Introduction
Let 
$X$ be a smooth projective geometrically connected curve over a finite field 
$\mathbb{F}_{q}$. We denote by 
$F$ its function field, by 
$\mathbb{A}$ the ring of adèles of 
$F$ and by 
$\mathbb{O}$ the ring of integral adèles.
Let 
$G$ be a connected split reductive group over 
$\mathbb{F}_{q}$. For simplicity, we assume in the introduction that the center of 
$G$ is finite.
We consider the space of automorphic forms 
$C_{c}(G(F)\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{C})$. On the one hand, there is the notion of cuspidal automorphic form. An automorphic form is said to be cuspidal if its image under the constant term morphism along any proper parabolic subgroup of 
$G$ is zero. A theorem of Harder [Reference HarderHar74, Theorem 1.2.1] says that the space of cuspidal automorphic forms has finite dimension. The proof uses the Harder–Narasimhan truncations and the contractibility of deep enough strata.
On the other hand, the space of automorphic forms is equipped with an action of the Hecke algebra 
$C_{c}(G(\mathbb{O})\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{Q})$ by convolution on the right. An automorphic form is said to be (rationally) Hecke-finite if it belongs to a finite-dimensional subspace that is stable under the action of the Hecke algebra.
In [Reference LafforgueLaf18, Proposition 8.23], Vincent Lafforgue proved that the space of cuspidal automorphic forms and the space of Hecke-finite automorphic forms are equal. In fact, the space of cuspidal automorphic forms is stable under the action of the Hecke algebra and is finite-dimensional, and thus it is included in the space of Hecke-finite automorphic forms. The converse direction follows from the following fact: any non-zero image of the constant term morphism along a proper parabolic subgroup 
$P$ with Levi quotient 
$M$ is supported on the components indexed by a cone in the lattice of the cocharacters of the center of 
$M$. Hence it generates an infinite-dimensional vector space under the action of the Hecke algebra of 
$M$. Thus a non-cuspidal automorphic form can not be Hecke-finite for the Hecke algebra of 
$M$.
Let 
$\ell$ be a prime number not dividing 
$q$. In [Reference DrinfeldDri78] and [Reference DrinfeldDri87], Drinfeld introduced the stacks classifying 
$\text{GL}_{n}$-shtukas for the representation 
$\text{St}\boxtimes \text{St}^{\ast }$ of 
$\text{GL}_{n}\times \text{GL}_{n}$, where 
$\text{St}$ is the standard representation of 
$\text{GL}_{n}$ and 
$\text{St}^{\ast }$ is its dual, and considered their 
$\ell$-adic cohomology. These were also used by Laurent Lafforgue in [Reference LafforgueLaf97]. Later in [Reference VarshavskyVar04], Varshavsky defined the stacks classifying 
$G$-shtukas 
$\operatorname{Cht}_{G,I,W}$ for general 
$G$ and for an arbitrary representation 
$W$ of 
$\widehat{G}^{I}$, where 
$\widehat{G}$ is the Langlands dual group of 
$G$ over 
$\mathbb{Q}_{\ell }$ and 
$I$ is a finite set (Drinfeld considered the case 
$G=\text{GL}_{n}$, 
$I=\{1,2\}$ and 
$W=\text{St}\boxtimes \text{St}^{\ast }$). Varshavsky also defined the degree 
$j$ cohomology group with compact support 
$H_{G,I,W}^{j}$ of the 
$\ell$-adic intersection complex of 
$\operatorname{Cht}_{G,I,W}$ (this stack is smooth in the case of Drinfeld but not in general). In particular, when 
$I=\emptyset$ and 
$W=\mathbf{1}$ is the one-dimensional trivial representation of the trivial group 
$\widehat{G}^{\emptyset }$, the cohomology group 
$H_{G,\emptyset ,\mathbf{1}}^{0}$ coincides with 
$C_{c}(G(F)\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{Q}_{\ell })$.
The Hecke algebra 
$C_{c}(G(\mathbb{O})\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{Z}_{\ell })$ acts on the cohomology group 
$H_{G,I,W}^{j}$. In [Reference LafforgueLaf18], Vincent Lafforgue defined the subspace 
$H_{G,I,W}^{j,\;\operatorname{Hf}}$ of 
$H_{G,I,W}^{j}$ which consists of the cohomology classes 
$c$ for which 
$C_{c}(G(\mathbb{O})\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{Z}_{\ell })\cdot c$ is a finitely generated 
$\mathbb{Z}_{\ell }$-submodule of 
$H_{G,I,W}^{j}$. When 
$I=\emptyset$ and 
$W=\mathbf{1}$, the space 
$H_{G,\emptyset ,\mathbf{1}}^{0,\,\operatorname{Hf}}$ coincides with the space of Hecke-finite automorphic forms, and thus coincides with the space of cuspidal automorphic forms. Vincent Lafforgue used 
$H_{G,I,W}^{0,\;\operatorname{Hf}}$ to construct the excursion operators on the space of cuspidal automorphic forms and obtained a canonical decomposition of this space indexed by the Langlands parameters.
We can also define a subspace 
$H_{G,I,W}^{j,\;\operatorname{Hf-rat}}$ of 
$H_{G,I,W}^{j}$ which consists of the cohomology classes 
$c$ for which 
$C_{c}(G(\mathbb{O})\backslash G(\mathbb{A})/G(\mathbb{O}),\mathbb{Q}_{\ell })\cdot c$ is a finite-dimensional 
$\mathbb{Q}_{\ell }$-vector subspace of 
$H_{G,I,W}^{j}$. By definition, we have 
$H_{G,I,W}^{j,\;\operatorname{Hf}}\subset H_{G,I,W}^{j,\;\operatorname{Hf-rat}}$. When 
$I=\emptyset$ and 
$W=\mathbf{1}$, it is easy to see that they are equal.
In this paper, we are interested in the constant term morphism of the cohomology of stacks of shtukas, analogous to the case of automorphic forms. For any parabolic subgroup 
$P$ of 
$G$, let 
$M$ be its Levi quotient. As in [Reference VarshavskyVar04], we can define the stack of 
$P$-shtukas 
$\operatorname{Cht}_{P,I,W}$ and the stack of 
$M$-shtukas 
$\operatorname{Cht}_{M,I,W}$. The morphisms 
$G{\hookleftarrow}P{\twoheadrightarrow}M$ induce a correspondence
$$\begin{eqnarray}\operatorname{Cht}_{G,I,W}\leftarrow \operatorname{Cht}_{P,I,W}\rightarrow \operatorname{Cht}_{M,I,W}\!.\end{eqnarray}$$From this we construct a constant term morphism
$$\begin{eqnarray}C_{G}^{P,\,j}:H_{G,I,W}^{j}\rightarrow H_{M,I,W}^{j}.\end{eqnarray}$$ Then we define the cuspidal cohomology 
$H_{G,I,W}^{j,\;\operatorname{cusp}}\subset H_{G,I,W}^{j}$ as the intersection of the kernels of the constant term morphisms for all proper parabolic subgroups.
This construction was suggested by Vincent Lafforgue. He also conjectured the following.
– The cuspidal cohomology is of finite dimension.
– The following three
$\mathbb{Q}_{\ell }$-vector subspaces of $H_{G,I,W}^{j}$ are equal: $$\begin{eqnarray}H_{G,I,W}^{j,\;\operatorname{Hf}}=H_{G,I,W}^{j,\;\operatorname{Hf-rat}}=H_{G,I,W}^{j,\;\operatorname{cusp}}.\end{eqnarray}$$
In this paper, we prove these conjectures except for the equality with 
$H_{G,I,W}^{j,\;\operatorname{Hf}}$, which we plan to treat in a future paper. The main results are as follows.
Theorem 0.0.1 (Theorem 5.0.1).
The 
$\mathbb{Q}_{\ell }$-vector space 
$H_{G,I,W}^{j,\;\operatorname{cusp}}$ has finite dimension.
Proposition 0.0.2 (Proposition 6.0.1).
The two 
$\mathbb{Q}_{\ell }$-vector subspaces 
$H_{G,I,W}^{j,\;\operatorname{cusp}}$ and 
$H_{G,I,W}^{j,\;\operatorname{Hf-rat}}$ of 
$H_{G,I,W}^{j}$ are equal.
As a consequence, 
$H_{G,I,W}^{j,\;\operatorname{Hf}}$ has finite dimension.
In particular, when 
$I=\emptyset$ and 
$W=\mathbf{1}$, the constant term morphism 
$C_{G}^{P,\,0}$ coincides with the usual constant term morphism for automorphic forms. In this case, Theorem 0.0.1 coincides with Theorem 1.2.1 in [Reference HarderHar74], and Proposition 0.0.2 coincides with [Reference LafforgueLaf18, Proposition 8.23] mentioned before.
Let 
$N\subset X$ be a finite subscheme. Theorem 0.0.1 and Proposition 0.0.2 are still true for the cohomology with level structure on 
$N$.
Structure of the paper
In §1 we construct the parabolic induction diagram and define Harder–Narasimhan truncations which are compatible with the parabolic induction. In §2 we recall the cohomology of the stacks of 
$G$-shtukas and define the cohomology of the stacks of 
$M$-shtukas. In §3 we construct the constant term morphism using the compatibility of the geometric Satake equivalence with the constant term functors for the Beilinson–Drinfeld affine grassmannians.
The idea of the proofs of Theorem 0.0.1 and Proposition 0.0.2 is analogous to the case of automorphic forms. The goal of §§4 and 5 is to prove Theorem 0.0.1. In §4 we prove the contractibility of deep enough horospheres. In §5 we use this result and an argument by induction on the semisimple rank to prove the finiteness of cuspidal cohomology. In §6 we show that the constant term morphism commutes with the action of the Hecke algebra, and we prove Proposition 0.0.2.
Notation and conventions
0.0.3
Let 
$G$ be a connected split reductive group over 
$\mathbb{F}_{q}$. Let 
$G^{\text{der}}$ be the derived group of 
$G$ and 
$G^{\text{ab}}:=G/G^{\text{der}}$ the abelianization of 
$G$. Let 
$Z_{G}$ be the center of 
$G$ and 
$G^{\text{ad}}$ the adjoint group of 
$G$ (equal to 
$G/Z_{G}$).
0.0.4
We fix a discrete subgroup 
$\unicode[STIX]{x1D6EF}_{G}$ of 
$Z_{G}(\mathbb{A})$ such that 
$\unicode[STIX]{x1D6EF}_{G}\cap Z_{G}(\mathbb{O})Z_{G}(F)=\{1\}$, the quotient 
$Z_{G}(F)\backslash Z_{G}(\mathbb{A})/Z_{G}(\mathbb{O})\unicode[STIX]{x1D6EF}_{G}$ is finite and the composition of morphisms 
$\unicode[STIX]{x1D6EF}_{G}{\hookrightarrow}Z_{G}(\mathbb{A}){\hookrightarrow}G(\mathbb{A}){\twoheadrightarrow}G^{\text{ab}}(\mathbb{A})$ is injective. Note that the volume of 
$G(F)\backslash G(\mathbb{A})/G(\mathbb{O})\unicode[STIX]{x1D6EF}_{G}$ is finite. We write 
$\unicode[STIX]{x1D6EF}:=\unicode[STIX]{x1D6EF}_{G}$.
0.0.5
We fix a Borel subgroup 
$B\subset G$. By a parabolic subgroup we will mean a standard parabolic subgroup (i.e. a parabolic subgroup containing 
$B$), unless explicitly stated otherwise.
0.0.6
Let 
$H$ be a connected split reductive group over 
$\mathbb{F}_{q}$ with a fixed Borel subgroup. Let 
$\unicode[STIX]{x1D6EC}_{H}$ (respectively 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}$) denote the weight (respectively coweight) lattice of 
$H$. Let 
$\langle ~,~\rangle :\widehat{\unicode[STIX]{x1D6EC}}_{H}\times \unicode[STIX]{x1D6EC}_{H}\rightarrow \mathbb{Z}$ denote the natural pairing between the two.
Let 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{+}\subset \widehat{\unicode[STIX]{x1D6EC}}_{H}$ denote the monoid of dominant coweights and 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{\text{pos}}\subset \widehat{\unicode[STIX]{x1D6EC}}_{H}$ the monoid generated by positive simple coroots. Let 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{\mathbb{Q}}:=\widehat{\unicode[STIX]{x1D6EC}}_{H}\underset{\mathbb{Z}}{\otimes }\mathbb{Q}$. Let 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{\text{pos},\mathbb{Q}}$ and 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{+,\mathbb{Q}}$ denote the rational cones of 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{\text{pos}}$ and 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{+}$. We use analogous notation for the weight lattice.
We use the partial order on 
$\widehat{\unicode[STIX]{x1D6EC}}_{H}^{\mathbb{Q}}$ defined by 
$\unicode[STIX]{x1D707}_{1}\leqslant \text{}^{H}\unicode[STIX]{x1D707}_{2}\Leftrightarrow \unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}\in \widehat{\unicode[STIX]{x1D6EC}}_{H}^{\text{pos},\mathbb{Q}}$ (i.e. 
$\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}$ is a linear combination of simple coroots of 
$H$ with coefficients in 
$\mathbb{Q}_{{\geqslant}0}$).
We will apply these notations to 
$H=G$, 
$H=G^{\text{ad}}$ or 
$H=$ some Levi quotient 
$M$ of 
$G$.
0.0.7
We denote by 
$\unicode[STIX]{x1D6E4}_{G}$ the set of simple roots of 
$G$ and by 
$\widehat{\unicode[STIX]{x1D6E4}}_{G}$ the set of simple coroots. The standard parabolic subgroups of 
$G$ are in bijection with the subsets of 
$\unicode[STIX]{x1D6E4}_{G}$ in the following way. To a parabolic subgroup 
$P$ with Levi quotient 
$M$, we associate the subset 
$\unicode[STIX]{x1D6E4}_{M}$ in 
$\unicode[STIX]{x1D6E4}_{G}$ equal to the set of simple roots of 
$M$.
0.0.8
Let 
$N\subset X$ be a finite subscheme. We denote by 
${\mathcal{O}}_{N}$ the ring of functions on 
$N$ and write 
$K_{G,N}:=\operatorname{Ker}(G(\mathbb{O})\rightarrow G({\mathcal{O}}_{N}))$.
Let 
$H$ be an algebraic group over 
$\mathbb{F}_{q}$. We denote by 
$H_{N}$ the Weil restriction 
$\operatorname{Res}_{{\mathcal{O}}_{N}/\mathbb{F}_{q}}H$.
0.0.9
If not specified, all schemes are defined over 
$\mathbb{F}_{q}$ and all the fiber products are taken over 
$\mathbb{F}_{q}$.
0.0.10
For any scheme 
$S$ over 
$\mathbb{F}_{q}$ and 
$x$ an 
$S$-point of 
$X$, we denote by 
$\unicode[STIX]{x1D6E4}_{x}\subset X\times S$ the graph of 
$x$.
0.0.11
For any scheme 
$S$ over 
$\mathbb{F}_{q}$, we denote by 
$\operatorname{Frob}_{S}:S\rightarrow S$ the Frobenius morphism over 
$\mathbb{F}_{q}$. For any 
$G$-bundle 
${\mathcal{G}}$ on 
$X\times S$, we denote by 
$^{\unicode[STIX]{x1D70F}}{\mathcal{G}}$ the 
$G$-bundle 
$(\operatorname{Id}_{X}\times _{\mathbb{F}_{q}}\operatorname{Frob}_{S})^{\ast }{\mathcal{G}}$.
0.0.12
We use [Reference Laumon and Moret-BaillyLMB99, Definitions 3.1 and 4.1] for prestacks, stacks and algebraic stacks.
0.0.13
As in [Reference Laumon and Moret-BaillyLMB99, §18], [Reference Laszlo and OlssonLO08] and [Reference Laszlo and OlssonLO09], for 
${\mathcal{X}}$ an algebraic stack locally of finite type over 
$\mathbb{F}_{q}$, we denote by 
$D_{c}^{b}({\mathcal{X}},\mathbb{Q}_{\ell })$ the bounded derived category of constructible 
$\ell$-adic sheaves on 
${\mathcal{X}}$. We have the notion of six operators and perverse sheaves.
If 
$f:{\mathcal{X}}_{1}\rightarrow {\mathcal{X}}_{2}$ is a morphism of finite type of schemes (respectively algebraic stacks) locally of finite type, we will denote by 
$f_{!}$, 
$f_{\ast }$, 
$f^{\ast }$, 
$f^{!}$ the corresponding functors between 
$D_{c}^{b}({\mathcal{X}}_{1},\mathbb{Q}_{\ell })$ and 
$D_{c}^{b}({\mathcal{X}}_{2},\mathbb{Q}_{\ell })$, always understood in the derived sense.
0.0.14
We will work with étale cohomology. So for any stack (respectively scheme) (for example 
$\operatorname{Cht}_{G,N,I,W}$ and 
$\operatorname{Gr}_{G,I,W}$), we consider only the reduced substack (respectively subscheme) associated to it.
1 Parabolic induction diagram of stacks of shtukas
The goal of this section is to introduce the parabolic induction diagram of stacks of shtukas without a bound on the modifications at paws in §§1.1–1.3 and to introduce the Harder–Narasimhan stratification for the parabolic induction diagram in §§1.4–1.7.
In §§1.1–1.3 we work in the context of prestacks (see 0.0.12).
1.1 Reminder of stacks of shtukas and Beilinson–Drinfeld affine grassmannians
This subsection is based on [Reference VarshavskyVar04, §2] and [Reference LafforgueLaf18, §§1 and 2]. All the results are well known.
Definition 1.1.1. We define 
$\operatorname{Bun}_{G,N}$ to be the prestack that associates to any affine scheme 
$S$ over 
$\mathbb{F}_{q}$ the groupoid
1.1.2
$\operatorname{Bun}_{G,N}$ is a smooth algebraic stack over 
$\mathbb{F}_{q}$, locally of finite type.
Definition 1.1.3. We define 
$\operatorname{Hecke}_{G,N,I}$ to be the prestack that associates to any affine scheme 
$S$ over 
$\mathbb{F}_{q}$ the groupoid 
$\operatorname{Hecke}_{G,N,I}(S)$ that classifies the following data:
(i)
$(x_{i})_{i\in I}\in (X\smallsetminus N)^{I}(S)$;(ii)
$({\mathcal{G}},\unicode[STIX]{x1D713}),({\mathcal{G}}^{\prime },\unicode[STIX]{x1D713}^{\prime })\in \operatorname{Bun}_{G,N}(S)$;(iii) an isomorphism of
$G$-bundles $\unicode[STIX]{x1D719}:\left.{\mathcal{G}}\vphantom{\big|}\right|_{(X\times S)\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}\overset{{\sim}}{\rightarrow }\left.{\mathcal{G}}^{\prime }\vphantom{\big|}\right|_{(X\times S)\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}$ which preserves the $N$-level structure, i.e. $\unicode[STIX]{x1D713}^{\prime }\circ \left.\unicode[STIX]{x1D719}\vphantom{\big|}\right|_{N\times S}=\unicode[STIX]{x1D713}$.
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.
1.1.7
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:
(i)
$(x_{i})_{i\in I}\in (X\smallsetminus N)^{I}(S)$;(ii)
$({\mathcal{G}},\unicode[STIX]{x1D713})\in \operatorname{Bun}_{G,N}(S)$;(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 
$d\rightarrow \infty$.
Definition 1.1.11. We define the Beilinson–Drinfeld affine grassmannian 
$\operatorname{Gr}_{G,I}$ to be the ind-scheme that associates to any affine scheme 
$S$ over 
$\mathbb{F}_{q}$ the set 
$\operatorname{Gr}_{G,I}(S)$ classifying the following data:
(i)
$(x_{i})_{i\in I}\in X^{I}(S)$;(ii)
${\mathcal{G}},{\mathcal{G}}^{\prime }$ two $G$-bundles on $\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}$;(iii) an isomorphism of
$G$-bundles $\unicode[STIX]{x1D719}:\left.{\mathcal{G}}\vphantom{\big|}\right|_{\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}\overset{{\sim}}{\rightarrow }\left.{\mathcal{G}}^{\prime }\vphantom{\big|}\right|_{\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}\smallsetminus (\bigcup _{i\in I}\unicode[STIX]{x1D6E4}_{x_{i}})}$ where the precise meaning is given in [Reference LafforgueLaf18, Notation 1.7];(iv) a trivialization
$\unicode[STIX]{x1D703}:{\mathcal{G}}^{\prime }\overset{{\sim}}{\rightarrow }G$ on $\unicode[STIX]{x1D6E4}_{\sum \infty x_{i}}$.
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.
(a) For any
$d\in \mathbb{N}$, we define $G_{I,d}$ to be the group scheme over $X^{I}$ that associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the set consisting of pairs $((x_{i})_{i\in I},f)$, where $(x_{i})_{i\in I}\in X^{I}(S)$ and $f$ is an automorphism of the trivial $G$-bundle on $\unicode[STIX]{x1D6E4}_{\sum dx_{i}}$.(b) We define the group scheme
$G_{I,\infty }:=\underset{\longleftarrow }{\lim }\,G_{I,d}$.
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:
Remark 1.1.17. The prestack 
$[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]$ is not an inductive limit of algebraic stacks. But we can still use it for the construction in §§1.2 and 1.3. We will construct a variant of morphism (1.2) for algebraic stacks in 2.4.1.
The following definition will be used in §4.
(a) We define
$\operatorname{Bun}_{G,N,I,d}$ to be the prestack that associates to any affine scheme $S$ over $\mathbb{F}_{q}$ the groupoid classifying the following data: (i)
$(x_{i})_{i\in I}\in (X\smallsetminus N)^{I}(S)$;(ii)
${\mathcal{G}}$: a $G$-bundle over $X\times S$;(iii) a level structure on the divisor
$(N\times S)+\unicode[STIX]{x1D6E4}_{\sum dx_{i}}$, i.e. an isomorphism of $G$-bundles: $\unicode[STIX]{x1D713}:\left.{\mathcal{G}}\vphantom{\big|}\right|_{(N\times S)+\unicode[STIX]{x1D6E4}_{\sum dx_{i}}}\overset{{\sim}}{\rightarrow }\left.G\vphantom{\big|}\right|_{(N\times S)+\unicode[STIX]{x1D6E4}_{\sum dx_{i}}}$.
(b) We define
$\operatorname{Bun}_{G,N,I,\infty }:=\underset{\longleftarrow }{\lim }~\operatorname{Bun}_{G,N,I,d}$.
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 Parabolic induction diagrams
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 
$\operatorname{Bun}_{M,N}$, 
$\operatorname{Cht}_{M,N,I}$, 
$\operatorname{Gr}_{M,I}$, 
$M_{I,\infty }$, 
$\unicode[STIX]{x1D716}_{M,N,I,\infty }$.
Remark 1.2.2. When 
$N$ is non-empty, the prestack 
$\operatorname{Cht}_{P,N,I}$ defined above is not the same as the one defined in [Reference VarshavskyVar04, 2.28]. We will describe the difference in Remark 3.4.4.
1.2.3
The morphisms of groups 
$G{\hookleftarrow}P{\twoheadrightarrow}M$ induce morphisms of prestacks over 
$\operatorname{Spec}\mathbb{F}_{q}$:
$$\begin{eqnarray}\operatorname{Bun}_{G,N}\xleftarrow[{}]{i^{\text{Bun}}}\operatorname{Bun}_{P,N}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\text{Bun}}}\operatorname{Bun}_{M,N}\!.\end{eqnarray}$$Construction 1.2.4. The morphisms of groups 
$G{\hookleftarrow}P{\twoheadrightarrow}M$ induce morphisms of prestacks over 
$(X\smallsetminus N)^{I}$.
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 
$X^{I}$:
$$\begin{eqnarray}\operatorname{Gr}_{G,I}\xleftarrow[{}]{i^{0}}\operatorname{Gr}_{P,I}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{0}}\operatorname{Gr}_{M,I}\!.\end{eqnarray}$$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.
1.3 Quotient by $\unicode[STIX]{x1D6EF}$
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
$$\begin{eqnarray}\unicode[STIX]{x1D716}_{G,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{G,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [G_{I,\infty }^{\text{ad}}\backslash \operatorname{Gr}_{G,I}].\end{eqnarray}$$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
$$\begin{eqnarray}\unicode[STIX]{x1D716}_{P,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{P,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [\overline{P}_{I,\infty }\backslash \operatorname{Gr}_{P,I}].\end{eqnarray}$$ 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
$$\begin{eqnarray}\unicode[STIX]{x1D716}_{M,N,I,\infty }^{\unicode[STIX]{x1D6EF}}:\operatorname{Cht}_{M,N,I}/\unicode[STIX]{x1D6EF}\rightarrow [\overline{M}_{I,\infty }\backslash \operatorname{Gr}_{M,I}].\end{eqnarray}$$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.
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 Harder–Narasimhan stratification of $\operatorname{Bun}_{G}$
In 1.4.1–1.4.10, we recall the Harder–Narasimhan stratification of 
$\operatorname{Bun}_{G}$ defined in [Reference SchiederSch15] and [Reference Drinfeld and GaitsgoryDG15, §7]. (In these papers, the group is reductive over an algebraically closed field. Since our group 
$G$ is split over 
$\mathbb{F}_{q}$, we use Galois descent to obtain the stratification over 
$\mathbb{F}_{q}$.)
In 1.4.11–1.4.17, we recall a variant of the Harder–Narasimhan stratification of 
$\operatorname{Bun}_{G}$ which is compatible with the quotient by 
$\unicode[STIX]{x1D6EF}$, as in [Reference VarshavskyVar04, §2] and [Reference LafforgueLaf18, §1].
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
$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{G,P}:=\widehat{\unicode[STIX]{x1D6EC}}_{G}/\widehat{\unicode[STIX]{x1D6EC}}_{[M,M]_{\text{sc}}}.\end{eqnarray}$$ 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:
$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}\overset{{\sim}}{\rightarrow }\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}.\end{eqnarray}$$We define the slope map to be the composition
$$\begin{eqnarray}\unicode[STIX]{x1D719}_{P}:\widehat{\unicode[STIX]{x1D6EC}}_{G,P}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}\cong \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}.\end{eqnarray}$$ We define 
$\operatorname{pr}_{P}$ to be the composition
$$\begin{eqnarray}\operatorname{pr}_{P}:\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}{\twoheadrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\mathbb{Q}}\simeq \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}.\end{eqnarray}$$ 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 
$\widehat{\unicode[STIX]{x1D6EC}}_{G}{\hookrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$.
Lemma 1.4.4 [Reference SchiederSch15, Proposition 3.1].
The slope map 
$\unicode[STIX]{x1D719}_{P}$ preserves the partial orders ‘
${\leqslant}\text{}^{G}$’ on 
$\widehat{\unicode[STIX]{x1D6EC}}_{G,P}$ and 
$\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}$ in the sense that it maps 
$\widehat{\unicode[STIX]{x1D6EC}}_{G,P}^{\text{pos}}$ to 
$\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\text{pos},\mathbb{Q}}$.
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 
${\mathcal{P}}\overset{P}{\times }G\simeq {\mathcal{G}}_{s}$.
(a) By [Reference SchiederSch15, Lemma 3.3], the above Definition 1.4.6 is equivalent to
(the argument repeats the proof in [Reference SchiederSch15, Lemma 3.3] by replacing$\unicode[STIX]{x1D719}_{G}(\check{\unicode[STIX]{x1D706}}_{G})$ by $\unicode[STIX]{x1D707}$).(b) By [Reference SchiederSch15, Proposition 3.2 and Remark 3.2.4], the definition of
$\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$ in (a) is equivalent to the Tannakian description:where${\mathcal{B}}_{\unicode[STIX]{x1D706}}$ is the line bundle associated to ${\mathcal{B}}$ and $B\rightarrow T\xrightarrow[{}]{\unicode[STIX]{x1D706}}\mathbb{G}_{m}$.(c) The reason why we use Definition 1.4.6 (rather than its equivalent forms) is that it will be useful for non-split groups in future works.
Lemma 1.4.8 [Reference Drinfeld and GaitsgoryDG15, 7.3.4, Proposition 7.3.5].
(a) For any
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$, the stack $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$ is an open substack of $\operatorname{Bun}_{G}$.(b) For any
$\unicode[STIX]{x1D707}_{1}\leqslant \text{}^{G}\unicode[STIX]{x1D707}_{2}$, we have an open immersion $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}_{1}}{\hookrightarrow}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}_{2}}$.(c) We have
$\operatorname{Bun}_{G}=\bigcup _{\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$.(d) The open substack
$\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G}\unicode[STIX]{x1D707}}$ is of finite type.
Definition 1.4.9. For any 
$\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$, let 
$\operatorname{Bun}_{G}^{(\unicode[STIX]{x1D706})}\subset \operatorname{Bun}_{G}$ be the quasi-compact locally closed reduced substack defined in [Reference SchiederSch15, Theorem 2.1] and [Reference Drinfeld and GaitsgoryDG15, Theorem 7.4.3]. It is called a Harder–Narasimhan stratum of 
$\operatorname{Bun}_{G}$.
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
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}_{G}:\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}.\end{eqnarray}$$ 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
Remark 1.4.13. For the same reason as in Remark 1.4.7, Definition 1.4.12 is equivalent to [Reference VarshavskyVar04, Notation 2.1(b)] and [Reference LafforgueLaf18, (1.3)].
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 
$\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$.
(a) For any
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, the stack $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ is an open substack of $\operatorname{Bun}_{G}$.(b) For any
$\unicode[STIX]{x1D707}_{1}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{2}$, we have an open immersion $\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{1}}{\hookrightarrow}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{2}}$.(c) The stack
$\operatorname{Bun}_{G}$ is the inductive limit of these open substacks: $\operatorname{Bun}_{G}=\bigcup _{\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$.(d) The stack
$\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}$ is of finite type.
Proof. Parts (a), (b) and (c) are induced by Lemma 1.4.8 (see also [Reference VarshavskyVar04, Lemme A.3)]. Part (d) follows from 1.4.14. ◻
Remark 1.4.17. See [Reference VarshavskyVar04, Lemmas 3.1 and 3.7] for another proof of Lemma 1.4.8(d) and Lemma 1.4.16(d).
1.5 Harder–Narasimhan stratification of $\operatorname{Bun}_{M}$
Let 
$P$ be a proper parabolic subgroup of 
$G$ and 
$M$ its Levi quotient.
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 
$\deg _{P^{\prime }}:\operatorname{Bun}_{P^{\prime }}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{M,P^{\prime }}$.
Definition 1.5.3. Applying Definition 1.4.9 to 
$M$, for any 
$\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}$, we define a quasi-compact locally closed substack 
$\operatorname{Bun}_{M}^{(\unicode[STIX]{x1D706})}\subset \operatorname{Bun}_{M}$, called a Harder–Narasimhan stratum of 
$\operatorname{Bun}_{M}$.
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
(a) For any
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$, the stack $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ is an open substack of $\operatorname{Bun}_{M}$.(b) For any
$\unicode[STIX]{x1D707}_{1}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{2}$, we have an open immersion $\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{1}}{\hookrightarrow}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}_{2}}$.(c) The stack
$\operatorname{Bun}_{M}$ is the inductive limit of these open substacks: $\operatorname{Bun}_{M}=\bigcup _{\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$.
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
$$\begin{eqnarray}\operatorname{Bun}_{M}\xrightarrow[{}]{\deg _{M}}\widehat{\unicode[STIX]{x1D6EC}}_{M,M}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{M,M}^{\mathbb{Q}}\simeq \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}}^{\mathbb{ Q}}{\twoheadrightarrow}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}},\end{eqnarray}$$ 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 
$\unicode[STIX]{x1D708}\in A_{M}$.
Definition 1.5.8. We define 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}$ to be the intersection of 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ and 
$\operatorname{Bun}_{M}^{\unicode[STIX]{x1D708}}$.
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
$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}\!.\end{eqnarray}$$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
$$\begin{eqnarray}\operatorname{pr}_{P}^{\text{ad}}:\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\mathbb{Q}}\rightarrow \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}}.\end{eqnarray}$$ 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
$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}=\underset{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}},\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}}{\bigcup }\mathop{\operatorname{Bun}}_{M}^{(\unicode[STIX]{x1D706})}\!.\end{eqnarray}$$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 
$\widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{\operatorname{pos},\mathbb{Q}}$.
Lemma 1.5.14. Let 
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$. Then the stack 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}$ is empty unless 
$\unicode[STIX]{x1D708}\in A_{M}$ defined in 1.5.7 and 
$\unicode[STIX]{x1D708}\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})$.
Proof. The first condition follows from 1.5.7. To prove the second condition, note that for the set 
$\{\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{M}^{+,\mathbb{Q}}\;|\;\unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\operatorname{pr}_{P}^{\text{ad}}\circ \unicode[STIX]{x1D6F6}_{G}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}\}$ to be non-empty, by 1.5.13 we must have 
$\unicode[STIX]{x1D708}\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})$.◻
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
$$\begin{eqnarray}\unicode[STIX]{x1D706}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}\quad \text{and}\quad \operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706})=\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})\;\Leftrightarrow \;\unicode[STIX]{x1D706}\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}.\end{eqnarray}$$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
$$\begin{eqnarray}\unicode[STIX]{x1D706}\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}\quad \text{and}\quad \operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D706})=\unicode[STIX]{x1D708}\;\Leftrightarrow \;\unicode[STIX]{x1D706}\leqslant \text{}^{\overline{M}}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D708}}.\end{eqnarray}$$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 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}$.
Lemma 1.5.19. The stack 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}$ is of finite type.
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 
$\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{Q}}$:
$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}:=\{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\mathbb{ Q}},\;\unicode[STIX]{x1D708}\leqslant \text{}^{G^{\text{ad}}}\operatorname{pr}_{P}^{\text{ad}}(\unicode[STIX]{x1D707})\}.\end{eqnarray}$$We deduce that
$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}\end{eqnarray}$$and
$$\begin{eqnarray}\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$1.6 Harder–Narasimhan stratification of parabolic induction
Recall that we have morphisms (1.3): 
$\operatorname{Bun}_{G}\xleftarrow[{}]{i^{\text{Bun}}}\operatorname{Bun}_{P}\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\text{Bun}}}\operatorname{Bun}_{M}$.
Definition 1.6.1. Let 
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$. We define 
$\operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ to be the inverse image of 
$\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ in 
$\operatorname{Bun}_{P}$.
Lemma 1.6.2. The image of 
$\operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ in 
$\operatorname{Bun}_{M}$ is included in 
$\operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$.
Proof. Let 
${\mathcal{P}}\in \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$ and let 
${\mathcal{M}}$ be its image in 
$\operatorname{Bun}_{M}$. We will check that 
${\mathcal{M}}\in \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$. For any parabolic subgroup 
$P^{\prime }$ of 
$M$, let 
$M^{\prime }$ be its Levi quotient. Let 
${\mathcal{P}}^{\prime }$ be a 
$P^{\prime }$-structure of 
${\mathcal{M}}$ and 
${\mathcal{M}}^{\prime }:={\mathcal{P}}^{\prime }\overset{P^{\prime }}{\times }M^{\prime }$. By Definition 1.5.4, we need to prove that 
$\unicode[STIX]{x1D6F6}_{G}\circ \unicode[STIX]{x1D719}_{P^{\prime }}\circ \operatorname{deg}_{P^{\prime }}({\mathcal{P}}^{\prime })\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}$.
Let 
$P^{\prime \prime }:=P\underset{M}{\times }P^{\prime }$. It is a parabolic subgroup of 
$G$ with Levi quotient 
$M^{\prime }$. We have the following.
By [Reference Drinfeld and GaitsgoryDG16, Lemma 2.5.8], we can define a 
$P^{\prime \prime }$-bundle 
${\mathcal{P}}^{\prime \prime }:={\mathcal{P}}\underset{{\mathcal{M}}}{\times }{\mathcal{P}}^{\prime }$. We have 
$\deg _{P^{\prime }}{\mathcal{P}}^{\prime }=\deg _{M^{\prime }}{\mathcal{M}}^{\prime }=\deg _{P^{\prime \prime }}{\mathcal{P}}^{\prime \prime }$. Taking into account that 
$\widehat{\unicode[STIX]{x1D6EC}}_{G}^{\mathbb{Q}}=\widehat{\unicode[STIX]{x1D6EC}}_{M}^{\mathbb{Q}}$, we deduce that 
$\unicode[STIX]{x1D6F6}_{G}\circ \unicode[STIX]{x1D719}_{P^{\prime }}\circ \operatorname{deg}_{P^{\prime }}({\mathcal{P}}^{\prime })=\unicode[STIX]{x1D6F6}_{G}\,\circ \,\unicode[STIX]{x1D719}_{P^{\prime \prime }}\,\circ \,\operatorname{deg}_{P^{\prime \prime }}({\mathcal{P}}^{\prime \prime })\leqslant \text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}$, where the last inequality follows from the definition of 
$\operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}$.◻
1.6.3
By Lemma 1.6.2, morphisms (1.3) induce morphisms:
$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}.\end{eqnarray}$$ The group 
$\unicode[STIX]{x1D6EF}$ acts on all these stacks. All the morphisms are 
$\unicode[STIX]{x1D6EF}$-equivariant. Thus morphisms (1.25) induce morphisms:
$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$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:
$$\begin{eqnarray}\operatorname{Bun}_{G}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Bun}_{P}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Bun}_{M}^{{\leqslant}\text{}^{G^{\text{ad}}}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$1.7 Harder–Narasimhan stratification of stack of shtukas
Notation 1.7.1. In the remaining part of the paper, we will only use the truncations indexed by ‘
${\leqslant}\text{}^{G^{\text{ad}}}$’ (rather than ‘
${\leqslant}\text{}^{G}$’). To simplify the notation, from now on, ‘
${\leqslant}$’ means ‘
${\leqslant}\text{}^{G^{\text{ad}}}$’.
Definition 1.7.2. Let 
$\unicode[STIX]{x1D707}\in \widehat{\unicode[STIX]{x1D6EC}}_{G^{\text{ad}}}^{+,\mathbb{Q}}$ (respectively 
$\unicode[STIX]{x1D706}\in \widehat{\unicode[STIX]{x1D6EC}}_{G}^{+,\mathbb{Q}}$). We define 
$\operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$ (respectively 
$\operatorname{Cht}_{G,N,I}^{(\unicode[STIX]{x1D706})}$) to be the inverse image of 
$\operatorname{Bun}_{G}^{{\leqslant}\,\unicode[STIX]{x1D707}}$ (respectively 
$\operatorname{Bun}_{G}^{(\unicode[STIX]{x1D706})}$) by the morphism
$$\begin{eqnarray}\operatorname{Cht}_{G,N,I}\rightarrow \operatorname{Bun}_{G},\quad ((x_{i})_{i\in I},({\mathcal{G}},\unicode[STIX]{x1D713})\xrightarrow[{}]{\unicode[STIX]{x1D719}}(\text{}^{\unicode[STIX]{x1D70F}}{\mathcal{G}},^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D713}))\mapsto {\mathcal{G}}.\end{eqnarray}$$ Similarly, we define 
$\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$ (respectively 
$\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}$, 
$\operatorname{Cht}_{M,N,I}^{(\unicode[STIX]{x1D706})}$) using the morphism 
$\operatorname{Cht}_{M,N,I}\rightarrow \operatorname{Bun}_{M}$ and 
$\operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}$ (respectively 
$\operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}$) using the morphism 
$\operatorname{Cht}_{P,N,I}\rightarrow \operatorname{Bun}_{P}$.
1.7.3
The following diagram is commutative
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 
$\operatorname{Cht}_{P,N,I}$.
Lemma 1.7.4. The image of 
$\operatorname{Cht}_{P,N,I}^{{\leqslant}\,\unicode[STIX]{x1D707}}$ in 
$\operatorname{Cht}_{M,N,I}$ is included in 
$\operatorname{Cht}_{M,N,I}^{{\leqslant}\,\unicode[STIX]{x1D707}}$.
1.7.5
Just as in 1.6.3 and 1.6.4, morphisms (1.4) induce morphisms:
$$\begin{eqnarray}\displaystyle & \operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \operatorname{Cht}_{G,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}\leftarrow \operatorname{Cht}_{P,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}\rightarrow \operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\,\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}. & \displaystyle\end{eqnarray}$$We deduce from (1.24) a decomposition:
$$\begin{eqnarray}\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6EF}=\underset{\unicode[STIX]{x1D708}\in \widehat{\unicode[STIX]{x1D6EC}}_{Z_{M}/Z_{G}}^{\unicode[STIX]{x1D707}}}{\bigsqcup }\operatorname{Cht}_{M,N,I}^{{\leqslant}\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D6EF}.\end{eqnarray}$$2 Cohomology of stacks of shtukas
In §§2.1–2.5 we recall the definition of the cohomology of stacks of 
$G$-shtukas with values in perverse sheaves coming from 
$[G_{I,\infty }\backslash \operatorname{Gr}_{G,I}]$ via 
$\unicode[STIX]{x1D716}_{G,N,I,\infty }$, i.e. coming from 
$G_{I,\infty }$-equivariant perverse sheaves over 
$\operatorname{Gr}_{G,I}$. These sections are based on [Reference LafforgueLaf18, §§1, 2 and 4].
In §2.6 we define the cohomology of stacks of 
$M$-shtukas.
Notation 2.0.1. Our results are of geometric nature, i.e. we will not consider the action of 
$\text{Gal}(\overline{\mathbb{F}_{q}}/\mathbb{F}_{q})$. From now on, we pass to the base change over 
$\overline{\mathbb{F}_{q}}$. We keep the same notations 
$X$, 
$\operatorname{Bun}_{G,N}$, 
$\operatorname{Cht}_{G,N,I}$, 
$\operatorname{Gr}_{G,I}$, etc., but now everything is over 
$\overline{\mathbb{F}_{q}}$ and the fiber products are taken over 
$\overline{\mathbb{F}_{q}}$.
2.1 Reminder of a generalization of the geometric Satake equivalence
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