}$

is identified with the analytification of the intersection sheaves
$\text{IC}_{\mu_{\bullet}}$ of the Witt vector affine Grassmannian
$Gr_{\mu_{\bullet}}$, see [1] section 3.4.1.

More precisely, $\text{IC}_{\mu_{\bullet}}$ being $L^{+}G$-equivariant
descends to the truncated Witt vector Shtukas
$Hecke^{loc(m)}_{\mu_{\bullet}}$, see appendix A.3 or [1] section 5.1 for
definition, and we denote it by $A_{\mu_{\bullet}}$. As in appendix A.1, we
have the analytification $a_{X}^{*}c_{X}^{*}A_{\mu_{\bullet}}$ defined on
$Hecke^{loc(m),\diamond}_{\mu_{\bullet}}$ with
$X=Hecke^{loc(m)}_{\mu_{\bullet}}$, which can be further viewed as sheaves on
$Hecke^{W}_{\mu_{\bullet}}$ through the identification in A.3, and this is
canonically identified with the pullback of
$\mathscr{S}_{U}^{(\\{1\\},\cdots,\\{n\\})}$ along

$Hecke^{W}_{\mu_{\bullet}}\longrightarrow
Hecke_{\mu_{\bullet},s}^{(\\{1\\},\cdots,\\{n\\})}.$

6) Lastly, there is a natural functor

$U\longrightarrow\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}$

from the category of representations of ${}^{L}G^{I}$ to the category
$D_{\text{ét}}^{b}(Hecke^{\text{loc},(I_{1},\cdots,I_{k})},\overline{\mathbb{Q}_{l}})$
which satisfies the local version of 1), 2), 3) and 4). For the comparison
with Witt vector Hecke stacks, we have a natural restriction map

$Hecke_{\mu_{\bullet},s}^{loc,(\\{1\\},\cdots,\\{n\\})}\longrightarrow
Hecke^{loc(m),\diamond\diamond}_{\mu_{\bullet}}$

along which the pullback of $c_{X}^{*}A_{\mu_{\bullet}}$ is identified with
$\mathscr{S}_{U}^{\text{loc},(\\{1\\},\cdots,\\{n\\})}$ with $U$ irreduccible
with highest weight $\mu_{\bullet}$.

There is a natural restriction map

$Hecke^{(I_{1},\cdots,I_{k})}\longrightarrow
Hecke^{\text{loc},(I_{1},\cdots,I_{k})},$

along which the pullback of
$\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}$ is identified with
$\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}$, and 1),2),3),4) and 5) is the
pullback of the corresponding local version.

######  Remark 3.2.

By the derived category of a small v-stack $X$ with
$\overline{\mathbb{Q}_{l}}$-coefficients, we take the naive definition

$D^{b}_{\text{ét}}(X,\overline{\mathbb{Q}_{l}}):=\underset{E}{\text{colim}}\
D^{b}_{\text{ét}}(X,E),$

where $E$ ranges over finite extensions of $\mathbb{Q}_{l}$,
$D^{b}_{\text{ét}}(X,E)$ is the idempotent completion of the
$D^{b}_{\text{ét}}(X,\mathcal{O}_{E})[\frac{1}{l}]$ and the colimit is taken
in the $\infty$-categorical sense. Recall that the derived category
$D^{b}_{\text{ét}}(X,\mathcal{O}_{E})$ with adic cofficients is defined in
[12] section 26. The six functor formalism generalises directly to this
setting. Moreover, the constructions in this paper works already with the
coefficient ring $\mathcal{O}_{\mathbb{Q}_{l}(\sqrt{q})}$, and we can take
that instead.

The definition given here compares favorably with the constructible derived
categories of schemes, see [15] section 6.8. or [16], which suffices for our
use in this paper.

######  Remark 3.3.

We can state 3) for more general partitions, at the cost of further
complicating the notation. The special cases as stated above suffices for our
purpose.

######  Remark 3.4.

We can refine the theorem using Langlands $L$-group. More precisely, let
$Hecke_{\text{Spd}(\mathbb{Z}_{p})^{I}}^{(I_{1},\cdots,I_{k})}$ be the Hecke
stack over $\text{Spd}(\mathbb{Z}_{p})^{I}$, see remark 2.10. Then there is a
functor from representations $U$ of the L-group ${}^{L}G$ to the complexes of
sheaves $\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}$ on
$Hecke_{\text{Spd}(\mathbb{Z}_{p})^{I}}^{(I_{1},\cdots,I_{k})}$ with similar
properties as in the theorem, which refines the functor in the theorem.

###### Proof.

Using the pullback from the canonical map

$Hecke^{(I_{1},\cdots,I_{k})}\longrightarrow
Hecke^{\text{loc},(I_{1},\cdots,I_{k})}$

it is enough to prove the local version. Moreover, it is enough to work with
coefficients with ring of integers of $\mathbb{Q}_{l}[\sqrt{q}]$, which is
treated in [4].

It is enough to consider irreducible $U$’s. From the geometric Satake proved
in [4] theorem<EMAIL_ADDRESS>there exists a canonical perverse sheaf
$\mathscr{S}_{U}^{\text{loc},(I)}$ on $Hecke^{\text{loc},(I)}$ for each
representation $U$ of $\hat{G}^{I}$. Now let $\\{I_{1},\cdots I_{k}\\}$ be a
partition of $I$, and $U$ be an irreduccible representation of $\hat{G}^{I}$.
Then $U=\underset{j}{\boxtimes}U_{j}$, where $U_{j}$ is a representation of
$\hat{G}^{I_{j}}$. We define

$\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}:=(\kappa_{(I_{1},\cdots,I_{k})}^{\text{loc},(I_{1},\cdots,I_{k})})^{*}\underset{j}{\boxtimes}\mathscr{S}_{U_{j}}^{\text{loc},(I_{j})},$

(so $\mathscr{S}_{U}^{(I_{1},\cdots,I_{k})}$ on global Hecke stacks is the
pullback of $\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}$ along
$Hecke^{(I_{1},\cdots,I_{k})}\rightarrow
Hecke^{\text{loc},(I_{1},\cdots,I_{k})}$) then 4) is automatic, using the
identification
$\kappa_{(I_{1},\cdots,I_{k})}^{\text{loc},(I_{1},\cdots,I_{k})}=(\underset{j}{\prod}\kappa_{(I_{i_{j}},\cdots,I_{i_{j+1}-1})}^{\text{loc},(I_{i_{j}},\cdots,I_{i_{j+1}-1})})\circ\kappa^{\text{loc},(I_{1},\cdots,I_{k})}_{(J_{1},\cdots,J_{l})}$.

By the proof of [4] proposition<EMAIL_ADDRESS>$(\pi^{\text{loc},(I_{1},\cdots,I_{k})})_{!}\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}$
is the fusion product of $\mathscr{S}_{U_{j}}^{\text{loc},(I_{j})}$, then [4]
proposition<EMAIL_ADDRESS>tells us that the corresponding representation is
$\underset{j}{\boxtimes}U_{j}=U$. In other words,

$(\pi^{\text{loc},(I_{1},\cdots,I_{k})})_{!}\mathscr{S}_{U}^{\text{loc},(I_{1},\cdots,I_{k})}\cong\mathscr{S}_{U}^{\text{loc},(I)},$

which proves 3).

For 1), we observe that

$(\kappa_{(I_{1},\cdots,I_{k})}^{\text{loc},(I_{1},\cdots,I_{k})})^{-1}\underset{j}{\prod}Hecke_{\mu_{j}}^{\text{loc},(I_{j})}=Hecke_{\mu_{\bullet}}^{\text{loc},(I_{1},\cdots,I_{k})}.$

It follows from the proof of [4] theorem<EMAIL_ADDRESS>that
$\mathscr{S}_{U}^{\text{loc},(\\{1\\})}$ is the intersection sheaf of
$Hecke_{\mu}^{\text{loc},(\\{1\\})}$, where $\mu$ is the highest weight of
$U$. In particular, $\mathscr{S}_{U}^{\text{loc},(\\{1\\})}$ is supported on
$Hecke_{\mu}^{\text{loc},(\\{1\\})}$, so the claim follows from our
observation.

To prove 2), we first prove that it holds for the trivial partition $(J)$,
namely
$i^{\text{loc}}_{*}p^{\text{loc},*}\mathscr{S}_{U}^{\text{loc},(I)}\cong\mathscr{S}_{\Delta_{\zeta}^{*}U}^{\text{loc},(J)}$.
We observe that both sides are natural with respect to compositions of
$\zeta$. We know that $\zeta$ can be written as composition of simple
degeneration and injections, i.e. a simple degeneration is a surjective map
collapsing only two elements, and a simple injection is an injective map
adding only one element. They can be written as the disjoint union of some
finite set with either $\\{1,2\\}\twoheadrightarrow\\{1\\}$ or
$\emptyset\hookrightarrow\\{1\\}$. Using [4] proposition<EMAIL_ADDRESS>we can
assume that $\zeta$ is either $\\{1,2\\}\twoheadrightarrow\\{1\\}$ or
$\emptyset\hookrightarrow\\{1\\}$.

We check $\\{1,2\\}\twoheadrightarrow\\{1\\}$ first. In this case, $i$ is an
isomorphism and $p$ is the closed immersion

$p^{\text{loc}}:Hecke^{\text{loc},(\\{1\\})}\hookrightarrow
Hecke^{\text{loc},(\\{1,2\\})}$

identifying $Hecke^{\text{loc},(\\{1\\})}$ with the substack of
$Hecke^{\text{loc},(\\{1,2\\})}$ where the two legs are the same. We can
assume that $U=U_{1}\boxtimes U_{2}$ is irreducible, then there is a canonical
identification

$p^{\text{loc},*}\mathscr{S}_{U}^{\text{loc},(\\{1,2\\})}\cong\mathscr{S}^{\text{loc},(\\{1\\})}_{\Delta^{*}U}$

where $\Delta:\hat{G}\hookrightarrow\hat{G}^{2}$ is the diagonal map and $U$
is a representation of $\hat{G}^{2}$. This is because
$\Delta^{*}U=U_{1}\otimes U_{2}$ and
$p^{*}\mathscr{S}_{U}^{\text{loc},(\\{1,2\\})}$ is the convolution product of
$\mathscr{S}_{U_{1}}^{\text{loc},(\\{1\\})}$ and
$\mathscr{S}_{U_{2}}^{\text{loc},(\\{2\\})}$ (using
$\mathscr{S}_{U}^{\text{loc},(\\{1,2\\})}$ is the fusion product of
$\mathscr{S}_{U_{1}}^{\text{loc},(\\{1\\})}$ and
$\mathscr{S}_{U_{2}}^{\text{loc},(\\{2\\})}$), see the paragraph following the
proof of [4] proposition<EMAIL_ADDRESS>Then the identification follows from the
geometric Satake ([4] theorem vi@.11.1).

Next, we assume that $\zeta$ is $\emptyset\hookrightarrow\\{1\\}$, then $p$ is

$p^{\text{loc}}:\text{Spd}(\mathcal{O}_{L})\rightarrow\ast,$

the unique map to the final object, and

$i^{\text{loc}}:\text{Spd}(\mathcal{O}_{L})\hookrightarrow
Hecke^{\text{loc},(\\{1\\})}$

is the section of the structure map corresponding to the trivial modification,
i.e. the modification is an isomorphism. Then

$i_{*}^{\text{loc}}p^{\text{loc},*}\mathscr{S}_{1}^{\text{loc},(\emptyset)}\cong\mathscr{S}_{1}^{\text{loc},(\\{1\\})}$

where $1$ is the trivial representation (of either $\hat{G}$ or the trivial
group $\hat{G}^{\emptyset}$). This follows from
$\mathscr{S}_{1}^{\text{loc},(\emptyset)}$ being the constant local system,
and $\mathscr{S}_{1}^{\text{loc},(\\{1\\})}$ being the constant local system
supported on $Hecke^{\text{loc},(\\{1\\})}_{0}$ (by 1)). The general $U$ are
direct sums of $1$, so the identification follows.

Now for general partition $(J_{1},\cdots,J_{k})$, we have a Cartesian diagram

${Hecke^{\text{loc},(I_{1},\cdots,I_{k})}}$${Hecke^{\text{loc},(I_{1},\cdots,I_{k})}\times_{Spd(\mathcal{O}_{L})^{I},\zeta}Spd(\mathcal{O}_{L})^{J}}$${Hecke^{\text{loc},(J_{1},\cdots,J_{k})}}$${\underset{j}{\prod}Hecke^{\text{loc},(I_{j})}}$${\underset{j}{\prod}Hecke^{\text{loc},(I_{j})}\times_{Spd(\mathcal{O}_{L})^{I_{j}},\zeta}Spd(\mathcal{O}_{L})^{J_{j}}}$${\underset{j}{\prod}Hecke^{\text{loc},(J_{j})}}$$\scriptstyle{\kappa^{\text{loc},(I_{1},\cdots,I_{k})}_{(I_{1},\cdots,I_{k})}}$$\scriptstyle{i^{\text{loc}}}$$\scriptstyle{\kappa^{\text{loc},(I_{1},\cdots,I_{k})}_{(I_{1},\cdots,I_{k})}\times\text{id}}$$\scriptstyle{p^{\text{loc}}}$$\scriptstyle{\kappa^{\text{loc},(J_{1},\cdots,J_{k})}_{(J_{1},\cdots,J_{k})}}$$\scriptstyle{\prod
i^{\text{loc}}}$$\scriptstyle{\prod p^{\text{loc}}}$

and we can pullback the identification for trivial partition to obtain the
desired one.

For the comparison with Witt vector Hecke stacks. We note that when
$I=\\{1\\}$, it follows from the paragraph before [12] proposition<EMAIL_ADDRESS>and
the proof of $loc.cit.$ that under the inclusion
$D_{\text{ét}}(Hecke_{\mu,s}^{\text{loc},{(\\{1\\})}},\Lambda)\subset
D_{\text{ét}}(Gr_{\mu}^{W,\diamond\diamond},\Lambda)$ with
$\Lambda=\mathcal{O}_{\mathbb{Q}_{l}(\sqrt{q})}$ (recall that
$Hecke_{\mu,s}^{\text{loc},{(\\{1\\})}}=L^{+}G\setminus
Gr_{\mu}^{W,\diamond\diamond}$),

$c_{X}^{*}IC_{\mu}=\mathscr{S}_{U}^{\text{loc},(\\{1\\})}$

with $X=Gr_{\mu}^{W}$ the Witt vector affine Grassmannian. Now we work with
the general $IC_{\mu_{\bullet}}$, recall that the construction of
$IC_{\mu_{\bullet}}$ is by taking the twisted product of $IC_{\mu_{i}}$ on
$Gr_{\mu_{\bullet}}=\overset{\sim}{\boxtimes}\ Gr_{\mu_{i}}$ , see [1] section
3.4.1. By passing to Hecke stacks, this corresponds precisely to
$(\kappa_{(\\{1\\},\cdots,\\{n\\}}^{\text{loc},(\\{1\\},\cdots,\\{n\\})})^{*}\underset{j}{\boxtimes}\mathscr{S}_{U_{j}}^{\text{loc},(\\{1\\})}$.
∎

###  3.2. Hecke Correspondence

We have a canonical map

$\epsilon_{\mu}^{(\\{1\\})}:Sht_{\mu,\eta}\longrightarrow Hecke_{\mu,\eta}$

that sends a Shtuka $\\{\mathcal{P},\varphi_{\mathcal{P}},S^{\sharp}\\}$ to
$\\{\mathcal{P},Frob_{S}^{*}\mathcal{P},\varphi_{\mathcal{P}},S^{\sharp}\\}$,
and we can use it to pullback the canonical sheaves in geometric Satake to
sheaves on Shtukas.

Let $U$ be an irreducible representation of $\hat{G}$ with highest weight
$\mu$. We have a diagram

${Sht^{\nu}_{\mu|\mu,\eta}}$${Sht_{\mu,\eta}}$${Sht_{\mu,\eta}}$${Hecke_{\mu,\eta}}$${Hecke_{\mu,\eta}^{\text{loc}}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$$\scriptstyle{r}$
(2)

where the last vertical map is the canonical restriction map. The square in
the diagram is not commutative, but they become commutative after composing
with vertical map. Indeed, recall from notation 2.8, the legs of Shtukas in
the diagram are in characteristic zero, while the modification bounded by
$\nu$ takes place at characteristic $p$, hence it does not affect the local
type of $\varphi_{\mathcal{P}}$, i.e. the diagram is commutative after
composing with the vertical map.

Now we have a cohomological correspondence $\mathscr{C}_{\nu}$ from
$(Sht_{\mu,\eta},\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})})$ to
itself supported on $Sht^{\nu}_{\mu|\mu,\eta}$, namely

$\mathscr{C}_{\nu}:p_{1}^{*}\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})}\cong
p_{1}^{*}\epsilon_{\mu}^{(\\{1\\}),*}r^{*}\mathscr{S}_{U}^{\text{loc},(\\{1\\})}\cong
p_{2}^{*}\epsilon_{\mu}^{(\\{1\\}),*}r^{*}\mathscr{S}_{U}^{\text{loc},(\\{1\\})}\cong
p_{2}^{*}\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})}\cong
p_{2}^{!}\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})},$ (3)

where we use
$r^{*}\mathscr{S}_{U}^{\text{loc},(\\{1\\})}=\mathscr{S}_{U}^{(\\{1\\})}$ (see
6) of theorem 3.1), and the étaleness of $p_{2}$ for the last isomorphism.

Now let $V$ be a representation of $\hat{G}$, and $h_{V}\in
C_{c}(G(L)//\mathcal{G}(\mathcal{O}_{L}),\overline{\mathbb{Q}_{l}})$ be the
function corresponding to $V$ through classical Satake. We know that
$C_{c}(G(L)//\mathcal{G}(\mathcal{O}_{L}),\overline{\mathbb{Q}_{l}})$ has a
basis indexed by the dominant cocharcters $\nu$ ($G$ is unramified, being
reductive over $\mathbb{Z}_{p}$), i.e.
$1_{\mathcal{G}(\mathcal{O}_{L})\nu(p)\mathcal{G}(\mathcal{O}_{L})}$ forms a
basis of
$C_{c}(G(L)//\mathcal{G}(\mathcal{O}_{L}),\overline{\mathbb{Q}_{l}})$. Let
$\Gamma_{V}$ be the union of $Sht^{\nu}_{\mu|\mu,\eta}$ with $\nu$ showing up
in $h_{V}$. Then we define the Hecke operator $T_{V}$ as the cohomological
correspondence supported on $\Gamma_{V}$, and is $h_{V}(\nu(p))$ times the
cohomological correspondence (3) on $Sht^{\nu}_{\mu|\mu,\eta}$. More
precisely, we define

$T_{V}:=\underset{h_{V}(\nu(p))\neq
0}{\sum}h_{V}(\nu(p))\cdot\mathscr{C}_{\nu}$

as a cohomological correspondence from
$(Sht_{\mu,\eta},\epsilon_{\mu}^{*}\mathscr{S}_{U})$ to itself supported on

$\Gamma_{V}:=\underset{h_{V}(\nu(p))\neq 0}{\cup}Sht^{\nu}_{\mu|\mu,\eta}.$

Note that when $V$ is irreducible with highest weight $\delta$, then
$h_{V}(\nu(p))\neq 0$ precisely when $\nu\leq\delta$, hence
$\Gamma_{V}=Sht^{\nu}_{\mu|\mu,\eta}$. Since $Sht^{\nu}_{\mu|\mu,\eta}\subset
Sht^{\delta}_{\mu|\mu,\eta}$ as closed substack, we can view
$\mathscr{C}_{\nu}$ as supported on $Sht^{\delta}_{\mu|\mu,\eta}$, whence the
sum in the definition of $T_{V}$ makes sense. In general $\Gamma_{V}$ is the
union of $Sht^{\delta}_{\mu|\mu,\eta}$ where $\delta$ ranges over highest
weights of irreduccible summands of $V$.

We observe that when $\mu$ is minuscule,
$\mathscr{S}_{U}=\overline{\mathbb{Q}_{l}}$ is the constant sheaf and the
cohomological correspondence $\mathscr{C}_{\nu}$ is the canonical isomorphism

$p_{1}^{*}\overline{\mathbb{Q}_{l}}\cong\overline{\mathbb{Q}_{l}}\cong
p_{2}^{*}\overline{\mathbb{Q}_{l}}\cong p_{2}^{!}\overline{\mathbb{Q}_{l}}.$

We assume from now on that $V$ is a representation of ${}^{L}G$, in other
words, $V$, as a representation of $\hat{G}$, is canonically isomorphic to its
Frobenius twist ${}^{\sigma}V$, where ${}^{\sigma}V$ is the representation of
$\hat{G}$ obtained by twisting the action of $G$ on $V$ with the inverse
Frobenius action on $\hat{G}$, see [1] section 3.4.5 . Then the (unramified)
classical Satake gives $h_{V}\in
C_{c}(G(\mathbb{Q}_{p})//\mathcal{G}(\mathbb{Z}_{p})),\overline{\mathbb{Q}_{l}})$.

###  3.3. Creation correspondence

We now start to define the excursion operators. First we define the creation
correspondence. We follow the same notation as above.

######  Notation 3.5.

We use subscript $V$ and $V^{*}$ to mean that we restrict to the special fiber
at the legs they correspond. For instance,

$Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}:=Hecke_{(\delta^{*},\delta)}^{(\\{1\\},\\{2\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\overline{\mathbb{F}_{p}})},$ $Hecke_{V^{*}\boxtimes
V\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\})}:=Hecke_{(\delta^{*},\delta,\mu)}^{(\\{1\\},\\{2\\},\\{3\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\overline{\mathbb{F}_{p}})\times Spd(\mathcal{O}_{L})},$
$Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}:=Hecke_{(\delta^{*},\delta,\mu)}^{(\\{1\\},\\{2\\},\\{3\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\overline{\mathbb{F}_{p}})\times Spd(L)}$

and similarly for others.

Let $\delta$ and $\delta^{*}$ be the highest weight of $V$ and the dual
representation $V^{*}$ respectively. We denote

$Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}:=Sht_{(\delta^{*},\delta,\mu)}^{(\\{1\\},\\{2\\},\\{3\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\overline{\mathbb{F}_{p}})\times Spd(L)}$

in other words, $Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$ parameterizes

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\mathcal{P}_{3}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

where the two modifications takes place at the characteristic $p$ untilt that
are bounded by $\delta^{*}$ and $\delta$ respectively, while the third
modification takes place at the characteristic 0 untilt and is bounded by
$\mu$.

Similarly, let

$Sht_{\mu|V^{*}\boxtimes V\boxtimes\mu,\eta}^{0}\subset
Sht^{(\\{1\\})|(\\{1\\},\\{2\\},\\{3\\}),0}_{\mu|(\delta^{*},\delta,\mu),\eta}$
(4)

be the v-substack parametrizes modifications

${\mathcal{P}}$${\text{Frob}_{S}^{*}\mathcal{P}}$${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\mathcal{P}_{3}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

with the first line in $Sht_{\mu,\eta}$ and the second line in
$Sht_{V^{*}\boxtimes V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$. By our
convention, $Sht_{\mu,\eta}$ has the leg in characteristic zero, while the
first two modifications of the second line takes place in characteristic $p$,
hence the third modification has the same as leg as the first line, and the
diagram is equivalent to

${\mathcal{P}}$${\mathcal{P}_{2}^{\prime}}$${\text{Frob}_{S}^{*}\mathcal{P}}$${\mathcal{P}_{1}^{\prime}}$$\scriptstyle{\sim}$
(5)

where the first two dashed arrows are modifications at characteristic $p$
untilt bounded by $\delta^{*}$ and $\delta$ respectively whose composition is
an isomorphism, and the last dashed arrow is a modification at characteristic
zero untilt bounded by $\mu$. It means that $Sht_{\mu|V^{*}\boxtimes
V\boxtimes\mu,\eta}^{0}$ is a clsoed v-substack of
$Sht_{(\delta^{*},\delta,\mu),\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$. More
precisely, it lies in $Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})},$ which parameterizes diagrams
as in (5) without the isomorphism condition.

From geometric Satake, we have a morphism

$\mathscr{S}_{1}^{(\\{1\\})}\longrightarrow\mathscr{S}_{V^{*}\otimes
V}^{(\\{1\\})}$ (6)

corresponding to the canonical morphism $1\rightarrow V^{*}\otimes V$. We note
that this defines a cohomological correspondence supported on

${Hekce^{0}_{0|V^{*}\otimes V}}$${Hecke_{0}^{(\\{1\\})}}$${Hecke_{V^{*}\otimes
V}^{(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$ (7)

where we adopt the convention that for a representation $W$ of $\hat{G}^{I}$,

$Hecke_{U}:=\underset{\mu_{\bullet}}{\cup}Hecke_{\mu_{\bullet}}$

where the index $\mu_{\bullet}$ ranges over the highest weight of irreducible
summand of $U$, which are exterior products of highest weights of $\hat{G}$.
Similarly for $Hecke_{U|U^{\prime}}^{0}$. Indeed, we have

######  Lemma 3.6.

Let $I$ be a finite set, $U$ and $U^{\prime}$ be two representations of
$\hat{G}^{I}$, then we have a natural identification

$\text{Hom}_{\hat{G}^{I}}(U,U^{\prime})\cong\text{Corr}_{Hecke_{U|U^{\prime}}^{0}}(\mathscr{S}_{U}^{(I)},\mathscr{S}_{U^{\prime}}^{(I)})$

where the right hand side denote the cohomological correspondence from
$(Hecke_{U},\mathscr{S}_{U}^{(I)})$ to
$(Hecke_{U^{\prime}},\mathscr{S}_{U^{\prime}}^{(I)})$ supported on
$Hecke_{U|U^{\prime}}^{0}$. Moreover, the identification is compatible with
composition on both sides.

###### Proof.

It suffices to assume that $W$ and $W^{\prime}$ are irreducible with highest
weight $\mu$ and $\mu^{\prime}$ respectively. We have a Cartesian diagram

${Hecke_{\mu_{\bullet}|\mu^{\prime}_{\bullet}}^{0}}$${Hecke_{\mu^{\prime}_{\bullet}}}$${Hecke_{\mu_{\bullet}}}$${Hekce^{(I)}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{i}$

hence by proper base change ([12] 22.19, $p_{2}$ is proper as it is the base
change of a closed immersion, so $p_{2,!}$ is defined, similarly for $p_{1}$)

$\text{Hom}(p_{1}^{*}\mathscr{S}_{U}^{(I)},p_{2}^{!}\mathscr{S}_{U^{\prime}}^{(I)})=\text{Hom}(p_{2,!}p_{1}^{*}\mathscr{S}_{U}^{(I)},\mathscr{S}_{U^{\prime}}^{(I)})=\text{Hom}(p_{2,!}p_{1}^{*}\mathscr{S}_{U}^{(I)},\mathscr{S}_{U^{\prime}}^{(I)})=\text{Hom}(i^{\prime*}i_{!}\mathscr{S}_{U}^{(I)},\mathscr{S}_{U^{\prime}}^{(I)})$
$=\text{Hom}(i_{!}\mathscr{S}_{U}^{(I)},i^{\prime}_{*}\mathscr{S}_{U^{\prime}}^{(I)})=\text{Hom}(\mathscr{S}_{U}^{(I)},\mathscr{S}_{U^{\prime}}^{(I)})=\text{Hom}_{\hat{G}^{I}}(U,U^{\prime})$

where the last identification is geometric Satake. The compatibility with
composition is easy to see from the naturality of the above chain of
identifications. ∎

Further, 2) of theorem 3.1 gives us a natural identification

$\mathscr{S}_{V^{*}\otimes
V}^{(\\{1\\})}\cong\Delta^{*}\mathscr{S}_{V^{*}\boxtimes V}^{(\\{1,2\\})}$ (8)

under the canonical injection

$\Delta:Hecke^{(\\{1\\})}\hookrightarrow Hecke^{(\\{1,2\\})},$

while 3) of theorem 3.1 provides a canonical isomorhpism

$(\pi^{(\\{1\\},\\{2\\})})_{!}\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\cong\mathscr{S}_{V^{*}\boxtimes V}^{(\\{1,2\\})}$ (9)

from the convolution map

$\pi^{(\\{1\\},\\{2\\})}:Hecke^{(\\{1\\},\\{2\\})}\longrightarrow
Hecke^{(\\{1,2\\})}.$

We can view (8) and (9) as cohomological correspondences. Combining (6), (8)
and (9), we obtain

$\mathscr{S}_{1}^{(\\{1\\})}\longrightarrow\mathscr{S}_{V^{*}\otimes
V}^{(\\{1\\})}\cong\Delta^{*}\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1,2\\})}\cong\Delta^{*}(\pi^{(\\{1\\},\\{2\\})})_{!}\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}$

which can be view as a cohomological correspondence

$\mathscr{C}_{\sharp}^{+,0}:\pi^{*}\mathscr{S}_{1}^{(\\{1\\})}\longrightarrow
i^{!}\mathscr{S}_{V\boxtimes V^{*}}^{(\\{1\\},\\{2\\})}$ (10)

from $(\mathscr{S}_{1}^{(\\{1\\})},Hecke_{0,s})$ to
$(\mathscr{S}_{V^{*}\boxtimes V}^{(\\{1\\},\\{2\\})},Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})})$ supported on

${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\}),0}}$${Hecke_{0,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})},}$$\scriptstyle{i}$$\scriptstyle{\pi}$ (11)

where

$Hecke_{V^{*}\boxtimes V}^{(\\{1\\},\\{2\\}),0}\subset Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}$

is defined to be the closed substack parametrizing

${\mathcal{P}_{1}}$${\mathcal{P}_{3}}$${\mathcal{P}_{2}}$$\scriptstyle{\sim}$

i.e. it is the closed substack of $Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}$ defined by the condition that
$\varphi_{2}\circ\varphi_{1}$ is an isomorphism, in other words, it is
$Hecke_{0|V^{*}\boxtimes V}^{(\\{1\\})|(\\{1\\},\\{2\\}),0}$ and the
correspondence diagram is exactly the defining diagram of
$Hecke_{0|V^{*}\boxtimes V}^{(\\{1\\})|(\\{1\\},\\{2\\}),0}$. Indeed, we can
write down the support of (6), (8) and (9), being

${Hekce^{0}_{0|V^{*}\otimes V}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}}$${Hecke_{0,s}^{(\\{1\\})}}$${Hecke_{V^{*}\otimes
V}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1,2\\})},}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi^{(\\{1\\},\\{2\\})}}$$\scriptstyle{\Delta}$

and $Hecke_{V^{*}\boxtimes V}^{(\\{1\\},\\{2\\}),0}$ is precisely the fiber
product of $\pi^{(\\{1\\},\\{2\\})}$ and $\Delta\circ p_{2}$.

We can repeat the last paragraph for local Hecke stacks to obtain a
cohomological correspondence

$\mathscr{C}_{\sharp}^{\text{loc},+,0}:\pi^{\text{loc},*}\mathscr{S}_{1}^{\text{loc},(\\{1\\})}\longrightarrow
i^{\text{loc},!}\mathscr{S}_{V\boxtimes V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}$
(12)

supported on

${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),0}}$${Hecke_{0,s}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})}.}$$\scriptstyle{i^{\text{loc}}}$$\scriptstyle{\pi^{\text{loc}}}$

Moreover, by 6) of theorem 3.1, $\mathscr{C}_{\sharp}^{+,0}$ is the pullback
of $\mathscr{C}_{\sharp}^{\text{loc},+,0}$ along the diagram

${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\}),0}}$${Hecke_{0,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),0}}$${Hecke_{0,s}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})},}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{i^{\text{loc}}}$$\scriptstyle{\pi^{\text{loc}}}$

which clearly have Cartesian squares. Note that it is important that we
restrict to legs at the characteristic $p$ untilt, as otherwise the
correspondence Hecke stacks $Hekce^{0}_{0|V^{*}\otimes V}$ need not exist, see
remark 2.14.

We observe that we can repeat the construction of
$\mathscr{C}_{\sharp}^{\text{loc},+,0}$ using truncated Witt Hecke stacks, and
obtain $\mathscr{C}_{\sharp}^{\text{loc}(m),+,0}$, and it follows essentially
from 5) of theorem 3.1 that

$\mathscr{C}_{\sharp}^{\text{loc},+,0}=\mathscr{C}_{\sharp}^{\text{loc}(m,n),+,0,\diamond\diamond}$

under the identification in appendix A.3 between sheaves on truncated and
untruncated Shtukas, i.e. $\mathscr{C}_{\sharp}^{\text{loc},+,0}$ is the
$\diamond\diamond$-analytification of
$\mathscr{C}_{\sharp}^{\text{loc}(m,n),+,0}$, see appendix A.2 for the
notation.

We now consider the correspondence from definition 2.12

${Hecke_{(0,\mu)|(V^{*},V,\mu),\eta}^{(\\{1\\},\\{2\\})|(\\{1\\},\\{2\\},\\{3\\}),0}}$${Hecke_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Hecke_{(V^{*},V,\mu),\eta}^{(\\{1\\},\\{2\\},\\{3\\})}.}$$\scriptstyle{p}$
(13)

Observe that when the leg of $\mu$ is different from the legs of $V$ and
$V^{\prime}$,

$Hecke_{(0,\mu)|(V^{*},V,\mu),\eta}^{(\\{1\\},\\{2\\})|(\\{1\\},\\{2\\},\\{3\\}),0}=Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}$

where

$Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}\overset{i}{\hookrightarrow}Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$

is the substack parametrizing

${\mathcal{P}_{1}}$${\mathcal{P}_{3}}$${\mathcal{P}_{4}}$${\mathcal{P}_{2}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\sim}$$\scriptstyle{\varphi_{3}}$$\scriptstyle{\varphi_{2}}$

with $\varphi_{1}$ and $\varphi_{2}$ bounded by (highest weights of
irreducible summand of) $V^{*}$ and $V$ respectively, and $\varphi_{3}$
bounded by $\mu$. The morphism $p$ sends the above diagram to the first line

${\mathcal{P}_{1}}$${\mathcal{P}_{3}}$${\mathcal{P}_{4}.}$$\scriptstyle{\sim}$$\scriptstyle{\varphi_{3}}$

Obviously we have $Hecke_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}\cong
Hecke_{\mu,\eta}^{(\\{1\\})}$, and the isomorphism is induced by
$\kappa^{(\\{1\\},\\{2\\})}_{(\\{1\\},\\{2\\})}$. From 4) of theorem 3.1, we
have

$\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})}\cong\kappa^{(\\{1\\},\\{2\\}),*}_{(\\{1\\},\\{2\\})}(\mathscr{S}_{1}^{(\\{1\\})}\boxtimes\mathscr{S}_{U}^{(\\{2\\})})\cong\kappa^{(\\{1\\},\\{2\\}),*}_{(\\{1\\},\\{2\\})}(\overline{\mathbb{Q}_{l}}\boxtimes\mathscr{S}_{U}^{(\\{2\\})})$
(14)

which means that $\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})}\cong\mathscr{S}_{U}^{(\\{1\\})}$ under the isomorphism
$Hecke_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}\cong Hecke_{\mu,\eta}^{(\\{1\\})}$.
From 4) of theorem 3.1 again, we have

$\mathscr{S}_{V^{*}\boxtimes V\boxtimes
U}^{(\\{1\\},\\{2\\},\\{3\\})}\cong\kappa^{(\\{1\\},\\{2\\},\\{3\\}),*}_{(\\{1,2\\},\\{3\\})}(\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\boxtimes\mathscr{S}_{U}^{(\\{3\\})}).$ (15)

By definition of $p$, we have
$\kappa^{(\\{1\\},\\{2\\})}_{(\\{1\\},\\{2\\})}\circ
p=(\pi^{(\\{1\\},\\{2\\})}\boxtimes
id)\circ\kappa^{(\\{1\\},\\{2\\},\\{3\\})}_{(\\{1,2\\},\\{3\\})}\circ i$.

We have a commutative diagram

${Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Hecke_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\}),0}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{0}^{(\\{1\\})}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\times
Hecke_{\mu,\eta}^{(\\{1\\})}.}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\})}_{(\\{1\\},\\{2\\})}}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\},\\{3\\})}_{(\\{1,2\\},\\{3\\})}}$$\scriptstyle{i\times\text{id}}$$\scriptstyle{\pi\times\text{id}}$

It is clear that the two squares are Cartesian, thus we can pullback (see [1]
A.2.13, which is true in any six functor formalism) the cohomological
correspondence (10) times identity correspondence from
$(\mathscr{S}_{U}^{(\\{1\\})},Hecke_{\mu}^{(\\{1\\})})$ to itself, i.e. we
pullback

$\mathscr{C}_{\sharp}^{+,0}\boxtimes\text{id}:(\pi\times\text{id})^{*}\mathscr{S}_{1}^{(\\{1\\})}\boxtimes\mathscr{S}_{U}^{(\\{1\\})}\longrightarrow(i\times\text{id})^{!}\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\boxtimes\mathscr{S}_{U}^{(\\{1\\})}$

along $\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}$ to obtain
the cohomological correspondence

$\mathscr{C}_{\sharp}^{+}:p^{*}\mathscr{S}_{U}^{(\\{1\\})}\longrightarrow
i^{!}\mathscr{S}_{V^{*}\boxtimes V\boxtimes U}^{(\\{1\\},\\{2\\},\\{3\\})}$
(16)

supported on $Hecke_{V^{*}\boxtimes
V\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),0}$. More precisely, it is the
composition

$\mathscr{C}_{\sharp}^{+}:p^{*}\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})}\overset{(\ref{980})}{\cong}p^{*}\kappa^{(\\{1\\},\\{2\\}),*}_{(\\{1\\},\\{2\\})}(\mathscr{S}_{1}^{(\\{1\\})}\boxtimes\mathscr{S}_{U}^{(\\{2\\})})\cong(\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})})^{*}(\pi\times\text{id})^{*}(\mathscr{S}_{1}^{(\\{1\\})}\boxtimes\mathscr{S}_{U}^{(\\{2\\})})$
$\xrightarrow{(\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})})^{*}\mathscr{C}_{\sharp}^{+}\boxtimes\text{id}}(\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})})^{*}(i\times\text{id})^{!}(\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\boxtimes\mathscr{S}_{U}^{(\\{3\\})})$
$\xrightarrow{\text{BC}}i^{!}\kappa^{(\\{1\\},\\{2\\},\\{3\\}),*}_{(\\{1,2\\},\\{3\\})}(\mathscr{S}_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\boxtimes\mathscr{S}_{U}^{(\\{3\\})})\overset{(\ref{1222221})}{\cong}i^{!}\mathscr{S}_{V^{*}\boxtimes
V\boxtimes U}^{(\\{1\\},\\{2\\},\\{3\\})}.$

We have a canonical map

$\epsilon_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}:Sht_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}\longrightarrow
Hecke_{\mu_{\bullet}}^{(I_{1},\cdots,I_{k})}$

which is simply viewing
$\text{Frob}_{S}^{*}\mathcal{P}_{1}|_{\mathcal{O}^{\land}_{\mathcal{Y}_{[0,\infty)}(S),\sum
S_{i}^{\sharp}}}$ as $\mathcal{P}_{k+1}$. Similarly we have

$\epsilon_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}:Sht^{0}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}\longrightarrow
Hecke^{0}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}},$

which forms a commutative diagram

${Sht^{0}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$${Sht_{\mu_{\bullet}}}$${Sht_{\mu^{\prime}_{\bullet}}}$${Hecke^{0}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$${Hecke_{\mu_{\bullet}}}$${Hecke_{\mu^{\prime}_{\bullet}}.}$$\scriptstyle{\epsilon_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$$\scriptstyle{\epsilon_{\mu_{\bullet}}^{(I)}}$$\scriptstyle{\epsilon_{\mu^{\prime}_{\bullet}}^{(I)}}$

with the two squares Cartesian.

Consider the commutative diagram

${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Sht_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Hecke_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\}),0}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{0}^{(\\{1\\})}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{(\\{1\\},\\{2\\})}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),0}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{0}^{\text{loc},(\\{1\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\epsilon^{(\\{1\\},\\{2\\},\\{3\\}),0}_{V^{*}\boxtimes
V\boxtimes\mu}}$$\scriptstyle{\epsilon^{(\\{1\\},\\{2\\})}_{(0,\mu)}}$$\scriptstyle{\epsilon^{(\\{1\\},\\{2\\},\\{3\\})}_{V^{*}\boxtimes
V\boxtimes\mu}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\})}_{(\\{1\\},\\{2\\})}}$$\scriptstyle{\kappa^{(\\{1\\},\\{2\\},\\{3\\})}_{(\\{1,2\\},\\{3\\})}}$$\scriptstyle{i\times\text{id}}$$\scriptstyle{\pi\times\text{id}}$$\scriptstyle{i^{\text{loc}}\times\text{id}}$$\scriptstyle{\pi^{\text{loc}}\times\text{id}}$

Clearly the squares are Cartesian, so we can pullback cohomological
correspondence $\mathscr{C}_{\sharp}^{+}$ supported on $Hecke_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}$ to a cohomological
correspondence

$\mathscr{C}_{\sharp}:p_{1}^{*}\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})}\longrightarrow
p_{2}^{!}\epsilon^{(\\{1\\},\\{2\\},\\{3\\}),*}_{V^{*}\boxtimes
V\boxtimes\mu}\mathscr{S}_{V^{*}\boxtimes V\boxtimes
U}^{(\\{1\\},\\{2\\},\\{3\\})}$ (17)

supported on $Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}$. As
$\mathscr{C}_{\sharp}^{+}$ is by definition the pullback along
$\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}$ of
$\mathscr{C}_{\sharp}^{+,0}\boxtimes\text{id}$ supported on
$Hecke_{V^{*}\boxtimes V}^{(\\{1\\},\\{2\\}),0}\times
Hecke_{\mu,\eta}^{(\\{1\\})}$, we have that $\mathscr{C}_{\sharp}$ is the
pullbakc of $\mathscr{C}_{\sharp}^{+,0}\boxtimes\text{id}$ along
$\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}\circ\epsilon^{(\\{1\\},\\{2\\},\\{3\\}),0}_{V^{*}\boxtimes
V\boxtimes\mu}$, i.e.

$(\kappa^{(\\{1\\},\\{2\\},\\{3\\}),0}_{(\\{1,2\\},\\{3\\})}\circ\epsilon^{(\\{1\\},\\{2\\},\\{3\\}),0}_{V^{*}\boxtimes
V\boxtimes\mu})^{*}\mathscr{C}_{\sharp}^{+,0}\boxtimes\text{id}=\mathscr{C}_{\sharp}.$

Further, we know that $\mathscr{C}_{\sharp}^{+,0}$ is the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},+,0}$ along the canonical restriction map,
so $\mathscr{C}_{\sharp}$ is the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},+,0}\boxtimes\text{id}$ all the way from the
bottom of the diagram.

The key observation for us is that the above diagram factorizes through local
Shtukas at the special fiber, namely we have the following commutative diagram
whose composition is the same as that of the above diagram.

${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Sht_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc},W,0}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{0}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),0}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{0}^{\text{loc},(\\{1\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}\times\text{id}}$$\scriptstyle{p_{1}\times\text{id}}$$\scriptstyle{\epsilon^{\text{loc},(\\{1\\},\\{2\\}),0}_{V^{*}\boxtimes
V}\times\text{id}}$$\scriptstyle{\epsilon^{\text{loc},(\\{1\\})}_{0}\times\text{id}}$$\scriptstyle{\epsilon^{\text{loc},(\\{1\\},\\{2\\})}_{V^{*}\boxtimes
V}\times\text{id}}$$\scriptstyle{i^{\text{loc}}\times\text{id}}$$\scriptstyle{\pi^{\text{loc}}\times\text{id}}$

where the vertical maps along the first lines are restriction to the local
Shtukas at the special fiber.

The restriction map

$Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}\longrightarrow
Sht_{V^{*}\boxtimes V}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}$

in the diagram is a Shtuka version of
$\kappa_{(\\{1,2\\},\\{3\\})}^{(\\{1\\},\\{2\\},\\{3\\})}$, namely sending

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\mathcal{P}_{3}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$

to

${\mathcal{P}_{1}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})}}$${\mathcal{P}_{2}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{3}\circ\varphi_{2}}$

and

${\mathcal{P}_{3}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}})}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}|_{\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S^{\sharp}})},}$$\scriptstyle{\varphi_{3}}$

and similarly for the other two. Since the modification $\varphi_{3}$ has leg
at characteristic 0 untilt, it is an isomorphism after restricting to
$\text{Spec}(\mathcal{O}^{\wedge}_{\mathcal{Y}_{[0,\infty)}(S),S})$.

Similar to the $p$-adic Shtuka case, we can pullback
$\mathscr{C}_{\sharp}^{\text{loc},+,0}$ in (12) along

${Sht_{V^{*}\boxtimes
V}^{\text{loc},W,0}}$${Sht_{0}^{\text{loc},W}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc},W}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),0}}$${Hecke_{0}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})}}$$\scriptstyle{p_{2}^{\text{loc}}}$$\scriptstyle{p_{1}^{\text{loc}}}$$\scriptstyle{\epsilon^{\text{loc},W,0}_{V^{*}\boxtimes
V}}$$\scriptstyle{\epsilon^{\text{loc},W}_{0}}$$\scriptstyle{\epsilon^{\text{loc},W}_{V^{*}\boxtimes
V}}$$\scriptstyle{i^{\text{loc}}}$$\scriptstyle{\pi^{\text{loc}}}$ (18)

to obtain the Witt vector creation cohomological correspondence

$\mathscr{C}_{\sharp}^{\text{loc},0,W}:p_{1}^{\text{loc},*}\epsilon^{\text{loc},W,*}_{0}\mathscr{S}_{1}^{(\\{1\\})}\longrightarrow
p_{2}^{\text{loc},!}\epsilon^{\text{loc},W,*}_{V^{*}\boxtimes
V}\mathscr{S}_{V^{*}\boxtimes V}^{(\\{1\\},\\{2\\})}$

supported on $Sht_{V^{*}\boxtimes V}^{\text{loc},W,0}$.

One important thing to notice is that the truncated version of diagram (18) is
really the $\diamond\diamond$-analytification of the diagram below of perfect
stacks

${Sht_{V^{*}\boxtimes
V}^{\text{loc}(m,n),0}}$${Sht_{0}^{\text{loc}(m,n)}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc}(m,n)}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc}(m),0}}$${Hecke_{0}^{\text{loc}(m)}}$${Hecke_{V^{*}\boxtimes
V}^{\text{loc}(m)}}$$\scriptstyle{p_{2}^{\text{loc}(m,n)}}$$\scriptstyle{p_{1}^{\text{loc}(m,n)}}$$\scriptstyle{\epsilon^{\text{loc}(m,n),0}_{V^{*}\boxtimes
V}}$$\scriptstyle{\epsilon^{\text{loc}(m,n)}_{0}}$$\scriptstyle{\epsilon^{\text{loc}(m,n)}_{V^{*}\boxtimes
V}}$$\scriptstyle{i^{\text{loc}(m)}}$$\scriptstyle{\pi^{\text{loc}(m)}}$

which defines $\mathscr{C}_{\sharp}^{\text{loc}(m,n),0}$ by pullback of
$\mathscr{C}_{\sharp}^{\text{loc}(m),+,0}$. We know that
$\mathscr{C}_{\sharp}^{\text{loc}(m),+,0,\diamond\diamond}=\mathscr{C}_{\sharp}^{\text{loc},+,0}$
so it follows from proposition A.4 that

$\mathscr{C}_{\sharp}^{\text{loc},0,W}=\mathscr{C}_{\sharp}^{\text{loc}(m,n),0,\diamond\diamond}$

under the identification in appendix A.3.

Then by what we have just seen, we have

######  Proposition 3.7.

The creation cohomological correspondence $\mathscr{C}_{\sharp}$ of (17) is
naturally identified with the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},0,W}\boxtimes\text{id}$ along

${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Sht_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc},W,0}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{0}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{V^{*}\boxtimes
V}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}^{\text{loc}}\times\text{id}}$$\scriptstyle{p_{1}^{\text{loc}}\times\text{id}}$

Moreover, $\mathscr{C}^{\text{loc}}$ is the $\diamond\diamond$-analytification
of the corresponding truncated Witt vector version as considered in [1].

We have defined the creation correspondence on $Sht_{\mu,\eta}$, we next make
the observation that it can be extended to $Sht_{\mu}^{(\\{1\\})}$. Let

$Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}:=Sht_{(\delta^{*},\mu,\delta)}^{(\\{1,2\\},\\{3\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\mathcal{O}_{L})\times Spd(\overline{\mathbb{F}_{p}})},$

i.e. it parameterizes

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

where the second modification takes place at the characteristic $p$ untilt and
is bounded by $\delta$. The first modification has two poles, the first is
fixed at the characteristic $p$ untilt where the modification is bounded by
$\delta^{*}$, while the second varies along $Spd(\mathcal{O}_{L})$ over which
the modification is bounded by $\mu$.

We first observe that $Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}$ extends $Sht_{V^{*}\boxtimes
V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$.

######  Lemma 3.8.

There is a natural identification

$Sht_{V^{*}\boxtimes\mu\boxtimes V}^{(\\{1,2\\},\\{3\\})}|_{Spd(L)}\cong
Sht_{V^{*}\boxtimes V\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$

###### Proof.

When the two legs of $\mathcal{P}_{1}\dashrightarrow\mathcal{P}_{2}$ are
different, it is equivalent to

$\mathcal{P}_{1}\dashrightarrow\mathcal{P}_{3}\dashrightarrow\mathcal{P}_{2}$

where the first modification takes places at characteristic $p$ untilt, and
the second is at $Spd(L)$. Now as the two legs of

$\mathcal{P}_{3}\dashrightarrow\mathcal{P}_{2}\dashrightarrow\text{Frob}_{S}^{*}\mathcal{P}_{1}$

are different, it is equivalent to

$\mathcal{P}_{3}\dashrightarrow\mathcal{P}_{2}^{\prime}\dashrightarrow\text{Frob}_{S}^{*}\mathcal{P}_{1}$

where the first modification takes place at the characteristic $p$ untilt, and
the second takes place at characteristic zero untilt. ∎

Let

$Sht_{\mu|V^{*}\boxtimes\mu\boxtimes V}^{0}\subset
Sht^{(\\{1\\})|(\\{1,2\\},\\{3\\}),0}_{\mu|(\delta^{*},\mu,\delta)}$ (19)

be the v-substack parameterizes modifications

${\mathcal{P}}$${\text{Frob}_{S}^{*}\mathcal{P}}$${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

with the first line in $Sht_{\mu}$ and the second line in
$Sht_{V^{*}\boxtimes\mu\boxtimes V}^{(\\{1,2\\},\\{3\\})}$.

We have a commutative diagram

${Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V}^{0}}$${Sht_{\mu}^{(\\{1\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}}$${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V}}$${Hecke_{\mu}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}.}$$\scriptstyle{\epsilon_{\mu|V^{*}\boxtimes\mu\boxtimes
V}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$$\scriptstyle{\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}}$ (20)

We identify $Hecke_{\mu}^{(\\{1\\})}$ with
$Hecke_{(0,\mu)}^{(\\{1\\},\\{2\\})}$, then we have a cartesian diagram

${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V}}$${Hecke_{(0,\mu)}^{(\\{1\\},\\{2\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}}$${Hecke^{(\\{1,2\\})}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi^{(\\{1\\},\\{2\\})}}$$\scriptstyle{\pi^{(\\{1,2\\},\\{3\\})}}$

where we abuse the notation by writing

$\pi^{(\\{1,2\\},\\{3\\})}:Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}\rightarrow Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2,3\\})}\subset Hecke^{(\\{1,2\\})}$

where we can view $Hecke_{V^{*}\boxtimes\mu\boxtimes V}^{(\\{1,2,3\\})}$ as a
subspace of $Hecke^{(\\{1,2\\})}$ because the legs indexed by $V$ and $V^{*}$
are fixed at the characteristic $p$ untilt. Then

$Corr_{Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes V}}(\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})},\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})=Hom(p_{1}^{*}\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})},p_{2}^{!}\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})$ $=Hom(p_{2,!}p_{1}^{*}\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})},\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})=Hom(\pi^{(\\{1,2\\},\\{3\\}),*}(\pi^{(\\{1\\},\\{2\\})})_{!}\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})},\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})$
$=Hom((\pi^{(\\{1\\},\\{2\\})})_{!}\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})},(\pi^{(\\{1,2\\},\\{3\\})})_{*}\mathscr{S}_{V^{*}\boxtimes
U\boxtimes V}^{(\\{1,2\\},\\{3\\})})$ $=Hom(\mathscr{S}_{1\boxtimes
U}^{(\\{1,2\\})},\mathscr{S}_{(V^{*}\otimes V)\boxtimes
U}^{(\\{1,2\\})})=Hom_{\hat{G}^{2}}(1\boxtimes U,(V^{*}\otimes V)\boxtimes U)$

where we use 2) and 3) of theorem 3.1 in the second last equality, and
geometric Satake of [4] theorem<EMAIL_ADDRESS>in the last equality. More precisely,
3) of theorem 3.1 tells us that

$(\pi^{(\\{1,2\\},\\{3\\})})_{!}\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})}\cong\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2,3\\})}$

on $Hecke_{V^{*}\boxtimes U\boxtimes V}^{(\\{1,2,3\\})}$, then 2) of theorem
3.1 gives us

$\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2,3\\})}\cong\mathscr{S}_{(V^{*}\otimes V)\boxtimes
U}^{(\\{1,2\\})}$

under the inclusion $Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2,3\\})}\subset Hecke^{(\\{1,2\\})}$.

Now under the above identification, the canonical morphism $1\rightarrow
V\otimes V^{*}$ times the identity of $U$ gives us a cohomological
correspondence

$\mathscr{C}_{\sharp}^{+}:(Hecke_{(0,\mu)}^{(\\{1\\},\\{2\\})},\mathscr{S}_{1\boxtimes
U}^{(\\{1\\},\\{2\\})})\longrightarrow(Hecke_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})},\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})$ (21)

supported on $Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes V}$. Since the squares
in diagram (20) are cartesian, we can pullback $\mathscr{C}_{\sharp}^{+}$
along $\epsilon_{\mu|V^{*}\boxtimes\mu\boxtimes V}$ to obtain the desired
creation correspondence

$\mathscr{C}_{\sharp}:(Sht_{\mu}^{(\\{1\\})},\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})})\longrightarrow(Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})},\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})$

supported on $Sht_{\mu|V^{*}\boxtimes\mu\boxtimes V}^{0}$. To summarize, we
have the proposition.

######  Proposition 3.9.

The creation correspondence (17) extends naturally to a cohomological
correspondence

$\mathscr{C}_{\sharp}:(Sht_{\mu}^{(\\{1\\})},\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})})\longrightarrow(Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})},\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1,2\\},\\{3\\})})$ (22)

supported on $Sht_{\mu|V^{*}\boxtimes\mu\boxtimes V}^{0}$.

######  Remark 3.10.

The reason we use $Sht_{V^{*}\boxtimes\mu\boxtimes V}^{(\\{1,2\\},\\{3\\})}$
instead of $Sht_{V^{*}\boxtimes\mu\boxtimes V}^{(\\{1\\},\\{2\\},\\{3\\})}$ is
explained in the paragraph before [1] remark 6.1.10, namely, we want the
corresponding (integral) excursion operators have the same support as the
Hecke operators, so we can pullback the cohomological correspondence to
Shimura varieties.

Lastly, we identify the creation correspondence at the special fiber. As
usual, let

$Sht_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}:=Sht_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}|_{Spd(\overline{\mathbb{F}_{p}})},$

and

$Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}^{0}:=Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V}^{0}|_{Spd(\overline{\mathbb{F}_{p}})},$

i.e. it parameterizes

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1},}$

and

${\mathcal{P}_{1}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

respectively, where the modifications all take place at the characteristic $p$
untilt, $\mathcal{P}_{1}\dashrightarrow\mathcal{P}_{2}$ is bounded by
$V^{*}+\mu$ (the highest weights of irreducible components of $V^{*}$ \+
$\mu$), $\mathcal{P}_{2}\dashrightarrow\text{Frob}_{S}^{*}\mathcal{P}_{1}$ is
bounded by $V$, and
$\mathcal{P}_{1}\dashrightarrow\text{Frob}_{S}^{*}\mathcal{P}_{1}$ is bounded
by $\mu$.

As before, there is the diagram

${Sht^{0,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$${Sht_{\mu_{\bullet}}^{W}}$${Sht_{\mu^{\prime}_{\bullet}}^{W}}$${Hecke^{0,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$${Hecke_{\mu_{\bullet}}^{W}}$${Hecke_{\mu^{\prime}_{\bullet}}^{W},}$$\scriptstyle{\epsilon_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}^{W}}$$\scriptstyle{\epsilon_{\mu_{\bullet}}^{W}}$$\scriptstyle{\epsilon_{\mu^{\prime}_{\bullet}}^{W}}$

where the vertical maps are identifying $\mathcal{P}_{k+1}$ as
$\text{Frob}_{S}^{*}\mathcal{P}_{1}$.

On the other hand, we have the map

$Sht^{W}_{\mu_{\bullet}}\longrightarrow
Sht_{\mu_{\bullet},s}^{(\\{1\\},\\{2\\},\cdots,\\{k\\})}$

which is the restriction of Shtukas from $W(R^{+})$ to
$\mathcal{Y}_{[0,\infty)}(S)$, and similarly for
$Sht_{\mu_{\bullet}|\mu_{\bullet}}^{0,W}$, which forms a commutative diagram

${Sht^{0,W}_{\mu_{\bullet}|\mu_{\bullet}^{\prime}}}$${Sht_{\mu_{\bullet}}^{W}}$${Sht_{\mu^{\prime}_{\bullet}}^{W}}$${Sht^{(\\{1\\},\cdots,\\{k\\})|(\\{1\\},\cdots,\\{l\\}),0}_{\mu_{\bullet}|\mu_{\bullet}^{\prime},s}}$${Sht_{\mu_{\bullet},s}^{(\\{1\\},\\{2\\},\cdots,\\{k\\})}}$${Sht_{\mu^{\prime}_{\bullet},s}^{(\\{1\\},\\{2\\},\cdots,\\{l\\})}.}$

The diagram (20) restricted on the special fiber is

${Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}^{0}}$${Sht_{\mu,s}^{(\\{1\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}}$${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}.}$$\scriptstyle{\epsilon_{\mu|V^{*}\boxtimes\mu\boxtimes
V}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$$\scriptstyle{\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}}$

We can compose it with the previous one to obtain

${Sht^{0,W}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{W}}$${Sht_{V^{*}+\mu,V}^{W}}$${Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}^{0}}$${Sht_{\mu,s}^{(\\{1\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}}$${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})},}$$\scriptstyle{\epsilon_{\mu|V^{*}\boxtimes\mu\boxtimes
V}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$$\scriptstyle{\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1,2\\},\\{3\\})}}$

and we want to understand the pullback of (restriction to the special fiber
of) creation correspondence (22) along the upper half of the diagram. We note
that the diagram factorizes as

${Sht^{0,W}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{W}}$${Sht_{V^{*}+\mu,V}^{W}}$${Hecke_{\mu|(V^{*}+\mu,V)}^{0,W}}$${Hecke_{\mu}^{W}}$${Hecke_{V^{*}+\mu,V}^{W}}$${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})},}$$\scriptstyle{\epsilon_{\mu|(V^{*}+\mu,V)}^{W}}$$\scriptstyle{\epsilon_{\mu}^{W}}$$\scriptstyle{\epsilon_{V^{*}+\mu,V}^{W}}$$\scriptstyle{r_{\mu}}$$\scriptstyle{r_{V^{*}+\mu,V}}$
(23)

where the vertical maps of the second row are restriction of torsors from
$W(R^{+})$ to $\mathcal{Y}_{[0,\infty)}(R,R^{+})$. We abuse the notation as
usual by writing $V^{*}+\mu$ the highest weights of $V^{*}$ plus $\mu$.

We can pullback the cohomological correspondence (21) (restricting to the
special fiber) along the last diagram, which defines the creation
correspondence on the global Witt vector shtuka

$\mathscr{C}_{\sharp}^{W}:(Sht_{\mu}^{W},\epsilon_{\mu}^{W,*}r_{\mu}^{*}\mathscr{S}_{U}^{(\\{1\\})})\longrightarrow(Sht_{V^{*}+\mu,V}^{W},(\epsilon_{V^{*}+\mu,V}^{W})^{*}(r_{V^{*}+\mu,V})^{*}\mathscr{S}_{(V^{*}\otimes
U)\boxtimes V}^{(\\{1\\},\\{2\\})}).$ (24)

Moreover, the factorization tells us that $\mathscr{C}_{\sharp}^{W}$ is the
pullback of creation correspondence (22) along

${Sht^{0,W}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{W}}$${Sht_{V^{*}+\mu,V}^{W}}$${Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}^{0}}$${Sht_{\mu,s}^{(\\{1\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}.}$

The importance of $\mathscr{C}_{\sharp}^{W}$ is that it is analytification of
the creation correspondence for Witt vector Shtukas considered in [1]. Indeed,
we first observe that the restriction of $\mathscr{C}_{\sharp}^{+}$, as
defined in (21), to the special fiber has a local counter part
$\mathscr{C}_{\sharp}^{\text{loc},+}$, and $\mathscr{C}_{\sharp}^{+}$ is the
pullback of $\mathscr{C}_{\sharp}^{\text{loc},+}$ along the canonical
restriction diagram

${Hecke^{0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}}$${Hecke^{\text{loc},0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{\text{loc},(\\{1,2\\},\\{3\\})}.}$

The restriction to the special fiber is important here, as otherwise
$Hecke^{\text{loc},0}_{\mu|V^{*}\boxtimes\mu\boxtimes V,s}$ does not exist,
see remark 2.14. As always, $\mathscr{C}_{\sharp}^{+}$ being the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},+}$ follows ultimately from 6) of theorem
3.1. Now we can compose the diagram (23) with the above restriction diagram,
and obtain

${Sht^{0,W}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{W}}$${Sht_{V^{*}+\mu,V}^{W}}$${Hecke_{\mu|(V^{*}+\mu,V)}^{0,W}}$${Hecke_{\mu}^{W}}$${Hecke_{V^{*}+\mu,V}^{W}}$${Hecke^{\text{loc},0}_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}}$${Hecke_{\mu,s}^{\text{loc},(\\{1\\})}}$${Hecke_{V^{*}\boxtimes\mu\boxtimes
V,s}^{\text{loc},(\\{1,2\\},\\{3\\})},}$$\scriptstyle{\epsilon_{\mu|(V^{*}+\mu,V)}^{W}}$$\scriptstyle{\epsilon_{\mu}^{W}}$$\scriptstyle{\epsilon_{V^{*}+\mu,V}^{W}}$

and $\mathscr{C}_{\sharp}^{W}$ is the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},+}$ along the diagram. Up to truncation,
this is precisely

${Sht^{0,\text{loc}(m,n),\diamond}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{\text{loc}(m,n),\diamond}}$${Sht_{V^{*}+\mu,V}^{\text{loc}(m,n),\diamond}}$${Hecke_{\mu|(V^{*}+\mu,V)}^{0,\text{loc}(m),\diamond}}$${Hecke_{\mu}^{\text{loc}(m),\diamond}}$${Hecke_{V^{*}+\mu,V}^{\text{loc}(m),\diamond}}$${Hecke^{0,\text{loc}(m),\diamond\diamond}_{\mu|(V^{*}+\mu,V)}}$${Hecke_{\mu}^{\text{loc}(m),\diamond\diamond}}$${Hecke_{V^{*}+\mu,V}^{\text{loc}(m),\diamond\diamond},}$$\scriptstyle{\epsilon_{\mu|(V^{*}+\mu,V)}^{\text{loc}(m,n),\diamond}}$$\scriptstyle{\epsilon_{\mu}^{\text{loc}(m,n),\diamond}}$$\scriptstyle{\epsilon_{V^{*}+\mu,V}^{\text{loc}(m,n),\diamond}}$$\scriptstyle{a_{Hecke_{\mu|(V^{*}+\mu,V)}^{0,\text{loc}(m)}}}$$\scriptstyle{a_{Hecke_{\mu}^{\text{loc}(m)}}}$$\scriptstyle{a_{Hecke_{V^{*}+\mu,V}^{\text{loc}(m)}}}$

see appendix A.1 for the notation of $a_{?}$, and A.3 for the truncated Witt
vector Hecke stacks and Shtukas. We know from appendix A.3 that the truncated
version does not change cohomology theory, so we can naturally view the
cohomological correspondences on the (analytification of) truncated Hecke
stacks or Shtukas as being defined on the untruncated ones.

We can perform the same construction as of $\mathscr{C}_{\sharp}^{+}$ (in the
theory of perfect stacks) to obtain

$\mathscr{C}_{\sharp}^{\text{loc}(m),+}:(Hecke_{\mu}^{\text{loc}(m)},IC_{\mu})\longrightarrow(Hecke_{V^{*}+\mu,V}^{\text{loc}(m)},IC_{V^{*}+\mu,V})$

supported on

${Hecke^{0,\text{loc}(m)}_{\mu|(V^{*}+\mu,V)}}$${Hecke_{\mu}^{\text{loc}(m)}}$${Hecke_{V^{*}+\mu,V}^{\text{loc}(m)},}$

which is precisely the creation correspondence considered in [1]. Now 6) of
theorem 3.1 (and the comparison results in appendices A.1 and A.2) tells us
that

$\mathscr{C}_{\sharp}^{\text{loc}(m),+,\diamond\diamond}=\mathscr{C}_{\sharp}^{\text{loc},+}$

under the identification of sheaves on truncated and untruncated Hecke stacks
in A.3. By definition of $\mathscr{C}_{\sharp}^{\text{loc}(m),+,\diamond}$,
see appendix A.2, and $\mathscr{C}_{\sharp}^{+}$ being the pullback of
$\mathscr{C}_{\sharp}^{\text{loc},+}$, we have

$\mathscr{C}_{\sharp}^{\text{loc}(m),+,\diamond}=\mathscr{C}_{\sharp}^{+}.$

Then the commutation between the analytification and pullback of cohomological
correspondence (proposition A.4) tells us that

$\mathscr{C}_{\sharp}^{\text{loc}(m,n),\diamond}=\mathscr{C}_{\sharp}^{W}$

again under the identification of sheaves truncated and untruncated Hecke
Shtukas, where $\mathscr{C}_{\sharp}^{\text{loc}(m,n)}$ is the pullback of
$\mathscr{C}_{\sharp}^{\text{loc}(m)}$ along

${Sht^{0,\text{loc}(m,n)}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{\text{loc}(m,n)}}$${Sht_{V^{*}+\mu,V}^{\text{loc}(m,n)}}$${Hecke_{\mu|(V^{*}+\mu,V)}^{0,\text{loc}(m)}}$${Hecke_{\mu}^{\text{loc}(m)}}$${Hecke_{V^{*}+\mu,V}^{\text{loc}(m)}.}$$\scriptstyle{\epsilon_{\mu|(V^{*}+\mu,V)}^{\text{loc}(m,n)}}$$\scriptstyle{\epsilon_{\mu}^{\text{loc}(m,n)}}$$\scriptstyle{\epsilon_{V^{*}+\mu,V}^{\text{loc}(m,n)}}$

As usual, we use $\epsilon_{?}$ to denote the (in the sense of perfect stack)
forgetting map (viewing the Frobenius twist of $\mathcal{P}_{1}$ as
$\mathcal{P}_{2}$). $\mathscr{S}_{\sharp}^{\text{loc}(m,n)}$ is exactly the
construction of creation correspondence on the Witt vector Shtukas in [1], and
we have proved that $\mathscr{C}_{\sharp}^{W}$ is the analytification of the
creation correspondence in [1].

In summary, we have

######  Proposition 3.11.

$\mathscr{C}_{\sharp}^{W}$ is the pullback of the restriction of the creation
correspondence $\mathscr{C}_{\sharp}^{+}$ in (22) to
$Spd(\overline{\mathbb{F}_{p}})$ along

${Sht^{0,W}_{\mu|(V^{*}+\mu,V)}}$${Sht_{\mu}^{W}}$${Sht_{V^{*}+\mu,V}^{W}}$${Sht_{\mu|V^{*}\boxtimes\mu\boxtimes
V,s}^{0}}$${Sht_{\mu,s}^{(\\{1\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,s}^{(\\{1,2\\},\\{3\\})}.}$

Moreover,
$\mathscr{C}_{\sharp}^{W}=\mathscr{C}_{\sharp}^{\text{loc}(m,n),\diamond}$
being the analytification of the creation correspondence in [1].

###  3.4. Annihilation correspondence

Dually, we have the annihilation cohomological correspondence

$\mathscr{C}_{\flat}:p_{1}^{*}\epsilon^{(\\{1\\},\\{2\\},\\{3\\}),*}_{V\boxtimes
V^{*}\boxtimes\mu}\mathscr{S}_{V\boxtimes V^{*}\boxtimes
U}^{(\\{1\\},\\{2\\},\\{3\\})}\longrightarrow
p_{2}^{!}\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})}$ (25)

supported on

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})},}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

together with the local Witt vector version

$\mathscr{C}_{\flat}^{\text{loc},0,W}:p_{1}^{*}\epsilon^{\text{loc},W,*}_{V\boxtimes
V^{*}}\mathscr{S}_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}\longrightarrow
p_{2}^{!}\epsilon^{\text{loc},W,*}_{0}\mathscr{S}_{1}^{\text{loc},(\\{1\\})}$

supported on

${Sht_{V\boxtimes V^{*}}^{\text{loc},W,0}}$${Sht_{V\boxtimes
V^{*}}^{\text{loc},W}}$${Sht_{0}^{\text{loc},W},}$$\scriptstyle{p_{2}^{\text{loc}}}$$\scriptstyle{p_{1}^{\text{loc}}}$

which come from the evaluation map

$V\otimes V^{*}\longrightarrow 1.$

We have the similar proposition

######  Proposition 3.12.

The creation cohomological correspondence $\mathscr{C}_{\flat}$ of (25) is
naturally identified with the pullback of

$\mathscr{C}_{\flat}^{\text{loc},0,W}\boxtimes\text{id}:(Sht_{V\boxtimes
V^{*}}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})},(\epsilon_{V\boxtimes
V^{*}}^{\text{loc},W,*}\mathscr{S}_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})})\boxtimes\mathscr{S}_{U}^{\text{loc},(\\{1\\})})\longrightarrow$
$(Sht_{0}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})},(\epsilon_{0}^{\text{loc},W,*}\mathscr{S}_{1}^{\text{loc},(\\{1\\})})\boxtimes\mathscr{S}_{U}^{\text{loc},(\\{1\\})})$

along the restriction diagram

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\}),0}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{(0,\mu),\eta}^{(\\{1\\},\\{2\\})}}$${Sht_{V\boxtimes
V^{*}}^{\text{loc},W,0}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{V\boxtimes
V^{*}}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{0}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}^{\text{loc}}\times\text{id}}$$\scriptstyle{p_{1}^{\text{loc}}\times\text{id}}$

Moreover, $\mathscr{C}_{\flat}^{\text{loc},0,W}$ is the
$\diamond\diamond$-analytification of the corresponding truncated Witt vector
version $\mathscr{C}_{\flat}^{\text{loc}(m,n),0}$ as considered in [1].

Moreover, the annihilacotion correspondence extends integrally. Let

$Sht_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})}:=Sht_{(\delta,\delta^{*},\mu)}^{(\\{1\\},\\{2,3\\})}|_{Spd(\overline{\mathbb{F}_{p}})\times
Spd(\overline{\mathbb{F}_{p}})\times Spd(\mathcal{O}_{L}))},$

and

$Sht_{V\boxtimes V^{*}\boxtimes\mu|\mu}^{0}\subset
Sht^{(\\{1\\},\\{2,3\\})|(\\{1\\}),0}_{(\delta,\delta^{*},\mu)|\mu}$

be the v-substack parameterizes modifications

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\mathcal{P}_{1}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$

with the first line in $Sht_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})}$, and the second line in
$Sht_{\mu}^{(\\{1\\})}$.

######  Proposition 3.13.

The annihilation correspondence (25) extends to a cohomological correspondence

$\mathscr{C}_{\flat}:(Sht_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})},\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\}),*}\mathscr{S}_{V\boxtimes
V^{*}\boxtimes
U}^{(\\{1\\},\\{2,3\\})})\longrightarrow(Sht_{\mu}^{(\\{1\\})},\epsilon_{\mu}^{(\\{1\\}),*}\mathscr{S}_{U}^{(\\{1\\})})$
(26)

supported on $Sht_{\mu|V\boxtimes V^{*}\boxtimes\mu}^{0}$. More precisely,
$\mathscr{C}_{\flat}$ is the pullback along

${Sht_{V\boxtimes V^{*}\boxtimes\mu|\mu}^{0}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})}}$${Sht_{\mu}^{(\\{1\\})}}$${Hecke^{0}_{V\boxtimes
V^{*}\boxtimes\mu|\mu}}$${Hecke_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})}}$${Hecke_{\mu}^{(\\{1\\})}.}$$\scriptstyle{\epsilon_{\mu|V\boxtimes
V^{*}\boxtimes\mu}}$$\scriptstyle{\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})}}$$\scriptstyle{\epsilon_{\mu}^{(\\{1\\})}}$

of the correspondence

$\mathscr{C}_{\flat}^{+}:(Hecke_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2,3\\})},\mathscr{S}_{V\boxtimes
V^{*}\boxtimes
U}^{(\\{1\\},\\{2,3\\})})\longrightarrow(Hecke_{\mu}^{(\\{1\\})},\mathscr{S}_{U}^{(\\{1\\})})$
(27)

supported on $Hecke^{0}_{\mu|V\boxtimes V^{*}\boxtimes\mu}$ corresponding to
the evaluation map

$(V\otimes V^{*})\boxtimes U\longrightarrow 1\boxtimes U.$

We can identify the annihilation correspondence at the special fiber as well.

######  Proposition 3.14.

$\mathscr{C}_{\flat}^{W}$ is the pullback of the restriction of the
annihilation correspondence (26) to $Spd(\overline{\mathbb{F}_{p}})$ along the
diagram

${Sht^{0,W}_{(V,V^{*}+\mu)|\mu}}$${Sht_{V,V^{*}+\mu}^{W}}$${Sht_{\mu}^{W}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu|\mu,s}^{0}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,s}^{(\\{1\\},\\{2,3\\})}}$${Sht_{\mu,s}^{(\\{1\\})}}$

where

$\mathscr{C}_{\flat}^{W}:(Sht_{V,V^{*}+\mu}^{W},(\epsilon^{W}_{V,V^{*}+\mu})^{*}(r_{V,V^{*}+\mu})^{*}\mathscr{S}_{V\boxtimes(V^{*}\otimes
U)}^{(\\{1\\},\\{2\\})})\longrightarrow(Sht_{\mu}^{W},(\epsilon_{\mu}^{W})^{*}r_{\mu}^{*}\mathscr{S}_{U}^{(\\{1\\})})$
(28)

is the pullback of (27) (restricted to $Spd(\overline{\mathbb{F}_{p}})$) along

${Sht_{(V,V^{*}+\mu)|\mu}^{0,W}}$${Sht_{V,V^{*}+\mu}^{W}}$${Sht_{\mu}^{W}}$${Hecke_{(V,V^{*}+\mu)|\mu}^{0,W}}$${Hecke_{V,V^{*}+\mu}^{W}}$${Hecke_{\mu}^{W}}$${Hecke^{0}_{V\boxtimes(V^{*}\otimes\mu)|\mu,s}}$${Hecke_{V\boxtimes(V^{*}\otimes\mu),s}^{(\\{1\\},\\{2\\})}}$${Hecke_{\mu,s}^{(\\{1\\})}.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\epsilon^{W}_{(V,V^{*}+\mu)|\mu}}$$\scriptstyle{\epsilon^{W}_{V,V^{*}+\mu}}$$\scriptstyle{\epsilon_{\mu}^{W}}$$\scriptstyle{r_{V,V^{*}+\mu}}$$\scriptstyle{r_{\mu}}$

Moreover, $\mathscr{C}_{\flat}^{W}$ is the analytification of the annihilation
correspondence in [1], i.e.

$\mathscr{C}_{\flat}^{W}=\mathscr{C}_{\flat}^{\text{loc}(m,n),\diamond}$

under the identification in A.2 between sheaves on truncated and untruncated
Shtukas.

###  3.5. Partial Frobenius

Lastly, we have the partial Frobenius morphism

$F:Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}\longrightarrow
Sht_{V^{*}\boxtimes\mu\boxtimes V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$

sending

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\mathcal{P}_{3}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$

to

${\mathcal{P}_{2}}$${\mathcal{P}_{3}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\text{Frob}_{S}^{*}\mathcal{P}_{2},}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$$\scriptstyle{\text{Frob}_{S}^{*}\varphi_{1}}$

which is the same as

${\mathcal{P}_{2}}$${\mathcal{P}_{3}^{\prime}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\text{Frob}_{S}^{*}\mathcal{P}_{2},}$$\scriptstyle{\varphi_{3}^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\text{Frob}_{S}^{*}\varphi_{1}}$

where $\varphi_{3}^{\prime}$ is bounded by $\mu$ at a characteristic 0 leg,
and $\varphi_{2}^{\prime}$ is bounded by $V^{*}$ at a characteristic $p$ leg,
i.e. it defines an element of $Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}$. Note that
$\text{Frob}_{S}^{*}\varphi_{1}$ has modification type ${}^{\sigma}V$, which
is equal to $V$ by our assumption that $V$ is a representation of ${}^{L}G$.

Now we have a commutative diagram

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Hecke_{V}^{(\\{1\\})}\times
Hecke_{V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\})}}$${Hecke_{V}^{(\\{1\\})}\times
Hecke_{V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\})}}$$\scriptstyle{F}$$\scriptstyle{\kappa_{(\\{1\\},\\{2,3\\})}^{(\\{1\\},\\{2\\},\\{3\\})}\circ\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\})}}$$\scriptstyle{\kappa_{(\\{1,2\\},\\{3\\})}^{(\\{1\\},\\{2\\},\\{3\\})}\circ\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1\\},\\{2\\},\\{3\\})}}$$\scriptstyle{\text{Frob}\times\text{id}}$
(29)

using the canonical isomorphism

$\mathscr{C}_{F}^{+}:\text{Frob}^{*}\mathscr{S}_{V}^{(\\{1\\})}\cong\mathscr{S}_{V}^{(\\{1\\})}$

we obtain

$F^{*}\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1\\},\\{2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes U\boxtimes
V}^{(\\{1\\},\\{2\\},\\{3\\})}\cong F^{*}\epsilon_{V^{*}\boxtimes\mu\boxtimes
V}^{(\\{1\\},\\{2\\},\\{3\\}),*}\kappa_{(\\{1,2\\},\\{3\\})}^{(\\{1\\},\\{2\\},\\{3\\}),*}(\mathscr{S}_{V}^{(\\{1\\})}\boxtimes\mathscr{S}_{V^{*}\boxtimes
U}^{(\\{1\\},\\{2\\})})$ $\cong\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),*}\kappa_{(\\{1\\},\\{2,3\\})}^{(\\{1\\},\\{2\\},\\{3\\}),*}(\text{Frob}\times\text{id})^{*}(\mathscr{S}_{V}^{(\\{1\\})}\boxtimes\mathscr{S}_{V^{*}\boxtimes
U}^{(\\{1\\},\\{2\\})})$
$\overset{\mathscr{C}_{F}^{+}\boxtimes\text{id}}{\cong}\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),*}\kappa_{(\\{1\\},\\{2,3\\})}^{(\\{1\\},\\{2\\},\\{3\\}),*}(\mathscr{S}_{V}^{(\\{1\\})}\boxtimes\mathscr{S}_{V^{*}\boxtimes
U}^{(\\{1\\},\\{2\\})})\cong\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes
U\boxtimes V}^{(\\{1\\},\\{2\\},\\{3\\})}$

which defines a pullback cohomological correspondence

$\mathscr{C}_{F,\eta}:(Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})},\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes
U\boxtimes
V}^{(\\{1\\},\\{2\\},\\{3\\})})\longrightarrow(Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})},\epsilon_{V\boxtimes
V^{*}\boxtimes\mu}^{(\\{1\\},\\{2\\},\\{3\\}),*}\mathscr{S}_{V^{*}\boxtimes
U\boxtimes V}^{(\\{1\\},\\{2\\},\\{3\\})})$

supported on

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}.}$$\scriptstyle{F}$

Similarly, we have the local Witt vector partial Frobenius

$F^{\text{loc},W}:Sht^{\text{loc},W}_{V\boxtimes V^{*}}\longrightarrow
Sht^{\text{loc},W}_{V^{*}\boxtimes V}$

defined by sending

${\mathcal{P}_{1}}$${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$

to

${\mathcal{P}_{2}}$${\text{Frob}_{S}^{*}\mathcal{P}_{1}}$${\text{Frob}_{S}^{*}\mathcal{P}_{2},}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\text{Frob}_{S}^{*}\varphi_{1}}$

and there is a commutative diagram

${Sht^{\text{loc},W}_{V\boxtimes V^{*}}}$${Sht^{\text{loc},W}_{V^{*}\boxtimes
V}}$${Hecke_{V}^{\text{loc},(\\{1\\})}\times
Hecke_{V^{*}}^{\text{loc},(\\{1\\})}}$${Hecke_{V}^{\text{loc},(\\{1\\})}\times
Hecke_{V^{*}}^{\text{loc},(\\{1\\})}}$$\scriptstyle{F^{\text{loc},W}}$$\scriptstyle{\kappa_{(\\{1\\},\\{2\\})}^{\text{loc},(\\{1\\},\\{2\\})}\circ\epsilon_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}}$$\scriptstyle{\kappa_{(\\{1\\},\\{2\\})}^{\text{loc},(\\{1\\},\\{2\\})}\circ\epsilon_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}}$$\scriptstyle{\text{Frob}\times\text{id}}$

which induces a pullback cohomological correspondence

$\mathscr{C}_{F}^{\text{loc},0,W}:F^{\text{loc},W*}\epsilon_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\}),*}\mathscr{S}_{V^{*}\boxtimes
V}^{\text{loc},(\\{1\\},\\{2\\})}\cong\epsilon_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\}),*}\mathscr{S}_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}$

supported on

${Sht^{\text{loc},W}_{V\boxtimes V^{*}}}$${Sht^{\text{loc},W}_{V^{*}\boxtimes
V}}$${Sht^{\text{loc},W}_{V\boxtimes V^{*}}.}$$\scriptstyle{F^{\text{loc},W}}$

As usual, this is the $\diamond\diamond$-analytification of the corresponding
truncated Witt vector version $\mathscr{C}_{F}^{\text{loc}(m,n),0}$.

Moreover, we have the commutative diagram

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht^{\text{loc},W}_{V\boxtimes
V^{*}}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht^{\text{loc},W}_{V^{*}\boxtimes
V}\times Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{V\boxtimes
V^{*}}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{F}$$\scriptstyle{F^{\text{loc},W}\times\text{id}}$

where the vertical arrows are as before restriction maps. The two squares are
Cartesian, so we can pullback the cohomological correspondence
$\mathscr{C}_{F}^{\text{loc},0,W}\boxtimes\text{id}$, and this is exactly
$\mathscr{C}_{F,\eta}$. Indeed, the composition of (30) with
$\text{id}\times\kappa_{(\\{1\\},\\{2\\})}^{(\\{1\\},\\{2\\})}$ factorizes
through

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht^{\text{loc},W}_{V\boxtimes
V^{*}}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht^{\text{loc},W}_{V^{*}\boxtimes
V}\times
Hecke_{\mu,\eta}^{(\\{1\\})}}$${Hecke_{V}^{\text{loc},(\\{1\\})}\times
Hecke_{V^{*}}^{\text{loc},(\\{1\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Hecke_{V}^{\text{loc},(\\{1\\})}\times
Hecke_{V^{*}}^{\text{loc},(\\{1\\})}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$$\scriptstyle{F}$$\scriptstyle{F^{\text{loc},W}\times\text{id}}$$\scriptstyle{\kappa_{(\\{1\\},\\{2\\})}^{\text{loc},(\\{1\\},\\{2\\})}\circ\epsilon_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}\times\text{id}}$$\scriptstyle{\kappa_{(\\{1\\},\\{2\\})}^{\text{loc},(\\{1\\},\\{2\\})}\circ\epsilon_{V\boxtimes
V^{*}}^{\text{loc},(\\{1\\},\\{2\\})}\times\text{id}}$$\scriptstyle{\text{Frob}\times\text{id}\times\text{id}}$

from which it is easy to deduce the identification of the cohomological
correspondence. To summarize, we have

######  Proposition 3.15.

The cohomological correspondence $\mathscr{C}_{F,\eta}$ is the same as the
pullback of $\mathscr{C}_{F}^{\text{loc},0,W}\boxtimes\text{id}$ along the
diagram

${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V^{*}\boxtimes\mu\boxtimes
V,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht_{V\boxtimes
V^{*}\boxtimes\mu,\eta}^{(\\{1\\},\\{2\\},\\{3\\})}}$${Sht^{\text{loc},W}_{V\boxtimes
V^{*}}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht^{\text{loc},W}_{V^{*}\boxtimes
V}\times Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}}$${Sht_{V\boxtimes
V^{*}}^{\text{loc},W}\times
Hecke_{\mu,\eta}^{\text{loc},(\\{1\\})}.}$$\scriptstyle{F}$$\scriptstyle{F^{\text{loc},W}\times\text{id}}$

Moreover,
$\mathscr{C}_{F}^{\text{loc},0,W}=\mathscr{C}_{F}^{\text{loc}(m,n),0,\diamond\diamond}$,
being the $\diamond\diamond$-analytification of the truncated Witt vector
version considered in [1].