1.1.2

$\operatorname{Bun}_{G,N}$ is a smooth algebraic stack over $\mathbb{F}_{q}$, locally of finite type.

1.1.4

The prestack $\operatorname{Hecke}_{G,N,I}$ is an inductive limit of algebraic stacks over $(X\smallsetminus N)^{I}$. We define the morphism of paws $\operatorname{Hecke}_{G,N,I}\rightarrow (X\smallsetminus N)^{I}$ by sending $((x_{i})_{i\in I},({\mathcal{G}},\unicode[STIX]{x1D713})\xrightarrow[{}]{\unicode[STIX]{x1D719}}({\mathcal{G}}^{\prime },\unicode[STIX]{x1D713}^{\prime }))$ to $(x_{i})_{i\in I}$.

1.1.5

We denote by $\operatorname{pr}_{0}$ (respectively $\operatorname{pr}_{1}$) the projection $\operatorname{Hecke}_{G,N,I}\rightarrow \operatorname{Bun}_{G,N}$ which sends $((x_{i})_{i\in I},({\mathcal{G}},\unicode[STIX]{x1D713})\xrightarrow[{}]{\unicode[STIX]{x1D719}}({\mathcal{G}}^{\prime },\unicode[STIX]{x1D713}^{\prime }))$ to $({\mathcal{G}},\unicode[STIX]{x1D713})$ (respectively to $({\mathcal{G}}^{\prime },\unicode[STIX]{x1D713}^{\prime })$).

Let $\operatorname{Frob}:\operatorname{Bun}_{G,N}\rightarrow \operatorname{Bun}_{G,N}$ be the Frobenius morphism over $\mathbb{F}_{q}$. With the notation in 0.0.11, for any affine scheme $S$ over $\mathbb{F}_{q}$, the morphism $\operatorname{Frob}:\operatorname{Bun}_{G,N}(S)\rightarrow \operatorname{Bun}_{G,N}(S)$ is given by $({\mathcal{G}},\unicode[STIX]{x1D713})\rightarrow (\text{}^{\unicode[STIX]{x1D70F}}{\mathcal{G}},^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D713})$.

Definition 1.1.6. We define the prestack of shtukas $\operatorname{Cht}_{G,N,I}$ to be the following fiber product.

(1.1)

1.1.7

Concretely, $\operatorname{Cht}_{G,N,I}$ is the prestack which associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the groupoid $\operatorname{Cht}_{G,N,I}(S)$ classifying the following data:

1. (i) $(x_{i})_{i\in I}\in (X\smallsetminus N)^{I}(S)$;

2. (ii) $({\mathcal{G}},\unicode[STIX]{x1D713})\in \operatorname{Bun}_{G,N}(S)$;

3. (iii) an isomorphism of $G$-bundles $\unicode[STIX]{x1D719}:\left.{\mathcal{G}}\vphantom{\big|}\right|_{(X\times S)\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}\overset{{\sim}}{\rightarrow }\left.\text{}^{\unicode[STIX]{x1D70F}}{\mathcal{G}}\vphantom{\big|}\right|_{(X\times S)\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}$ which preserves the $N$-level structure, i.e. $^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D713}\circ \left.\unicode[STIX]{x1D719}\vphantom{\big|}\right|_{N\times S}=\unicode[STIX]{x1D713}$.

We define the morphism of paws $\mathfrak{p}_{G}:\operatorname{Cht}_{G,N,I}\rightarrow (X\smallsetminus N)^{I}$ by sending $((x_{i})_{i\in I},({\mathcal{G}},\unicode[STIX]{x1D713})\xrightarrow[{}]{\unicode[STIX]{x1D719}}(\text{}^{\unicode[STIX]{x1D70F}}{\mathcal{G}},^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D713}))$ to $(x_{i})_{i\in I}$.

1.1.8

The prestack $\operatorname{Cht}_{G,N,I}$ is an inductive limit of algebraic stacks over $(X\smallsetminus N)^{I}$.

1.1.9

We will omit the index $N$ if $N=\emptyset$.

We will need a local model of $\operatorname{Cht}_{G,N,I}$. For this, we recall the definition of Beilinson–Drinfeld affine grassmannians.

1.1.10

For $(x_{i})_{i\in I}\in X^{I}(S)$, $d\in \mathbb{N}$, we denote by $\unicode[STIX]{x1D6E4}_{\sum dx_{i}}$ the closed subscheme of $X\times S$ whose ideal is generated by $(\prod _{i\in I}t_{i})^{d}$ locally for the Zariski topology, where $t_{i}$ is an equation of the graph $\unicode[STIX]{x1D6E4}_{x_{i}}$. We define $\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}:=\mathop{\varinjlim }\nolimits_{d}\unicode[STIX]{x1D6E4}_{\sum dx_{i}}$ to be the formal neighborhood of $\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}}$ in $X\times S$.

A $G$-bundle on $\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}$ is a projective limit of $G$-bundles on $\unicode[STIX]{x1D6E4}_{\sum dx_{i}}$ as $d\rightarrow \infty$.

1.1.12

We have the morphism of paws: $\operatorname{Gr}_{G,I}\rightarrow X^{I}$. The fiber over $(x_{i})_{i\in I}\in X_{\overline{\mathbb{F}_{q}}}^{I}$ is $\prod _{y\in \{x_{i}|i\in I\}}\operatorname{Gr}_{G,y}$, where $\operatorname{Gr}_{G,y}$ is the usual affine grassmannian, i.e. the fpqc quotient $G_{{\mathcal{K}}_{y}}/G_{{\mathcal{O}}_{y}}$, where ${\mathcal{O}}_{y}$ is the complete local ring on $y$ and ${\mathcal{K}}_{y}$ is its field of fractions.

1.1.14

The fiber of $G_{I,\infty }$ over $(x_{i})_{i\in I}\in X_{\overline{\mathbb{F}_{q}}}^{I}$ is $\prod _{y\in \{x_{i}|i\in I\}}G_{{\mathcal{O}}_{y}}$.

1.1.15

The group scheme $G_{I,\infty }$ acts on $\operatorname{Gr}_{G,I}$ by changing the trivialization $\unicode[STIX]{x1D703}$. We denote by $[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]$ the quotient prestack. For any affine scheme $S$ over $\mathbb{F}_{q}$, $[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}](S)$ is the groupoid classifying the data (i), (ii) and (iii) in Definition 1.1.11.

1.1.16

We have a morphism of prestacks:

(1.2)

The following definition will be used in §4.

1.1.19

$\operatorname{Bun}_{G,N,I,d}$ is a smooth algebraic stack over $(X\smallsetminus N)^{I}$. Its fiber over a point $(x_{i})_{i\in I}\in (X\smallsetminus N)^{I}(\mathbb{F}_{q})$ is $\operatorname{Bun}_{G,N+\sum dx_{i}}$.

1.1.20

The definitions and constructions in this subsection work for all affine smooth geometrically connected algebraic groups over $\mathbb{F}_{q}$ (we will use these for parabolic subgroups of $G$ and their Levi quotients).

1.2.1

Let $P$ be a parabolic subgroup of $G$ and let $M$ be its Levi quotient. Applying the definitions and constructions in §1.1 to $P$ and $M$, respectively, we define $\operatorname{Bun}_{P,N}$, $\operatorname{Cht}_{P,N,I}$, $\operatorname{Gr}_{P,I}$, $P_{I,\infty }$, $\unicode[STIX]{x1D716}_{P,N,I,\infty }$ and $\operatorname{Bun}_{M,N}$, $\operatorname{Cht}_{M,N,I}$, $\operatorname{Gr}_{M,I}$, $M_{I,\infty }$, $\unicode[STIX]{x1D716}_{M,N,I,\infty }$.

1.2.3

The morphisms of groups $G{\hookleftarrow}P{\twoheadrightarrow}M$ induce morphisms of prestacks over $\operatorname{Spec}\mathbb{F}_{q}$:

(1.3)$$\begin{eqnarray}\operatorname{Bun}_{G,N}\xleftarrow[{}]{i^{\text{Bun}}}\operatorname{Bun}_{P,N}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\text{Bun}}}\operatorname{Bun}_{M,N}\!.\end{eqnarray}$$

Construction 1.2.4. The morphisms of groups $G{\hookleftarrow}P{\twoheadrightarrow}M$ induce morphisms of prestacks over $(X\smallsetminus N)^{I}$.

(1.4)

More concretely, for any affine scheme $S$ over $\mathbb{F}_{q}$:

$i:\operatorname{Cht}_{P,N,I}(S)\rightarrow \operatorname{Cht}_{G,N,I}(S)$ is given by $({\mathcal{P}}\rightarrow ^{\unicode[STIX]{x1D70F}}{\mathcal{P}})\mapsto ({\mathcal{P}}\overset{P}{\times }G\rightarrow ^{\unicode[STIX]{x1D70F}}{\mathcal{P}}\overset{P}{\times }G)$ where the level structure $\unicode[STIX]{x1D713}:\left.{\mathcal{P}}\vphantom{\big|}\right|_{N\times S}\overset{{\sim}}{\rightarrow }\left.P\vphantom{\big|}\right|_{N\times S}$ is sent to $\unicode[STIX]{x1D713}\overset{P}{\times }G$;

$\unicode[STIX]{x1D70B}:\operatorname{Cht}_{P,N,I}(S)\rightarrow \operatorname{Cht}_{M,N,I}(S)$ is given by $({\mathcal{P}}\rightarrow ^{\unicode[STIX]{x1D70F}}{\mathcal{P}})\mapsto ({\mathcal{P}}\overset{P}{\times }M\rightarrow ^{\unicode[STIX]{x1D70F}}{\mathcal{P}}\overset{P}{\times }M)$ where the level structure $\unicode[STIX]{x1D713}$ is sent to $\unicode[STIX]{x1D713}\overset{P}{\times }M$.

1.2.5

The morphisms of groups $G{\hookleftarrow}P{\twoheadrightarrow}M$ induce morphisms of ind-schemes over $X^{I}$:

(1.5)$$\begin{eqnarray}\operatorname{Gr}_{G,I}\xleftarrow[{}]{i^{0}}\operatorname{Gr}_{P,I}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{0}}\operatorname{Gr}_{M,I}\!.\end{eqnarray}$$

1.2.6

Let ${\mathcal{X}}$ (respectively ${\mathcal{Y}}$) be an (ind-)scheme over a base $S$ that is equipped with an action of a group scheme $A$ (respectively $B$) over $S$ from the right. Let $A\rightarrow B$ be a morphism of group schemes over $S$. Let ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ be a morphism of (ind-)schemes over $S$ which is $A$-equivariant (where $A$ acts on ${\mathcal{Y}}$ via $A\rightarrow B$). This morphism induces a morphism of quotient prestacks

$$\begin{eqnarray}[A\backslash {\mathcal{X}}]\rightarrow [B\backslash {\mathcal{Y}}].\end{eqnarray}$$

1.2.7

Applying 1.2.6 to $i^{0}:\operatorname{Gr}_{P,I}\rightarrow \operatorname{Gr}_{G,I}$ and $P_{I,\infty }{\hookrightarrow}G_{I,\infty }$, we obtain a morphism of prestacks:

$$\begin{eqnarray}\overline{i^{0}}:[P_{I,\infty }\backslash \operatorname{Gr}_{P,I}]\rightarrow [G_{I,\infty }\backslash \operatorname{Gr}_{G,I}].\end{eqnarray}$$

Applying 1.2.6 to $\unicode[STIX]{x1D70B}^{0}:\operatorname{Gr}_{P,I}\rightarrow \operatorname{Gr}_{M,I}$ and $P_{I,\infty }{\twoheadrightarrow}M_{I,\infty }$, we obtain a morphism of prestacks:

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D70B}^{0}}:[P_{I,\infty }\backslash \operatorname{Gr}_{P,I}]\rightarrow [M_{I,\infty }\backslash \operatorname{Gr}_{M,I}].\end{eqnarray}$$

1.2.8

The following diagram of prestacks is commutative.

(1.6)

1.3.1

Let $Z_{G}$ be the center of $G$ as defined in 0.0.3. We have an action of $\operatorname{Bun}_{Z_{G}}$ on $\operatorname{Bun}_{G,N}$ by twisting a $G$-bundle by a $Z_{G}$-bundle, i.e. the action of ${\mathcal{T}}_{Z}\in \operatorname{Bun}_{Z_{G}}$ is given by ${\mathcal{G}}\mapsto ({\mathcal{G}}\times {\mathcal{T}}_{Z})/Z_{G}$. Similarly, $\operatorname{Bun}_{Z_{G}}$ acts on $[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]$, i.e. the action of ${\mathcal{T}}_{Z}\in \operatorname{Bun}_{Z_{G}}$ is given by

$$\begin{eqnarray}({\mathcal{G}}\xrightarrow[{}]{\unicode[STIX]{x1D719}}{\mathcal{G}}^{\prime })\mapsto (({\mathcal{G}}\times \left.{\mathcal{T}}_{G}\vphantom{\big|}\right|_{\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}})/Z_{G}\xrightarrow[{}]{\unicode[STIX]{x1D719}}({\mathcal{G}}^{\prime }\times \left.{\mathcal{T}}_{G}\vphantom{\big|}\right|_{\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}})/Z_{G}).\end{eqnarray}$$

For ${\mathcal{T}}_{Z}\in \operatorname{Bun}_{Z_{G}}(\mathbb{F}_{q})$, we have a canonical identification ${\mathcal{T}}_{Z}\simeq ^{\unicode[STIX]{x1D70F}}{\mathcal{T}}_{Z}$. Thus $\operatorname{Bun}_{Z_{G}}(\mathbb{F}_{q})$ acts on $\operatorname{Cht}_{G,N,I}$ by twisting a $G$-bundle by a $Z_{G}$-bundle, i.e. the action of ${\mathcal{T}}_{Z}\in \operatorname{Bun}_{Z_{G}}(\mathbb{F}_{q})$ is given by $({\mathcal{G}}\xrightarrow[{}]{\unicode[STIX]{x1D719}}\text{}^{\unicode[STIX]{x1D70F}}{\mathcal{G}})\mapsto (({\mathcal{G}}\times {\mathcal{T}}_{Z})/Z_{G}\xrightarrow[{}]{\unicode[STIX]{x1D719}}\text{}^{\unicode[STIX]{x1D70F}}({\mathcal{G}}\times {\mathcal{T}}_{Z})/Z_{G})$.

The group $\unicode[STIX]{x1D6EF}$ defined in 0.0.4 acts on $\operatorname{Bun}_{G,N}$, $\operatorname{Cht}_{G,N,I}$ and $[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]$ via $\unicode[STIX]{x1D6EF}\rightarrow Z_{G}(\mathbb{A})\rightarrow \operatorname{Bun}_{Z_{G}}(\mathbb{F}_{q})$.

1.3.2

Note that the morphism $\unicode[STIX]{x1D716}_{G,N,I,\infty }$ defined in (1.2) is $\unicode[STIX]{x1D6EF}$-equivariant.

Now applying Definition 1.1.13 to $Z_{G}$ (respectively $G^{\text{ad}}$), we define a group scheme $(Z_{G})_{I,\infty }$ (respectively $G_{I,\infty }^{\text{ad}}$) over $X^{I}$. We have $G_{I,\infty }^{\text{ad}}=G_{I,\infty }/(Z_{G})_{I,\infty }$. The group scheme $(Z_{G})_{I,\infty }$ acts trivially on $\operatorname{Gr}_{G,I}$, so the action of $G_{I,\infty }$ on $\operatorname{Gr}_{G,I}$ factors through $G_{I,\infty }^{\text{ad}}$. We use this action to define the quotient prestack $[G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}]$. The morphism $G_{I,\infty }{\twoheadrightarrow}G_{I,\infty }^{\text{ad}}$ induces a morphism $[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]\rightarrow [G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}]$, which is $\unicode[STIX]{x1D6EF}$-equivariant for the trivial action of $\unicode[STIX]{x1D6EF}$ on $[G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}]$.

Hence the composition of morphisms

$$\begin{eqnarray}\operatorname{Cht}_{G,N,I}\xrightarrow[{}]{\unicode[STIX]{x1D716}_{G,N,I,\infty }}[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]\rightarrow [G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}]\end{eqnarray}$$

is $\unicode[STIX]{x1D6EF}$-equivariant. Thus it factors through

(1.7)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{G,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{G,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}].\end{eqnarray}$$

We will construct a variant of morphism (1.7) for algebraic stacks in 2.4.1.

1.3.3

$Z_{G}$ acts on a $P$-bundle via $Z_{G}{\hookrightarrow}P$. Just as in 1.3.1, we have an action of $\operatorname{Bun}_{Z_{G}}$ on $\operatorname{Bun}_{P,N}$ by twisting a $P$-bundle by a $Z_{G}$-bundle. This leads to an action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{P,N}$, $\operatorname{Cht}_{P,N,I}$ and $[P_{I,\infty }\backslash \operatorname{Gr}_{P,I}]$ via $\unicode[STIX]{x1D6EF}\rightarrow Z_{G}(\mathbb{A})\rightarrow \operatorname{Bun}_{Z_{G}}(\mathbb{F}_{q})$.

Using the morphism $Z_{G}{\hookrightarrow}M$, we similarly obtain an action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{M,N}$, $\operatorname{Cht}_{M,N,I}$ and $[M_{I,\infty }\backslash \operatorname{Gr}_{M,I}]$.

1.3.4

Applying Definition 1.1.13 to $\overline{P}:=P/Z_{G}$ (respectively $\overline{M}:=M/Z_{G}$), we define a group scheme $\overline{P}_{I,\infty }$ (respectively $\overline{M}_{I,\infty }$) over $X^{I}$. We have $\overline{P}_{I,\infty }=P_{I,\infty }/(Z_{G})_{I,\infty }$ and $\overline{M}_{I,\infty }=M_{I,\infty }/(Z_{G})_{I,\infty }$.

The morphism $\unicode[STIX]{x1D716}_{P,N,I,\infty }$ defined in 1.2.1 is $\unicode[STIX]{x1D6EF}$-equivariant. Since the group scheme $(Z_{G})_{I,\infty }$ acts trivially on $\operatorname{Gr}_{P,I}$, the action of $P_{I,\infty }$ on $\operatorname{Gr}_{P,I}$ factors through $\overline{P}_{I,\infty }$. We denote by $[\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}]$ the resulting quotient prestack. The morphism $P_{I,\infty }{\twoheadrightarrow}\overline{P}_{I,\infty }$ induces a morphism $[P_{I,\infty }\backslash \operatorname{Gr}_{P,I}]\rightarrow [\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}]$, which is $\unicode[STIX]{x1D6EF}$-equivariant for the trivial action of $\unicode[STIX]{x1D6EF}$ on $[\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}]$. Hence the composition of morphisms $\operatorname{Cht}_{P,N,I}\xrightarrow[{}]{\unicode[STIX]{x1D716}_{P,N,I,\infty }}[P_{I,\infty }\backslash \operatorname{Gr}_{P,I}]\rightarrow [\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}]$ is $\unicode[STIX]{x1D6EF}$-equivariant. Thus it factors through

(1.8)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{P,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{P,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}].\end{eqnarray}$$

Similarly, the composition of morphisms $\operatorname{Cht}_{M,N,I}\xrightarrow[{}]{\unicode[STIX]{x1D716}_{M,N,I,\infty }}[M_{I,\infty }\backslash \operatorname{Gr}_{M,I}]\rightarrow [\overline{M}_{I,\infty }\backslash \operatorname{Gr}_{M,I}]$ is $\unicode[STIX]{x1D6EF}$-equivariant for the trivial action of $\unicode[STIX]{x1D6EF}$ on $[\overline{M}_{I,\infty }\backslash \operatorname{Gr}_{M,I}]$. Thus it factors through

(1.9)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{M,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{M,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [\overline{M}_{I,\infty }\backslash \operatorname{Gr}_{M,I}].\end{eqnarray}$$

1.3.5

The morphisms $i$ and $\unicode[STIX]{x1D70B}$ in (1.6) are $\unicode[STIX]{x1D6EF}$-equivariant. Diagram (1.6) induces a commutative diagram of prestacks.

(1.10)

In the remaining part of §1, we introduce the Harder–Narasimhan stratification (compatible with the action of 𝛯) for the parabolic induction diagram (1.4). In order to do so, we use the Harder–Narasimhan stratification for the parabolic induction diagram (1.3). From now on we work in the context of algebraic (ind-)stacks.

In §1.4, we recall the usual Harder–Narasimhan stratification $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}\subset \operatorname{Bun}_{G}$ and a variant $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\subset \operatorname{Bun}_{G}$ which is compatible with the action by $\unicode[STIX]{x1D6EF}$.

In §1.5, we introduce the Harder–Narasimhan stratification $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\subset \operatorname{Bun}_{M}$, which allows us to construct in §1.6 the truncated parabolic induction diagrams (1.26):

$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

In §1.7, we define the Harder–Narasimhan stratification on the stacks of shtukas using §§1.4–1.6.

1.4.1

Applying 0.0.6 to group $G$, we define $\widehat{\unicode[STIX]{x1D6EC}}_{G}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{+}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos},\mathbb{Q}}$ and the partial order ‘${\leqslant}\text{}^{G}$’ on $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$.

1.4.2

[Reference SchiederSch15, 2.1.2] Let $P$ be a parabolic subgroup of $G$ and $M$ its Levi quotient. Consider the sublattice $\widehat{\unicode[STIX]{x1D6EC}}_{[M,M]_{\text{sc}}}\subset \widehat{\unicode[STIX]{x1D6EC}}_{G}$ spanned by the simple coroots of $M$. We define

(1.11)$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{G,P}:=\widehat{\unicode[STIX]{x1D6EC}}_{G}/\widehat{\unicode[STIX]{x1D6EC}}_{[M,M]_{\text{sc}}}.\end{eqnarray}$$

Let $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}:=\widehat{\unicode[STIX]{x1D6EC}}_{G,P}\otimes _{\mathbb{Z}}\mathbb{Q}$. We denote by $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\text{pos}}$ the image of $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos}}$ in $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}$, and by $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\text{pos},\mathbb{Q}}$ the image of $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos},\mathbb{Q}}$ in $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}$. We introduce the partial order on $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}$ by

1.4.3

[Reference SchiederSch15, 2.1.3], [Reference Drinfeld and GaitsgoryDG15, 7.1.3, 7.1.5] Let $Z_{M}$ be the center of $M$. Let $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}$ be the coweight lattice of $Z_{M}$, i.e. $\operatorname{Hom}(\mathbb{G}_{m},Z_{M})$. Note that it equals to $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}^{0}}=\operatorname{Hom}(\mathbb{G}_{m},Z_{M}^{0})$, where $Z_{M}^{0}$ is the neutral connected component of $Z_{M}$.

We have a natural inclusion $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}\subset \widehat{\unicode[STIX]{x1D6EC}}_{G}$ (because $Z_{M}$ is included in the image of $B{\hookrightarrow}P{\twoheadrightarrow}M$). The composition $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{Q}}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}{\twoheadrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}$ is an isomorphism:

(1.12)$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}\overset{{\sim}}{\rightarrow }\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}.\end{eqnarray}$$

We define the slope map to be the composition

(1.13)$$\begin{eqnarray}\unicode[STIX]{x1D719}_{P}:\widehat{\unicode[STIX]{x1D6EC}}_{G,P}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}\cong \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}.\end{eqnarray}$$

We define $\operatorname{pr}_{P}$ to be the composition

(1.14)$$\begin{eqnarray}\operatorname{pr}_{P}:\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}{\twoheadrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}\simeq \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}.\end{eqnarray}$$

By definition, we have $\widehat{\unicode[STIX]{x1D6EC}}_{G,G}^{\mathbb{Q}}=\widehat{\unicode[STIX]{x1D6EC}}_{Z_{G}}^{\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}=\widehat{\unicode[STIX]{x1D6EC}}_{M,M}$ and $\widehat{\unicode[STIX]{x1D6EC}}_{G,B}=\widehat{\unicode[STIX]{x1D6EC}}_{G}$. So $\unicode[STIX]{x1D719}_{B}$ is just the inclusion $\widehat{\unicode[STIX]{x1D6EC}}_{G}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$.

1.4.5

[Reference VarshavskyVar04, Lemma 2.2], [Reference SchiederSch15, 2.2.1, 2.2.2], [Reference Drinfeld and GaitsgoryDG15, 7.2.3] The map $\operatorname{Bun}_{P}\rightarrow \operatorname{Bun}_{M}$ in 1.2.3 induces a bijection on the set of connected components of $\operatorname{Bun}_{P}$ and $\operatorname{Bun}_{M}$. We have $\unicode[STIX]{x1D70B}_{0}(\operatorname{Bun}_{P})\cong \unicode[STIX]{x1D70B}_{0}(\operatorname{Bun}_{M})\cong \widehat{\unicode[STIX]{x1D6EC}}_{G,P}$. Let $\deg _{M}:\operatorname{Bun}_{M}\rightarrow \unicode[STIX]{x1D70B}_{0}(\operatorname{Bun}_{M})\cong \widehat{\unicode[STIX]{x1D6EC}}_{G,P}$ and $\deg _{P}:\operatorname{Bun}_{P}\rightarrow \operatorname{Bun}_{M}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G,P}$.

Definition 1.4.6 [Reference Drinfeld and GaitsgoryDG15, 7.3.3, 7.3.4].

For any $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$, we define $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$ to be the stack that associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the groupoid

where a $P$-structure of ${\mathcal{G}}_{s}$ is a $P$-bundle ${\mathcal{P}}$ on $X_{s}$ such that ${\mathcal{P}}\overset{P}{\times }G\simeq {\mathcal{G}}_{s}$.

1.4.10

[Reference Drinfeld and GaitsgoryDG15, Corollary 7.4.5] We have

where $\operatorname{pr}_{P}$ is defined in (1.14) and $\unicode[STIX]{x1D704}:\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{Q}}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$ is the inclusion. For any $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$, we have

The set $\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}\,|\,\unicode[STIX]{x1D706}\leqslant \text{}^{G}\unicode[STIX]{x1D707}\text{ and }\operatorname{Bun}_{G}^{(\unicode[STIX]{x1D706})}\neq \emptyset \}$ is finite. This gives another proof of Lemma 1.4.8(d).

The above open substack $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$ is not preserved by the action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{G}$. Now we introduce open substacks which are preserved by the action of $\unicode[STIX]{x1D6EF}$.

1.4.11

Applying 0.0.6 to group $G^{\text{ad}}$, we define $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\text{pos}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\text{pos},\mathbb{Q}}$ and the partial order ‘${\leqslant}\text{}^{G^{\text{ad}}}$’ on $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}$.

The morphism $G{\twoheadrightarrow}G/Z_{G}=G^{\text{ad}}$ induces a morphism

(1.15)$$\begin{eqnarray}\unicode[STIX]{x1D6F6}_{G}:\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}.\end{eqnarray}$$

It maps $\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos},\mathbb{Q}}$ to $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\text{pos},\mathbb{Q}}$.

Definition 1.4.12. For any $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, we define $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ to be the stack that associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the groupoid

1.4.14

Just as in 1.4.10, for $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, we have

$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\mathop{\bigcup }_{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}},\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\operatorname{Bun}_{G}^{(\unicode[STIX]{x1D706})}\!.\end{eqnarray}$$

The set $\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}\,|\,\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}\text{ and }\operatorname{Bun}_{G}^{(\unicode[STIX]{x1D706})}\neq \emptyset \}$ is finite modulo $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{G}}$.

1.4.15

The action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{G}$ preserves $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$. We define the quotient $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$.

1.5.1

Applying 0.0.6 to group $M$, we define $\widehat{\unicode[STIX]{x1D6EC}}_{M}$, $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{+}$, $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\text{pos}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}$, $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\text{pos},\mathbb{Q}}$ and the partial order ‘${\leqslant}\text{}^{M}$’ on $\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\mathbb{Q}}$.

1.5.2

Sections 1.4.2–1.4.10 work also for $M$. In particular, let $P^{\prime }$ be a parabolic subgroup of $M$; we have the slope map $\unicode[STIX]{x1D719}_{P^{\prime }}:\widehat{\unicode[STIX]{x1D6EC}}_{M,P^{\prime }}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{M}^{\mathbb{Q}}$ and $\deg _{P^{\prime }}:\operatorname{Bun}_{P^{\prime }}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{M,P^{\prime }}$.

Now we introduce $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\subset \operatorname{Bun}_{M}$ which will be used to construct diagram (1.26).

Definition 1.5.4. For any $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, we define $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ to be the stack that associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the groupoid $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}(S):=$

where $\unicode[STIX]{x1D6F6}_{G}:\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\mathbb{Q}}=\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$ is defined in (1.15).

Similarly to Lemma 1.4.16, we have

1.5.6

The action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{M}$ preserve $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$. We define the quotient $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$. Note that $\unicode[STIX]{x1D6EF}$ is a lattice in $Z_{G}(F)\backslash Z_{G}(\mathbb{A})$. However, $\unicode[STIX]{x1D6EF}$ is only a discrete subgroup but not a lattice in $Z_{M}(F)\backslash Z_{M}(\mathbb{A})$ (since $P\subsetneq G$). We will see that $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$ is locally of finite type but not necessarily of finite type.

1.5.7

Note that $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}=\widehat{\unicode[STIX]{x1D6EC}}_{M,M}$. Consider the composition of morphisms

(1.16)$$\begin{eqnarray}\operatorname{Bun}_{M}\xrightarrow[{}]{\deg _{M}}\widehat{\unicode[STIX]{x1D6EC}}_{M,M}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{M,M}^{\mathbb{Q}}\simeq \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}{\twoheadrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}},\end{eqnarray}$$

where $\deg _{M}$ is defined in 1.4.5. We denote by $A_{M}$ the image of $\widehat{\unicode[STIX]{x1D6EC}}_{M,M}$ in $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$. For any $\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$, we denote by $\operatorname{Bun}_{M}^{\unicode[STIX]{x1D708}}$ its inverse image in $\operatorname{Bun}_{M}$. It is non-empty if and only if $\unicode[STIX]{x1D708}\in A_{M}$.

1.5.9

The stack $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}$ is open and closed in $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ and is open in $\operatorname{Bun}_{M}^{\unicode[STIX]{x1D708}}$. We have a decomposition

(1.17)$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}\!.\end{eqnarray}$$

1.5.10

Just as in 1.4.14, we have

$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\underset{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}},\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}{\bigcup }\mathop{\operatorname{Bun}}_{M}^{(\unicode[STIX]{x1D706})}\!.\end{eqnarray}$$

1.5.11

Similarly to (1.14), we define

(1.18)$$\begin{eqnarray}\operatorname{pr}_{P}^{\text{ad}}:\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}}.\end{eqnarray}$$

Taking into account that $\widehat{\unicode[STIX]{x1D6EC}}_{G}=\widehat{\unicode[STIX]{x1D6EC}}_{M}$ and $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}=\widehat{\unicode[STIX]{x1D6EC}}_{M,M}$, for any $\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}},$ we deduce that $\operatorname{Bun}_{M}^{(\unicode[STIX]{x1D706})}\subset \operatorname{Bun}_{M}^{\unicode[STIX]{x1D708}}$ if and only if $\unicode[STIX]{x1D708}=\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})$.

1.5.12

We deduce from 1.5.10 and 1.5.11 that

(1.19)$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}=\underset{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}},\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}}{\bigcup }\mathop{\operatorname{Bun}}_{M}^{(\unicode[STIX]{x1D706})}\!.\end{eqnarray}$$

1.5.13

We denote by $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\operatorname{pos},\mathbb{Q}}:=\operatorname{pr}_{P}^{\text{ad}}(\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\text{pos},\mathbb{Q}})$. We introduce the partial order on $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$ by

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{1}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{2}\Leftrightarrow \unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\operatorname{pos},\mathbb{Q}}.\end{eqnarray}$$

By definition, for $\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{M}$, we have $\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})=0$. By [Reference SchiederSch15, Proposition 3.1], for $\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}-\widehat{\unicode[STIX]{x1D6E4}}_{M}$ we have $\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})>0$ and these $\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})$ are linearly independent. Thus for $\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$ and $\unicode[STIX]{x1D706}_{1}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D706}_{2}$, we have $\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706}_{1})\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706}_{2})$. Also, the inclusion $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}\subset \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$ maps $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\operatorname{pos},\mathbb{Q}}$ to $\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\operatorname{pos},\mathbb{Q}}$.

1.5.15

Let $\overline{M}=M/Z_{G}$ as in 1.3.4. For $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$, we define $\unicode[STIX]{x1D706}\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}$ if and only if $\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}$ is a linear combination of simple coroots of $M$ with coefficients in $\mathbb{Q}_{{\geqslant}0}$ modulo $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{G}}^{\mathbb{Q}}$.

1.5.16

Let $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}$ and $\unicode[STIX]{x1D706}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}$. We write $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D707}-\sum _{\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}}c_{\check{\unicode[STIX]{x1D6FE}}}\unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})$ for some $c_{\check{\unicode[STIX]{x1D6FE}}}\in \mathbb{Q}_{{\geqslant}0}$. We deduce from 1.5.13 that $\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706})=\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})$ if and only if $c_{\check{\unicode[STIX]{x1D6FE}}}=0$ for all $\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}-\widehat{\unicode[STIX]{x1D6E4}}_{M}$. Hence

(1.20)$$\begin{eqnarray}\unicode[STIX]{x1D706}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}\quad \text{and}\quad \operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706})=\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})\;\Leftrightarrow \;\unicode[STIX]{x1D706}\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}.\end{eqnarray}$$

1.5.17

Let $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$ and $\unicode[STIX]{x1D708}\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})$. For every $\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}-\widehat{\unicode[STIX]{x1D6E4}}_{M}$, let $c_{\unicode[STIX]{x1D6FE}}\in \mathbb{Q}_{{\geqslant}0}$ be the unique coefficient such that

$$\begin{eqnarray}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})-\mathop{\sum }_{\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}-\widehat{\unicode[STIX]{x1D6E4}}_{M}}c_{\unicode[STIX]{x1D6FE}}\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})=\unicode[STIX]{x1D708}.\end{eqnarray}$$

We define $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D708}}:=\unicode[STIX]{x1D707}-\sum _{\check{\unicode[STIX]{x1D6FE}}\in \widehat{\unicode[STIX]{x1D6E4}}_{G}-\widehat{\unicode[STIX]{x1D6E4}}_{M}}c_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F6}_{G}(\check{\unicode[STIX]{x1D6FE}})$. As in 1.5.16, we deduce that

(1.21)$$\begin{eqnarray}\unicode[STIX]{x1D706}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}\quad \text{and}\quad \operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}\;\Leftrightarrow \;\unicode[STIX]{x1D706}\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D708}}.\end{eqnarray}$$

1.5.18

The action of $\unicode[STIX]{x1D6EF}$ on $\operatorname{Bun}_{M}$ preserves $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}$. We define the quotient $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}$.

Proof. By (1.21), we have

$$\begin{eqnarray}\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}\;|\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}\}=\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}\;|\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D708}}\}.\end{eqnarray}$$

We deduce from 1.4.10 (applied to $M$) that the set $\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}\;|\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708},\;\operatorname{Bun}_{M}^{(\unicode[STIX]{x1D706})}\neq \emptyset \}$ is finite modulo $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{G}}$. By Definition 1.5.3, $\operatorname{Bun}_{M}^{(\unicode[STIX]{x1D706})}$ is of finite type. From 1.5.12 we deduce the lemma.◻

1.5.20

By Lemma 1.5.14, the decomposition (1.17) is in fact indexed by a translated cone in $\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$:

(1.22)$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}:=\{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}},\;\unicode[STIX]{x1D708}\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})\}.\end{eqnarray}$$

We deduce that

(1.23)$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}\end{eqnarray}$$

and

(1.24)$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

1.6.3

By Lemma 1.6.2, morphisms (1.3) induce morphisms:

(1.25)$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}.\end{eqnarray}$$

The group $\unicode[STIX]{x1D6EF}$ acts on all these stacks. All the morphisms are $\unicode[STIX]{x1D6EF}$-equivariant. Thus morphisms (1.25) induce morphisms:

(1.26)$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

1.6.4

For any $\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$, we define $\operatorname{Bun}_{P}^{\unicode[STIX]{x1D708}}$ to be the inverse image of $\operatorname{Bun}_{M}^{\unicode[STIX]{x1D708}}$ in $\operatorname{Bun}_{P}$. We define $\operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}:=\operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\cap \operatorname{Bun}_{P}^{\unicode[STIX]{x1D708}}$. Morphisms (1.26) induce morphisms:

(1.27)$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

1.7.3

The following diagram is commutative

(1.28)

where the first line is defined in (1.4). We deduce that $\operatorname{Cht}_{P,N,I}^{{\leqslant}\,\unicode[STIX]{x1D707}}$ is the inverse image of $\operatorname{Cht}_{G,N,I}^{{\leqslant}\,\unicode[STIX]{x1D707}}$ in $\operatorname{Cht}_{P,N,I}$.

1.7.5

Just as in 1.6.3 and 1.6.4, morphisms (1.4) induce morphisms:

(1.29)$$\begin{eqnarray}\displaystyle & \operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}, & \displaystyle\end{eqnarray}$$

(1.30)$$\begin{eqnarray}\displaystyle & \operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}. & \displaystyle\end{eqnarray}$$

We deduce from (1.24) a decomposition:

(1.31)$$\begin{eqnarray}\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

2.1.1

The geometric Satake equivalence for the affine grassmannian is established in [Reference Mirkovic and VilonenMV07] over the ground field $\mathbb{C}$. By [Reference Mirkovic and VilonenMV07, §14], [Reference GaitsgoryGai07, §1.6] and [Reference ZhuZhu17], the constructions in [Reference Mirkovic and VilonenMV07] carries over to the case of an arbitrary algebraically closed ground field of characteristic prime to $\ell$.

2.1.2

Let $\widehat{G}$ be the Langlands dual group of $G$ over $\mathbb{Q}_{\ell }$ defined by the geometric Satake equivalence for the affine grassmannian, as in [Reference Mirkovic and VilonenMV07, Theorem 7.3] and [Reference GaitsgoryGai07, Theorem 2.2].

2.1.3

[Reference Mirkovic and VilonenMV07, §2], [Reference GaitsgoryGai01, 1.1.1 and §6] The Beilinson–Drinfeld affine grassmannian $\operatorname{Gr}_{G,I}$ is an ind-scheme. Every finite-dimensional closed subscheme of $\operatorname{Gr}_{G,I}$ is contained in some finite-dimensional closed subscheme of $\operatorname{Gr}_{G,I}$ stable under the action of $G_{I,\infty }$.

We denote by $\operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell })$ the category of $G_{I,\infty }$-equivariant perverse sheaves with $\mathbb{Q}_{\ell }$-coefficients on $\operatorname{Gr}_{G,I}$ (for the perverse normalization relative to $X^{I}$).

2.1.4

As in [Reference GaitsgoryGai07, 2.5], we denote by $P^{\widehat{G},I}$ the category of perverse sheaves with $\mathbb{Q}_{\ell }$-coefficients on $X^{I}$ (for the perverse normalization relative to $X^{I}$) endowed with an extra structure given in [Reference GaitsgoryGai07].

Theorem 2.1.5 [Reference GaitsgoryGai07, Theorem 2.6].

There is a canonical equivalence of categories

$$\begin{eqnarray}\operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell })\overset{{\sim}}{\rightarrow }P^{\widehat{G},I},\end{eqnarray}$$

compatible with the tensor structures defined in [Reference GaitsgoryGai07].

2.1.6

We denote by $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})$ the category of finite-dimensional $\mathbb{Q}_{\ell }$-linear representations of $\widehat{G}^{I}$. We have a fully faithful functor $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})\rightarrow P^{\widehat{G},I}:W\mapsto W\otimes {\mathbb{Q}_{\ell }}_{X^{I}}$. The composition of this functor and the inverse functor $P^{\widehat{G},I}\overset{{\sim}}{\rightarrow }\operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell })$ in Theorem 2.1.5 gives the following.

Corollary 2.1.7. We have a canonical natural fully faithful $\mathbb{Q}_{\ell }$-linear fiber functor:

$$\begin{eqnarray}\text{Sat}_{G,I}:\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})\rightarrow \operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell }).\end{eqnarray}$$

2.1.9

When $W=W_{1}\oplus W_{2}$, by the functoriality of $\text{Sat}_{G,I}$, we have ${\mathcal{S}}_{G,I,W}={\mathcal{S}}_{G,I,W_{1}}\oplus {\mathcal{S}}_{G,I,W_{2}}$. Then $\operatorname{Gr}_{G,I,W}=\operatorname{Gr}_{G,I,W_{1}}\cup \operatorname{Gr}_{G,I,W_{2}}$.

2.1.10

By [Reference LafforgueLaf18, Théroème 1.17], the above definition of $\operatorname{Gr}_{G,I,W}$ is equivalent to [Reference LafforgueLaf18, Définition 1.12 and the definition after (1.14)] (which describes $\operatorname{Gr}_{G,I,W}$ as a generalization of the Zariski closure of the Schubert cell in affine grassmannian). It is well known that $\operatorname{Gr}_{G,I,W}$ is a closed subscheme of $\operatorname{Gr}_{G,I}$ and that it is projective (see [Reference Mirkovic and VilonenMV07, §§2–3], [Reference ZhuZhu17, Proposition 2.1.5]). The ind-scheme $\operatorname{Gr}_{G,I}$ is an inductive limit of $\operatorname{Gr}_{G,I,W}$. ‘

2.2.2

For $d$ as in Proposition 2.2.1, we define the quotient stack $[G_{I,d}\backslash \operatorname{Gr}_{G,I,W}]$. Since the group scheme $G_{I,d}$ is of finite dimension, the stack $[G_{I,d}\backslash \operatorname{Gr}_{G,I,W}]$ is algebraic.

2.2.3

Let ${\mathcal{S}}_{G,I,W}$ be the $G_{I,\infty }$-equivariant perverse sheaf on $\operatorname{Gr}_{G,I,W}$ defined in Definition 2.1.8. By Proposition 2.2.1, the action of $G_{I,\infty }$ on ${\mathcal{S}}_{G,I,W}$ factors through $G_{I,d}$. Since the kernel of $G_{I,\infty }{\twoheadrightarrow}G_{I,d}$ is connected, by [Reference Beilinson, Bernstein, Deligne and GabberBBDG82, Proposition 4.2.5], we deduce that ${\mathcal{S}}_{G,I,W}$ is also $G_{I,d}$-equivariant.

Let $\unicode[STIX]{x1D709}_{G,I,d}:\operatorname{Gr}_{G,I,W}\rightarrow [G_{I,d}\backslash \operatorname{Gr}_{G,I,W}]$ be the canonical morphism. It is smooth of dimension $\dim G_{I,d}$. By [Reference Beilinson, Bernstein, Deligne and GabberBBDG82, Corollaire 4.2.6.2] and the discussion after it, there exists a perverse sheaf (up to shift $[\dim G_{I,d}]$) (for the perverse normalization relative to $X^{I}$) ${\mathcal{S}}_{G,I,W}^{d}$ on $[G_{I,d}\backslash \operatorname{Gr}_{G,I,W}]$ such that ${\mathcal{S}}_{G,I,W}=\unicode[STIX]{x1D709}_{G,I,d}^{\ast }{\mathcal{S}}_{G,I,W}^{d}$.

2.2.4

Let $d\leqslant d^{\prime }$ be two integers large enough as in Proposition 2.2.1. Then the morphisms $G_{I,\infty }{\twoheadrightarrow}G_{I,d^{\prime }}{\twoheadrightarrow}G_{I,d}$ induce a commutative diagram.

(2.1)

We have $(\unicode[STIX]{x1D709}_{G,I,d^{\prime }})^{\ast }{\mathcal{S}}_{G,I,W}^{d^{\prime }}={\mathcal{S}}_{G,I,W}=(\unicode[STIX]{x1D709}_{G,I,d})^{\ast }{\mathcal{S}}_{G,I,W}^{d}=(\unicode[STIX]{x1D709}_{G,I,d^{\prime }})^{\ast }(\operatorname{pr}_{d^{\prime }}^{d})^{\ast }{\mathcal{S}}_{G,I,W}^{d}$. By [Reference Beilinson, Bernstein, Deligne and GabberBBDG82, Proposition 4.2.5], the functor $(\unicode[STIX]{x1D709}_{G,I,d^{\prime }})^{\ast }$ (up to shift) is fully faithful. We deduce that ${\mathcal{S}}_{G,I,W}^{d^{\prime }}=(\operatorname{pr}_{d^{\prime }}^{d})^{\ast }{\mathcal{S}}_{G,I,W}^{d}$.

2.2.5

By Proposition 2.2.1, the action of $G_{I,\infty }^{\text{ad}}$ on $\operatorname{Gr}_{G,I,W}$ factors through $G_{I,d}^{\text{ad}}$. We define the quotient stack $[G_{I,d}^{\text{ad}}\backslash \operatorname{Gr}_{G,I,W}]$.

As in the discussion after [Reference LafforgueLaf18, Définition 2.14], since $(Z_{G})_{I,\infty }$ acts trivially on $\operatorname{Gr}_{G,I,W}$, the $G_{I,\infty }$-equivariant perverse sheaf ${\mathcal{S}}_{G,I,W}$ on $\operatorname{Gr}_{G,I,W}$ is also $G_{I,\infty }^{\text{ad}}$-equivariant and $G_{I,d}^{\text{ad}}$-equivariant. Indeed, by 2.1.9 it is enough to prove this for $W$ irreducible. By Remark 2.1.11, in this case ${\mathcal{S}}_{G,I,W}$ is isomorphic to the intersection complex of $\operatorname{Gr}_{G,I,W}$, hence is $G_{I,\infty }^{\text{ad}}$-equivariant.

Just as in 2.2.3, let $\unicode[STIX]{x1D709}_{G,I,d}^{\text{ad}}:\operatorname{Gr}_{G,I,W}\rightarrow [G_{I,d}^{\text{ad}}\backslash \operatorname{Gr}_{G,I,W}]$ be the canonical morphism. There exists a perverse sheaf (up to shift $[\dim G_{I,d}^{\text{ad}}]$) (for the perverse normalization relative to $X^{I}$) ${\mathcal{S}}_{G,I,W}^{\text{ad},\,d}$ on $[G_{I,d}^{\text{ad}}\backslash \operatorname{Gr}_{G,I,W}]$ such that ${\mathcal{S}}_{G,I,W}=(\unicode[STIX]{x1D709}_{G,I,d}^{\text{ad}})^{\ast }{\mathcal{S}}_{G,I,W}^{\text{ad},\,d}$.

2.3.2

$\operatorname{Cht}_{G,N,I}$ is an inductive limit of closed subtacks $\operatorname{Cht}_{G,N,I,W}$.

2.3.3

Let $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$. We define $\operatorname{Cht}_{G,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}:=\operatorname{Cht}_{G,N,I,W}\cap \operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$, where $\operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$ is defined in Definition 1.7.2. We define the quotient $\operatorname{Cht}_{G,N,I,W}/\unicode[STIX]{x1D6EF}$ and $\operatorname{Cht}_{G,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$.

2.3.5

The stack $\operatorname{Cht}_{G,N,I,W}/\unicode[STIX]{x1D6EF}=\mathop{\varinjlim }\nolimits_{\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}}\operatorname{Cht}_{G,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$ is locally of finite type.

2.4.1

For any $d\in \mathbb{Z}_{{\geqslant}0}$ large enough as in Proposition 2.2.1, we define $\unicode[STIX]{x1D716}_{G,N,I,d}$ to be the composition of morphisms

(2.2)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{G,N,I,d}:\operatorname{Cht}_{G,N,I,W}\xrightarrow[{}]{\unicode[STIX]{x1D716}_{G,N,I,\infty }}[G_{I,\infty }\backslash \operatorname{Gr}_{G,I,W}]\rightarrow [G_{I,d}\backslash \operatorname{Gr}_{G,I,W}].\end{eqnarray}$$

This is morphism (2.3) in [Reference LafforgueLaf18].

Just as in 1.3.2, we define a morphism

(2.3)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{G,N,I,d}^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{G,N,I,W}/\unicode[STIX]{x1D6EF}\rightarrow [G_{I,d}^{\text{ad}}\backslash \operatorname{Gr}_{G,I,W}].\end{eqnarray}$$

This is morphism (2.10) in [Reference LafforgueLaf18].

2.4.2

We denote by $\dim _{X^{I}}G_{I,d}$ the relative dimension of $G_{I,d}$ over $X^{I}$ and by $|I|$ the cardinal of $I$. We have $\dim _{X^{I}}G_{I,d}=d\cdot |I|\cdot \operatorname{dim}G$.

2.4.4

For all $d\in \mathbb{Z}_{{\geqslant}0}$ large enough as in Proposition 2.2.1, we have morphisms over $(X\smallsetminus N)^{I}$.

We deduce from Proposition 2.4.3 that $\dim \operatorname{Cht}_{G,N,I,W}=\dim \operatorname{Gr}_{G,I,W}$. We refer to [Reference LafforgueLaf18, Proposition 2.11] for the fact that $\operatorname{Cht}_{G,N,I,W}$ is locally isomorphic to $\operatorname{Gr}_{G,I,W}$ for the étale topology. We will not use this result in this paper.

Just as in Remark 2.4.6, ${\mathcal{F}}_{G,N,I,W}^{\unicode[STIX]{x1D6EF}}$ is independent of $d$.

2.5.2

Let $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$ and $\unicode[STIX]{x1D707}_{1}\leqslant \unicode[STIX]{x1D707}_{2}$. We have an open immersion:

(2.4)$$\begin{eqnarray}\operatorname{Cht}_{G,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}_{1}}/\unicode[STIX]{x1D6EF}{\hookrightarrow}\operatorname{Cht}_{G,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}_{2}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

For any $j$, morphism (2.4) induces a morphism of sheaves:

$$\begin{eqnarray}{\mathcal{H}}_{G,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}_{1}}\rightarrow {\mathcal{H}}_{G,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}_{2}}.\end{eqnarray}$$

Definition 2.5.3. We define

as an inductive limit in the category of constructible sheaves on $(X\smallsetminus N)^{I}$.

2.5.4

Let $\overline{\unicode[STIX]{x1D702}^{I}}$ be a geometric point over the generic point $\unicode[STIX]{x1D702}^{I}$ of $X^{I}$.

Definition 2.5.5. We define

(2.5)$$\begin{eqnarray}H_{G,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}}:=\left.{\mathcal{H}}_{G,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}}\vphantom{\big|}\right|_{\overline{\unicode[STIX]{x1D702}^{I}}}\;,\quad H_{G,N,I,W}^{j}:=\left.{\mathcal{H}}_{G,N,I,W}^{j}\vphantom{\big|}\right|_{\overline{\unicode[STIX]{x1D702}^{I}}}\,.\end{eqnarray}$$

By definition $H_{G,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}}$ is a $\mathbb{Q}_{\ell }$-vector space of finite dimension. We have $H_{G,N,I,W}^{j}=\mathop{\varinjlim }\nolimits_{\unicode[STIX]{x1D707}}H_{G,N,I,W}^{j,\;\leqslant \unicode[STIX]{x1D707}}$.

2.6.1

Let $\widehat{M}$ be the Langlands dual group of $M$ over $\mathbb{Q}_{\ell }$ defined by the geometric Satake equivalence. The compatibility between the geometric Satake equivalence and the constant term functor along $P$ (that we will recall in Theorem 3.2.6 below) induces a canonical inclusion $\widehat{M}{\hookrightarrow}\widehat{G}$ (compatible with pinning).

2.6.2

We view $W\in \operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})$ as a representation of $\widehat{M}^{I}$ via $\widehat{M}^{I}{\hookrightarrow}\widehat{G}^{I}$. As in §§2.1–2.4, we define $\operatorname{Gr}_{M,I,W}$ and $\operatorname{Cht}_{M,N,I,W}$. For $d\in \mathbb{Z}_{{\geqslant}0}$ large enough such that the action of $M_{I,\infty }$ on $\operatorname{Gr}_{M,I,W}$ factors through $M_{I,d}$, we define

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D716}_{M,N,I,d}:\operatorname{Cht}_{M,N,I,W}\rightarrow [M_{I,d}\backslash \operatorname{Gr}_{M,I,W}], & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D716}_{M,N,I,d}^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{M,N,I,W}/\unicode[STIX]{x1D6EF}\rightarrow [\overline{M}_{I,d}\backslash \operatorname{Gr}_{M,I,W}]. & \displaystyle \nonumber\end{eqnarray}$$

We define perverse sheaf ${\mathcal{S}}_{M,I,W}$ on $\operatorname{Gr}_{M,I,W}$, perverse sheaves (up to shift) ${\mathcal{S}}_{M,I,W}^{d}$ on $[M_{I,d}\backslash \operatorname{Gr}_{M,I,W}]$ and ${\mathcal{S}}_{M,I,W}^{\text{ad},d}$ on $[\overline{M}_{I,d}\backslash \operatorname{Gr}_{M,I,W}]$. We define

$$\begin{eqnarray}{\mathcal{F}}_{M,N,I,W}:=\unicode[STIX]{x1D716}_{M,N,I,d}^{\ast }{\mathcal{S}}_{M,I,W}^{d}\quad \text{and}\quad {\mathcal{F}}_{M,N,I,W}^{\unicode[STIX]{x1D6EF}}:=(\unicode[STIX]{x1D716}_{M,N,I,d}^{\unicode[STIX]{x1D6EF}})^{\ast }{\mathcal{S}}_{M,I,W}^{\text{ad},d}.\end{eqnarray}$$

2.6.3

Applying [Reference VarshavskyVar04, Proposition 2.16] to $M$, we deduce that $\operatorname{Cht}_{M,I,N,W}$ is a Deligne–Mumford stack locally of finite type and that for $\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}$, the Deligne–Mumford stack $\operatorname{Cht}_{M,I,N,W}^{(\unicode[STIX]{x1D706})}$ (defined in Definition 1.7.2) is of finite type.

Let $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$. We define $\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}:=\operatorname{Cht}_{M,N,I,W}\cap \operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$, where $\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$ is defined in Definition 1.7.2. We define the quotient $\operatorname{Cht}_{M,N,I,W}/\unicode[STIX]{x1D6EF}$ and $\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$. As in 1.5.6, $\unicode[STIX]{x1D6EF}$ is a lattice in $Z_{G}(F)\backslash Z_{G}(\mathbb{A})$ but only a discrete subgroup in $Z_{M}(F)\backslash Z_{M}(\mathbb{A})$. The decomposition (1.31) induces a decomposition

(2.6)$$\begin{eqnarray}\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF},\end{eqnarray}$$

where each $\operatorname{Cht}_{M,I,N,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}$ is of finite type (just as in Lemma 1.5.19).

Recall that we have the morphism of paws $\mathfrak{p}_{M}:\operatorname{Cht}_{M,I,N}/\unicode[STIX]{x1D6EF}\rightarrow (X\smallsetminus N)^{I}$.

Definition 2.6.4. For any $\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$ and $\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$, we define

$$\begin{eqnarray}{\mathcal{H}}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}:=R(\mathfrak{p}_{M})_{!}(\left.{\mathcal{F}}_{M,I,N,W}^{\unicode[STIX]{x1D6EF}}\vphantom{\big|}\right|_{\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}})\in D_{c}^{b}((X\smallsetminus N)^{I},\mathbb{Q}_{\ell }).\end{eqnarray}$$

For any $j\in \mathbb{Z}$, we define degree $j$ cohomology sheaf

$$\begin{eqnarray}{\mathcal{H}}_{M,N,I,W}^{j,\;\leqslant \unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}:=R^{j}(\mathfrak{p}_{M})_{!}(\left.{\mathcal{F}}_{M,I,N,W}^{\unicode[STIX]{x1D6EF}}\vphantom{\big|}\right|_{\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}}).\end{eqnarray}$$

2.6.5

If $\unicode[STIX]{x1D708}\notin \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}$, by Lemma 1.5.14, $\operatorname{Cht}_{M,I,N,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}=\emptyset$. In this case ${\mathcal{H}}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}=0$.

Definition 2.6.6. Let $\overline{\unicode[STIX]{x1D702}^{I}}$ be the geometric generic point of $X^{I}$ fixed in 2.5.4. We define

(2.7)$$\begin{eqnarray}H_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}:=\left.{\mathcal{H}}_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}\vphantom{\big|}\right|_{\overline{\unicode[STIX]{x1D702}^{I}}}\,.\end{eqnarray}$$

This is a finite-dimensional $\mathbb{Q}_{\ell }$-vector space. We define

(2.8)$$\begin{eqnarray}H_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}}:=\mathop{\prod }_{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}H_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}.\end{eqnarray}$$

2.6.7

Let $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$ and $\unicode[STIX]{x1D707}_{1}\leqslant \unicode[STIX]{x1D707}_{2}$. We have an open immersion:

(2.9)$$\begin{eqnarray}\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}_{1}}/\unicode[STIX]{x1D6EF}{\hookrightarrow}\operatorname{Cht}_{M,N,I,W}^{{\leqslant}\unicode[STIX]{x1D707}_{2}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$

For any $j$, morphism (2.9) induces a morphism of vector spaces:

$$\begin{eqnarray}H_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}_{1}}\rightarrow H_{M,N,I,W}^{j,\,\leqslant \unicode[STIX]{x1D707}_{2}}.\end{eqnarray}$$

Definition 2.6.8. We define

as an inductive limit in the category of $\mathbb{Q}_{\ell }$-vector spaces.

3.1.2

We define $\operatorname{Gr}_{P,I,W}:=(i^{0})^{-1}(\operatorname{Gr}_{G,I,W})$. As a consequence of Proposition 3.1.1, morphisms (1.5) induce morphisms over $X^{I}$:

(3.6)$$\begin{eqnarray}\operatorname{Gr}_{G,I,W}\xleftarrow[{}]{i^{0}}\operatorname{Gr}_{P,I,W}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{0}}\operatorname{Gr}_{M,I,W}.\end{eqnarray}$$

3.1.3

We deduce from the commutative diagram (1.6) that

$$\begin{eqnarray}i^{-1}(\operatorname{Cht}_{G,N,I,W})\subset \unicode[STIX]{x1D70B}^{-1}(\operatorname{Cht}_{M,N,I,W}),\end{eqnarray}$$

where the inverse images are in the sense of reduced substacks in $\operatorname{Cht}_{P,N,I}$. We define $\operatorname{Cht}_{P,N,I,W}:=i^{-1}(\operatorname{Cht}_{G,N,I,W})$. Morphisms in (1.4) induce morphisms over $(X\smallsetminus N)^{I}$:

(3.7)$$\begin{eqnarray}\operatorname{Cht}_{G,N,I,W}\xleftarrow[{}]{i}\operatorname{Cht}_{P,N,I,W}\xrightarrow[{}]{\unicode[STIX]{x1D70B}}\operatorname{Cht}_{P,N,I,W}\!.\end{eqnarray}$$

3.1.4

Let $d\in \mathbb{Z}_{{\geqslant}0}$ large enough depending on $W$ as in Proposition 2.2.1 applied to $\operatorname{Gr}_{G,I,W}$ and to $\operatorname{Gr}_{M,I,W}$. To simplify the notations, we write $\unicode[STIX]{x1D716}_{G,d}$ for $\unicode[STIX]{x1D716}_{G,N,I,d}$ defined in 2.4.1 and $\unicode[STIX]{x1D716}_{M,d}$ for $\unicode[STIX]{x1D716}_{M,N,I,d}$ defined in 2.6.2. Similarly we define $\unicode[STIX]{x1D716}_{P,d}$ to be the composition

$$\begin{eqnarray}\operatorname{Cht}_{P,N,I,W}\rightarrow [P_{I,\infty }\backslash \operatorname{Gr}_{P,I,W}]\rightarrow [P_{I,d}\backslash \operatorname{Gr}_{P,I,W}].\end{eqnarray}$$

We deduce from the commutative diagram (1.6), morphisms (3.6) and (3.7) a commutative diagram of algebraic stacks.

(3.8)

3.1.5

The right square in (3.8) is not Cartesian. We have a commutative diagram, where the square is Cartesian.

(3.9)

The following proof was suggested to the author by a referee.

Proof. Proposition 2.4.3 works also for $P$ and $M$. Hence the morphism $\unicode[STIX]{x1D716}_{P,d}$ is smooth of relative dimension $\dim _{X^{I}}P_{I,d}$ and the morphism $\unicode[STIX]{x1D716}_{M,d}$ (hence $\widetilde{\unicode[STIX]{x1D716}_{M,d}}$) is smooth of relative dimension $\dim _{X^{I}}M_{I,d}$. Thus to prove that $\unicode[STIX]{x1D70B}_{d}$ is smooth, it is enough to show that it induces a surjective map between relative tangent spaces.

For any closed point $x_{P}=((x_{i}),{\mathcal{P}}\rightarrow ^{\unicode[STIX]{x1D70F}}{\mathcal{P}})$ of $\operatorname{Cht}_{P,N,I,W}$, let $x_{M}:=\unicode[STIX]{x1D70B}_{d}(x_{P})$. We have the canonical morphism

(3.10)$$\begin{eqnarray}T_{\unicode[STIX]{x1D716}_{P,d}}(x_{P})\rightarrow T_{\widetilde{\unicode[STIX]{x1D716}_{M,d}}}(x_{M}),\end{eqnarray}$$

where $T_{\unicode[STIX]{x1D716}_{P,d}}(x_{P})$ (respectively $T_{\widetilde{\unicode[STIX]{x1D716}_{M,d}}}(x_{M})$) is the tangent space of $\operatorname{Cht}_{P,N,I,W}$ (respectively $\widetilde{\operatorname{Cht}}_{M,N,I,W}$) at $x_{P}$ (respectively $x_{M}$) relative to $[P_{I,d}\backslash \operatorname{Gr}_{P,I,W}]$.

Let $y=\unicode[STIX]{x1D716}_{P,d}(x_{P})$. By the proof of [Reference LafforgueLaf18, Proposition 2.8], we have a Cartesian square

(3.11)

where $b_{1}^{P}$ is a smooth morphism (which is the forgetful morphism of the level structure on $d\sum x_{i}$) and $b_{2}^{P}$ has zero differential (because it is the composition of the Frobenius morphism with some other morphism). We have $T_{\unicode[STIX]{x1D716}_{P,d}}(x_{P})=T_{b_{1}^{p}}(x_{P})$ (see for example [Reference LafforgueLaf97, I. 2. Proposition 1]). It is well known that $\operatorname{Bun}_{P,N+d\sum x_{i}}\xrightarrow[{}]{b_{1}^{P}}\operatorname{Bun}_{P,N}$ is a $P_{d\sum x_{i}}$-torsor, where $P_{d\sum x_{i}}$ is defined in 0.0.8. We deduce that $T_{b_{1}^{p}}(x_{P})=\operatorname{Lie}(P_{d\sum x_{i}})$.

Similarly, we have a Cartesian square (taking into account that $\widetilde{\unicode[STIX]{x1D716}_{M,d}}^{-1}(y)=\unicode[STIX]{x1D716}_{M,d}^{-1}(\overline{\unicode[STIX]{x1D70B}_{d}^{0}}(y))$)

(3.12)

where $b_{1}^{M}$ is a smooth morphism (which is the forgetful morphism of the level structure on $d\sum x_{i}$) and $b_{2}^{M}$ has zero differential. We deduce that $T_{\widetilde{\unicode[STIX]{x1D716}_{M,d}}}(x_{M})=T_{b_{1}^{M}}(x_{M})=\operatorname{Lie}(M_{d\sum x_{i}}),$ where $M_{d\sum x_{i}}$ is defined in 0.0.8.

Morphism (3.10) is the canonical morphism $\operatorname{Lie}(P_{d\sum x_{i}})\rightarrow \operatorname{Lie}(M_{d\sum x_{i}})$ induced by $P{\twoheadrightarrow}M$. Hence it is surjective. We deduce also that the relative tangent space of $\unicode[STIX]{x1D70B}_{d}$ is $\operatorname{Lie}(U_{d\sum x_{i}})$.◻

3.2.1

We apply Definition 1.1.11 to $\mathbb{G}_{m}$ and denote by $\operatorname{Gr}_{\mathbb{G}_{m},I}$ the associated reduced ind-scheme. We denote by $\unicode[STIX]{x1D70C}_{G}$ (respectively $\unicode[STIX]{x1D70C}_{M}$) the half sum of positive roots of $G$ (respectively $M$). Since $2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})$ is a character of $M$, the morphism $2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M}):M\rightarrow \mathbb{G}_{m}$ induces a morphism $\operatorname{Gr}_{M,I}\rightarrow \operatorname{Gr}_{\mathbb{G}_{m},I}$ by sending a $M$-bundle ${\mathcal{M}}$ to the $\mathbb{G}_{m}$-bundle ${\mathcal{M}}\overset{M}{\times }\mathbb{G}_{m}$. We have a morphism $\operatorname{deg}:\operatorname{Gr}_{\mathbb{G}_{m},I}\rightarrow \mathbb{Z}$ by taking the degree of a $\mathbb{G}_{m}$-bundle. We have the composition of morphisms

(3.13)$$\begin{eqnarray}\operatorname{Gr}_{M,I}\rightarrow \operatorname{Gr}_{\mathbb{G}_{m},I}\xrightarrow[{}]{\operatorname{deg}}\mathbb{Z}.\end{eqnarray}$$

We define $\operatorname{Gr}_{M,I}^{n}$ as the inverse image of $n\in \mathbb{Z}$. It is open and closed in $\operatorname{Gr}_{M,I}$. We define $\operatorname{Gr}_{P,I}^{n}:=(\unicode[STIX]{x1D70B}^{0})^{-1}\operatorname{Gr}_{M,I}^{n}$. Morphism (1.5) induces a morphism

(3.14)$$\begin{eqnarray}\operatorname{Gr}_{G,I}\xleftarrow[{}]{i_{n}^{0}}\operatorname{Gr}_{P,I}^{n}\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{n}^{0}}\operatorname{Gr}_{M,I}^{n}.\end{eqnarray}$$

3.2.2

Recall that we have defined $\widehat{\unicode[STIX]{x1D6EC}}_{G,P}$ in 1.4.2. As in [Reference SchiederSch15, 2.1.2], we define $\unicode[STIX]{x1D6EC}_{G,P}:=\{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{G}|\langle \check{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D706}\rangle =0\text{ for all }\check{\unicode[STIX]{x1D6FC}}\in \widehat{\unicode[STIX]{x1D6E4}}_{M}\}$. The pairing $\langle ~,~\rangle$ in 1.4.1 induces a pairing $\langle ~,~\rangle :\widehat{\unicode[STIX]{x1D6EC}}_{G,P}\times \unicode[STIX]{x1D6EC}_{G,P}\rightarrow \mathbb{Z}$.

3.2.3

We denote by $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})$ the category of finite-dimensional $\mathbb{Q}_{\ell }$-linear representations of $\widehat{M}^{I}$. Let $W\in \operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})$. Then $Z_{\widehat{M}}$ acts on $W$ via $Z_{\widehat{M}}{\hookrightarrow}\widehat{M}^{I}$ diagonally. We have the decomposition as $Z_{\widehat{M}}$ representation: $W=\bigoplus _{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6EC}_{Z_{\widehat{M}}}}W^{\unicode[STIX]{x1D703}}$.

Since $\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6EC}_{Z_{\widehat{M}}}=\widehat{\unicode[STIX]{x1D6EC}}_{G,P}$ and $2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})\in \unicode[STIX]{x1D6EC}_{G,P}$, we can consider $\langle \unicode[STIX]{x1D703},2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})\rangle$. Let $W_{n}=\bigoplus _{\langle \unicode[STIX]{x1D703},2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})\rangle =n}W^{\unicode[STIX]{x1D703}}$. We have $W=\bigoplus _{n\in \mathbb{Z}}W_{n}$.

Let $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})^{\unicode[STIX]{x1D703}}$ be the category of finite-dimensional $\mathbb{Q}_{\ell }$-linear representations of $\widehat{M}^{I}$ such that $Z_{\widehat{M}}$ acts by $\unicode[STIX]{x1D703}$. We have $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})=\bigoplus _{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6EC}_{Z_{\widehat{M}}}}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})^{\unicode[STIX]{x1D703}}$. Let

$$\begin{eqnarray}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})_{n}=\bigoplus _{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6EC}_{Z_{\widehat{M}}},\,\langle \unicode[STIX]{x1D703},2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})\rangle =n}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})^{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

We have

$$\begin{eqnarray}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})=\bigoplus _{n\in \mathbb{Z}}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})_{n}.\end{eqnarray}$$

We define $(\operatorname{Res}_{\widehat{M}^{I}}^{\widehat{G}^{I}})_{n}$ to be the composition of morphisms $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})\xrightarrow[{}]{\operatorname{Res}_{\widehat{M}^{I}}^{\widehat{G}^{I}}}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I}){\twoheadrightarrow}\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})_{n}$.

3.2.4

In morphism (3.13), $\operatorname{Gr}_{M,I,W^{\unicode[STIX]{x1D703}}}$ is sent to $\langle \unicode[STIX]{x1D703},2(\unicode[STIX]{x1D70C}_{G}-\unicode[STIX]{x1D70C}_{M})\rangle$. We deduce that $\operatorname{Gr}_{M,I}^{n}\cap \operatorname{Gr}_{M,I,W}=\operatorname{Gr}_{M,I,W_{n}}$.

3.2.5

In Corollary 2.1.7, we defined a fully faithful functor

$$\begin{eqnarray}\text{Sat}_{G,I}:\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{G}^{I})\rightarrow \operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell })\end{eqnarray}$$

which sends $W$ to ${\mathcal{S}}_{G,I,W}$. We denote by $\operatorname{Perv}_{G_{I,\infty }}(\operatorname{Gr}_{G,I},\mathbb{Q}_{\ell })^{\operatorname{MV}}$ the subcategory of essential image of this functor. Similarly, we define

$$\begin{eqnarray}\text{Sat}_{M,I}:\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})\rightarrow \operatorname{Perv}_{M_{I,\infty }}(\operatorname{Gr}_{M,I},\mathbb{Q}_{\ell })^{\operatorname{MV}}.\end{eqnarray}$$

Let $\text{Sat}_{M,I,n}$ be the restriction of $\text{Sat}_{M,I}$ to $\operatorname{Rep}_{\mathbb{Q}_{\ell }}(\widehat{M}^{I})_{n}$.

Theorem 3.2.6 ([Reference Beilinson and DrinfeldBD99, 5.3.29], [Reference Braverman and GaitsgoryBG02, Theorem 4.3.4], [Reference Mirkovic and VilonenMV07, Theorem 3.6] (for $M=T$), [Reference Baumann and RicheBR18, Proposition 15.2]).

1. (a) For any $n\in \mathbb{Z}$, the complex

   $$\begin{eqnarray}(\unicode[STIX]{x1D70B}_{n}^{0})_{!}(i_{n}^{0})^{\ast }{\mathcal{S}}_{G,I,W}\otimes \big(\mathbb{Q}_{\ell }[1]({\textstyle \frac{1}{2}})\big)^{\otimes n}\end{eqnarray}$$
   is in $\operatorname{Perv}_{M_{I,\infty }}(\operatorname{Gr}_{M,I}^{n},\mathbb{Q}_{\ell })^{\operatorname{MV}}$.

2. (b) We denote by $((\unicode[STIX]{x1D70B}_{n}^{0})_{!}(i_{n}^{0})^{\ast })^{{\sim}}$ the shifted functor $(\unicode[STIX]{x1D70B}_{n}^{0})_{!}(i_{n}^{0})^{\ast }\otimes (\mathbb{Q}_{\ell }[1]({\textstyle \frac{1}{2}}))^{\otimes n}$. Then there is a canonical isomorphism of fiber functors

   (3.15)$$\begin{eqnarray}\text{Sat}_{M,I,n}\circ (\operatorname{Res}_{\widehat{M}^{I}}^{\widehat{G}^{I}})_{n}=((\unicode[STIX]{x1D70B}_{n}^{0})_{!}(i_{n}^{0})^{\ast })^{{\sim}}\circ \text{Sat}_{G,I}.\end{eqnarray}$$

In other words, the following diagram of categories canonically commutes.

(3.16)

Corollary 3.2.8. There is a canonical isomorphism

(3.17)$$\begin{eqnarray}{\mathcal{S}}_{M,I,W_{n}}\simeq (\unicode[STIX]{x1D70B}_{n}^{0})_{!}(i_{n}^{0})^{\ast }{\mathcal{S}}_{G,I,W}[n](n/2).\end{eqnarray}$$

3.2.9

For any $n$, denote by $\operatorname{Gr}_{P,I,W}^{n}=\operatorname{Gr}_{P,I}^{n}\cap \operatorname{Gr}_{P,I,W}$. We have a commutative diagram, where the first line is induced by (3.6).

The morphism

$$\begin{eqnarray}\operatorname{Gr}_{P,I,W}^{n}\rightarrow [P_{I,d}\backslash \operatorname{Gr}_{P,I,W}^{n}]\underset{[M_{I,d}\backslash \operatorname{Gr}_{M,I,W_{n}}]}{\times }\operatorname{Gr}_{M,I,W_{n}}=[U_{I,d}\backslash \operatorname{Gr}_{P,I,W}^{n}]\end{eqnarray}$$

is a $U_{I,d}$-torsor. Since the group scheme $U_{I,d}$ is unipotent over $X^{I}$, we deduce that