Perverse Sheaves on the Moduli of Local Shtukas

Chapter X: Moduli of Local Shtukas

10.2 Perverse Sheaves on the Moduli of Local Shtukas

Let𝜇_•be a sequence of dominant coweights and (𝑚₁, 𝑛₁), (𝑚₂, 𝑛₂)be two pairs of 𝜇_•-large integers such that𝑚₁ ≤ 𝑚₂,𝑛₁ ≤ 𝑛₂, and𝑚₂ ≠∞. Define the functor

Res^𝑚𝑚²₁^,𝑛,𝑛²₁ :=(res^𝑚𝑚²₁^,𝑛.𝑛²₁)^★: P(Sht^loc(𝑚𝜇_• ^2,𝑛2),Λ) →P(Sht^loc(𝑚_𝜇 ^1,𝑛1)

•,Λ ). (10.7) Then(10.4)yields

Res^𝑚𝑚²₁^,𝑛,𝑛²₁◦Res^𝑚𝑚³₂^,𝑛,𝑛³₃ =Res^𝑚𝑚³₁^,𝑛,𝑛³₁. (10.8) Like Res^𝑚𝑛, the functor Res^𝑚𝑚^𝑖𝑗^,𝑛,𝑛^𝑖𝑗 is also an equivalence of categories if𝑚_𝑗 >1.

We define the category of perverse sheaves on the moduli of local Shtukas as P(Sht^loc_¯

𝑘

,Λ):= Ê

𝜉∈𝜋₁(𝐺)

P(Sht^loc_𝜉 ,Λ), P(Sht^loc_𝜉 ,Λ) := lim

−−→

(𝑚,𝑛, 𝜇)

P(Sht^loc𝜇 ^(𝑚,𝑛),Λ) (10.9) where the limit is taken over the triples{(𝑚, 𝑛, 𝜇) ∈Z²×𝜉 | (𝑚, 𝑛) is𝜇large}with the product partial order. As in [XZ17], we call objects in P(Sht^loc_𝜉 ,Λ) connected objects. The connecting morphism is given by the composite of fully faithful functor

P(Sht^loc(𝜇₁ ^𝑚1,𝑛1),Λ) P(Sht^loc(𝜇₁ ^𝑚2,𝑛2),Λ) P(Sht^{loc(𝑚2,𝑛2)}

𝜇⁰

1

,Λ).

Res^𝑚𝑚^{2, 𝑛2} 1, 𝑛

1

𝑖𝜇 1, 𝜇0

1

For each dominant coweight 𝜇and a pair of 𝜇-large integers(𝑚, 𝑛), we define the natural pullback functor

Ψ^{loc(𝑚,𝑛)} :=Res^𝑚,𝑛_𝑚,0 : P(Hk^loc(𝑚)𝜇 ,Λ) →P(Sht^{loc(𝑚,𝑛)}𝜇 ,Λ). (10.10) We observe that Ψ^{loc(𝑚,𝑛)} commutes with the connecting morphism in (10.9) by (10.8) and the proper smooth base change. Then we can take the limit and direct sum ofΨ^{loc(𝑚,𝑛)} and derive the following well-defined functor

Ψ^loc : P(Hk^loc_¯

𝑘

,Λ) →P(Sht^loc_¯

𝑘

,Λ). (10.11)

LetF𝑖 ∈ P(Sht^loc_𝜉

𝑖

, 𝐸)be connected objects. It is realized asF_{𝑖, 𝜇}^{(𝑚𝑖,𝑛𝑖)}

𝑖

∈P(Sht^loc(𝑚𝜇𝑖 ^{𝑖,𝑛𝑖)}

, 𝐸) for some 𝜇_𝑖 and some pair of 𝜇_𝑖-large integers (𝑚_𝑖, 𝑛_𝑖). We define the set of coho- mological correspondences betweenF₁andF₂as

Corr_Sht^loc(F₁,F₂)

:= Ê

𝜉∈𝜋₁(𝐺)

lim−−→Corr

Sht^𝜆,_𝜇^{loc(𝑚1, 𝑛1)}

1|𝜇2

(Sht^loc𝜇₁^{(𝑚1,𝑛1)},F^{(𝑚1,𝑛1)}

1, 𝜇₁ ),(Sht^loc𝜇₂^{(𝑚2,𝑛2)},F^{(𝑚2,𝑛2)}

2, 𝜇₂ ) ,

where the limit is taken over all partially ordered sextuples(𝜇₁, 𝜇₂, 𝜆, 𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂) such that

• (𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂)is (𝜇₁+𝜆, 𝜆)and(𝜇₂+𝜆, 𝜆)-acceptable,

• 𝜇_𝑖 ∈𝜉_𝑖, for some𝜉_𝑖 ∈𝜋₁(𝐺),

• 𝜆∈𝜉.

Let (𝜇₁, 𝜇₂, 𝜆, 𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂) ≤ (𝜇⁰

1, 𝜇⁰

2, 𝜆⁰, 𝑚⁰

1, 𝑛⁰

1, 𝑚⁰

2, 𝑛⁰

2) be another such sextu- ple. The connecting morphism between the cohomological correspondences

CorrSht^𝜆,_𝜇^{loc(𝑚1, 𝑛1)}

1|𝜇2

(Sht^loc(𝑚𝜇₁ ^1,𝑛1),F₁, 𝜇₁),(Sht^loc(𝑚𝜇₂ ^2,𝑛2),F₂, 𝜇₂)

(10.12) and

CorrSht

𝜆0,loc(𝑚0 1, 𝑛0

1) 𝜇0

1|𝜇0 2

(Sht^loc(𝑚

0 1,𝑛⁰

1) 𝜇⁰

1

,F₁, 𝜇⁰

1),(Sht^loc(𝑚

0 2,𝑛⁰

2) 𝜇⁰

2

,F₂, 𝜇⁰

2)

(10.13)

is given by first pulling back(4.13)to the Hecke correspondence Sht^loc(𝑚

0 1,𝑛⁰

1)

𝜇₁ Sht^{𝜆,loc(𝑚}

0 1,𝑛⁰

1)

𝜇₁|𝜇₂ Sht^loc(𝑚

0 2,𝑛⁰

2)

𝜇₂ ,

along the restriction morphism, then pushing it forward to the Hecke correspondence Sht^loc(𝑚

0 1,𝑛⁰

1) 𝜇⁰

1

Sht^𝜆

0,loc(𝑚⁰₁,𝑛⁰

1) 𝜇⁰

1|𝜇⁰

2

Sht^loc(𝑚

0 2,𝑛⁰

2) 𝜇⁰

2

.

The connecting morphism is well-defined and can be composed. We refer to [XZ17,

§5.4.1] for more discussions.

C h a p t e r 11

KEY THEOREM FOR CONSTRUCTING THE JACQUET-LANGLANDS TRANSFER

In this chapter, we state and prove the key theorem for our construction of the Jacquet-Langlands transfer. We will make use of the theory of the cohomological correspondences throughout this chapter. Instead of explaining all the details, we refer to [XZ17, Appendix A.2] for a nice discussion.

11.1 Preliminaries

Fix a half Tate twist Λ(1/2). Recall notations h𝑑i and 𝑓^★ introduced in §1.3.

Throughout this section, we consider the Langlands dual group scheme ˆ𝐺_Λ overΛ of𝐺 and itsΛ-representations. The subscriptsΛwill be omitted for simplicity. We generalize a few notions introduced in previous sections for the sake of stating the key theorem.

More on Local Hecke Stacks

Let 𝑉_• := 𝑉₁𝑉₂ · · ·𝑉_𝑠 ∈ Rep(𝐺ˆ^𝑠) and assume that for each 𝑖, 𝑉_𝑖 has the Jordan-Holder factors¥ {𝑉_𝜇

𝑖 𝑗}𝑗.

The integral geometric Satake equivalence (Theorem 8.0.14) Sat𝐺^𝑠 sends𝑉_• to an (𝐿⁺𝐺 ⊗ 𝑘¯)^𝑠-equivariant perverse sheaf Sat𝐺^𝑠(𝑉_•) on (𝐺 𝑟_𝐺 ⊗ 𝑘¯)^𝑠. We write 𝐺 𝑟_𝑉

•

for the support of the external tensor product Sat(𝑉₁)˜Sat(𝑉₂)˜ · · ·˜Sat(𝑉_𝑠). Let 𝑚 be a non-negative integer. We call it𝑉_𝑖-large if 𝑚 is 𝜇_{𝑖 𝑗}-large for each 𝑗, and we call it 𝑉_•-large if 𝑚 = 𝑚₁ +𝑚₂ + · · · +𝑚_𝑠 such that 𝑚_𝑖 is 𝑉_𝑖-large for each 𝑖. For a 𝑉_•-large integer 𝑚, Sat𝐺^𝑛(𝑉_•) descends to a perverse sheaf supported on Hk_𝑉^loc(𝑚)

• := [𝐿^𝑚𝐺\𝐺 𝑟_𝑉

•]. We write 𝑆(𝑉)^loc(𝑚) for the twist of this perverse sheaf by h𝑚dim𝐺i. Note that 𝑆(𝑉_•)^loc(𝑚) is isomorphic to the "★"-pullback of 𝑆(𝑉₁)^loc(𝑚1)𝑆(𝑉₂)^loc(𝑚2)· · ·𝑆(𝑉_𝑠)^{loc(𝑚𝑠)} along the perfectly smooth morphism Hk_𝑉^loc(𝑚)

• → Î

𝑖Hk^loc(_𝑉 ^𝑚𝑖)

𝑖 constructed in(9.9). In the case𝑠=1, we have

𝐺 𝑟_𝑉

1 =∪𝑗𝐺 𝑟_𝜇

1𝑗, Hk^loc_𝑉 ^(𝑚)

1 =∪𝑗Hk^loc𝜇_1𝑗^𝑚. In general, Hk^loc(𝑚)_𝑉

• is of the form∪𝜇_•Hk^loc(𝑚)𝜇_• . Via descent, Corollary 8.0.15 gives

the following natural isomorphism:

Hom_𝐺ˆ(𝑉_•, 𝑊_•) Corr

Hk⁰_𝑉^,loc(𝑚)

• |𝑊•

(Hk_𝑉^loc(𝑚)

•

,Sat^loc_𝐺 ^(𝑚)(𝑉_•)),(Hk^loc_𝑊 ^(𝑚)

•

,Sat^loc_𝐺 ^(𝑚)(𝑊_•)) . (11.1) Here and below, we regard 𝑉_• and 𝑊_• as representations of ˆ𝐺 via the diagonal embedding ˆ𝐺 ↩→ 𝐺ˆ^𝑠.

Let𝑉_• and𝑊_• be two representations of ˆ𝐺^𝑠. We can similarly define 𝐺 𝑟⁰

𝑉_•|𝑊_• :=

𝐺 𝑟_𝑉

•×𝐺 𝑟𝐺 𝐺 𝑟_𝑊

• and Hk_𝑉^0,loc(𝑚)

•|𝑊_• = [𝐿^𝑚𝐺\𝐺 𝑟⁰

𝑉_•|𝑊_•]. More on Moduli of Local Shtukas

Let𝑉_•∈Rep(𝐺ˆ^𝑠). For a pair of non-negative integers(𝑚, 𝑛), we can generalize the notion of 𝜇_•-largeand define the notion of𝑉_•-large. Let(𝑚, 𝑛)be a pair of𝑉_•and 𝑊_•-large integers, we can define the moduli of restricted local Shtukas Sht^loc_𝑉 ^(𝑚,𝑛)

•

and Sht^{loc(𝑚,𝑛)}

𝑉_•|𝑊• . Similar to Hk_𝑉^loc

•, the stacks Sht_𝑉^{loc(𝑚,𝑛)}

• and Sht^{loc(𝑚,𝑛)}

𝑉_•|𝑊• can be regarded as unions of Sht^loc(𝜇_• ^𝑚,𝑛) and unions of Sht^{loc(𝑚,𝑛)}

𝜇_•|𝜈_• . We have the natural forgetful map 𝜓^{loc(𝑚,𝑛)} : Sht_𝑉^{loc(𝑚,𝑛)}

• →Hk_𝑉^loc(𝑚)

•

. (11.2)

Choose a pair of𝑉_•-large integers(𝑚, 𝑛) such that𝑛 > 0. Write 𝑆(𝑉e_•)^{loc(𝑚,𝑛)} := Ψ^{loc(𝑚,𝑛)}(Sat(𝑉_•)^loc(𝑚)) ∈P(Sht^{loc(𝑚,𝑛)},Λ)

for the pullback of Sat(𝑉)^loc(𝑚) along the morphism 𝜓^{loc(𝑚,𝑛)} (up to a shift and twist). For𝑠 =1,𝑆(e𝑉)^{loc(𝑚,𝑛)} represents the perverse sheaf𝑆(e𝑉) := Ψ(Sat𝐺(𝑉)) ∈ P(Sht^loc_¯

𝑘

,Λ).

Consider the front face of the diagram(10.5). The second and third vertical maps are perfectly smooth. Pulling back the cohomological correspondence on the right hand side of(11.1)to the upper edge and pre-composing it with(11.1), we get the map

C^{loc(𝑚,𝑛)} : Hom_𝐺ˆ(𝑉_•, 𝑊_•) → Corr

Sht⁰_𝑉^{,loc(𝑚, 𝑛)}

• |𝑊•

(𝑆(𝑉e_•)^{loc(𝑚,𝑛)}, 𝑆(𝑊f_•)^{loc(𝑚,𝑛)}). (11.3) The mapC^{loc(𝑚,𝑛)} is compatible with the compositions at the source and target, and we refer to [XZ17, Lemma 6.1.8] for the proof.

Let𝑉_• ∈ Rep(𝐺ˆ^𝑠) and𝑊 ∈Rep(𝐺ˆ). We call a quadruple of non-negative integers (𝑚₁, 𝑛₂, 𝑚₁, 𝑛₁)𝑉_•𝑊-acceptableif

• 𝑚₁−𝑚₂=𝑛₁−𝑛₂is𝑊-large,

• (𝑚₂, 𝑛₁)is𝑉_•-large.

For a quadruple of 𝑉_•𝑊-acceptable integers (𝑚₁, 𝑛₂, 𝑚₁, 𝑛₁), we can construct the partial Frobenius morphism

𝐹⁻¹

𝑉_•𝑊 : Sht^loc(_𝜎𝑊^𝑚_𝑉^1,𝑛1)

• → Sht_𝑉^{loc(𝑚2,𝑛2)}

•𝑊 (11.4)

similar to (10.1). Here,𝜎𝑊 is the Frobenius twist of𝑊 as in Remark 8.0.14.

Let𝑉₁, 𝑉₂ ∈Rep(𝐺ˆ). For any projective object𝑊 ∈Rep(𝐺ˆ), choose a quadruple of ( (𝑉₁⊗𝑉₂⊗𝑊)𝑊^∗)-acceptable integers(𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂). We define the following stack

Sht^{𝑊 ,}_𝑉 ^{loc(𝑚1,𝑛1)}

1|𝑉₂ :=Sht^loc(_𝑉 ^𝑚1,𝑛1)

1|𝜎𝑊^∗(𝜎𝑊⊗𝑉₁) ×

Sht^loc(₍ ^{𝑚2, 𝑛2)}

𝜎 𝑊⊗𝑉1)𝑊∗ Sht^loc(_(𝜎𝑊^𝑚_⊗¹_𝑉^.𝑛1)

1)𝑊^∗|𝑉₂

. (11.5) The CategoryCoh^𝐺ˆ(𝐺 𝜎ˆ )

Recall from Remark 8.0.14 that the Langlands dual group ˆ𝐺 is naturally equipped with an action of the arithmetic Frobenius 𝜎. Consider the𝜎-twisted conjugation action of ˆ𝐺 on ˆ𝐺. We denote by Coh^𝐺ˆ(𝐺 𝜎ˆ )the abelian category of ˆ𝐺-equivariant coherent sheaves on the (non-neutral) component ˆ𝐺 𝜎 ⊂ 𝐺ˆ o 𝜎. Equivalently, Coh^𝐺ˆ(𝐺 𝜎ˆ ) can be regarded as the abelian category of coherent sheaves on the quotient stack [𝐺 𝜎ˆ /𝐺ˆ]where ˆ𝐺acts on ˆ𝐺 𝜎by the usual conjugation action.

Let𝑉 ∈Rep(𝐺ˆ) be an algebraic representation of ˆ𝐺. There is an associated vector bundle on ˆ𝐺 𝜎with global section O_ˆ

𝐺 ⊗𝑉. Consider the following action of ˆ𝐺 on O_𝐺ˆ ⊗𝑉. For any 𝑔 ∈ 𝐺ˆ and (𝑓 , 𝑣) ∈ O_𝐺ˆ ⊗𝑉, 𝑔· (𝑓 , 𝑣) := (𝑔 𝑓 𝜎⁻¹(𝑔), 𝑔𝑣). The associated vector bundle thus gives an object𝑉e∈Coh^𝐺ˆ(𝐺 𝜎ˆ ).

11.2 Key Theorem

The following theorem is an analogue of [XZ17, Theorem 6.0.1].

Theorem 11.2.1. Let𝑉₁, 𝑉₂ ∈ Rep(𝐺ˆ) be two projective Λ-modules. Then there exists the following map

S𝑉₁,𝑉₂ : Hom

Coh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₂) −→Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₂)), (11.6) which is compatible with the natural composition maps in the source and target.

We prove this theorem in the rest of this section.

We give an explicit construction of S𝑉₁,𝑉₂. Consider the following canonical iso- morphisms

HomCoh^𝐺ˆ(𝐺 𝜎)ˆ (𝑉e₁,𝑉e₂) (11.7) HomO_{𝐺 𝜎ˆ} (O_{𝐺 𝜎ˆ} ⊗𝑉₁,O_{𝐺 𝜎ˆ} ⊗𝑉₂)^𝐺ˆ

Hom(𝑉₁,O_{𝐺 𝜎ˆ} ⊗𝑉₂)^𝐺ˆ (𝑉^∗

1 ⊗ O_ˆ

𝐺 𝜎 ⊗𝑉₂)^𝐺ˆ.

Let 𝑊 ∈ Rep_Λ(𝐺ˆ_Λ) be a projective Λ-module with Λ-basis {𝑒_𝑖}𝑖 and dual basis {𝑒^∗

𝑖}𝑖. We construct the map Θ𝑊 : Hom_𝐺ˆ

Λ(𝑉₁, 𝜎𝑊^∗⊗𝑉₂⊗𝑊) Hom_𝐺ˆ(𝑉₁,Hom(𝜎𝑊⊗𝑊^∗, 𝑉₂)) →Hom

Coh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₂), by sendinga ∈Hom_𝐺ˆ

Λ(𝑉₁, 𝜎𝑊^∗ ⊗𝑉₂⊗𝑊)to the𝑉^∗

1 ⊗𝑉₂-valued functionΘ𝑊(a) on ˆ𝐺 𝜎defined by

(Θ𝑊(a) (𝑔)) (𝑣₁) :=Õ

𝑖

(a(𝑣₁)) (𝑔 𝑒^∗

𝑖 ⊗𝑒_𝑖). It suffices to construct the map

C𝑊 : Hom_𝐺ˆ(𝑉₁, 𝜎𝑊^∗ ⊗𝑊 ⊗𝑉₂) →Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₂)).

for every 𝑊 ∈ Rep_Λ(𝐺ˆ_Λ). Let a ∈ Hom_𝐺ˆ(𝑉₁, 𝜎𝑊^∗ ⊗ 𝑊 ⊗ 𝑉₂). We have the following coevaluation and evaluation maps:

𝛿_𝜎𝑊 :1→𝜎𝑊^∗⊗ 𝜎𝑊 , 𝑒_𝑊 :𝑊 ⊗𝑊^∗ →1.

Choose a quadruple (𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂) of (𝑉₁ ⊗𝑉₂⊗𝑊)𝑊^∗-large integers. Then the mapC^{loc(𝑚1,𝑛1)} defined in (11.3)sendsato the cohomological correspondence C^{loc(𝑚1,𝑛1)}(a) :𝑆(𝑉e₁)^{loc(𝑚1,𝑛1)} −→ 𝑆(𝜎𝑊f^∗(𝑉e₂⊗𝑊e))^{loc(𝑚1,𝑛1)}. (11.8) The partial Frobenius morphism(11.4)gives rise to the cohomological correspon- dence (cf.[XZ17, A.2.3])

DΓ^∗

𝐹⁻¹

(𝑊⊗𝑉2)𝑊∗ :𝑆(𝜎𝑊f^∗(𝑉e₂⊗𝑊e))^{loc(𝑚1,𝑛1)} −→𝑆( (𝑉e₂⊗𝑊e)𝑊f^∗)^{loc(𝑚2,𝑛2)}. (11.9) Finally,C^{loc(𝑚2,𝑛2)} sends id⊗𝑒_𝑊 to the cohomological correspondence

C^{loc(𝑚2,𝑛2)}(id⊗𝑒_𝑊) :𝑆( (𝑉e₂⊗𝑊e)𝑊f^∗)^{loc(𝑚2,𝑛2)} −→ 𝑆(𝑉e₂)^{loc(𝑚2,𝑛2)}. (11.10)

The composition of cohomological correspondences (11.8), (11.9), and (11.10) yields a cohomological correspondence

C𝑊(a) ∈Corr

Sht^{𝑊 ,}_𝑉 ^{loc(𝑚1, 𝑛1)}

1|𝑉2

(𝑆(𝑉e₁)^{loc(𝑚1,𝑛1)}, 𝑆(𝑉e₂)^{loc(𝑚2,𝑛2)}).

The construction of the mapS𝑉₁,𝑉₂ can be summarized in the following diagram

HomCoh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₂) Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₂))

Hom_𝐺ˆ(𝑉₁, 𝜎𝑊^∗ ⊗𝑉₂⊗𝑊) .

S𝑉1,𝑉2

Θ𝑊

C𝑊

We prove that the cohomological correspondence constructed in the previous section is well-defined and can be composed.

Leta⁰ denote the image ofa under the canonical isomorphism Hom_𝐺ˆ(𝑉₁, 𝜎𝑊^∗ ⊗ 𝑉₂⊗𝑊) Hom_𝐺ˆ(𝜎𝑊 ⊗𝑉₁⊗𝑊^∗, 𝑉₂).

Lemma 11.2.2. Let 𝑋 , 𝑌 , 𝑊₁, 𝑊₂, 𝑊⁰

1, 𝑊⁰

2 be representations of 𝐺ˆ, and 𝑓₁ ⊗ 𝑓₂ : 𝑊₁⊗𝑊₂→𝑊⁰

1⊗𝑊⁰

2be a𝐺ˆ×𝐺ˆ-module homomorphism. Letb∈Hom_𝐺ˆ(𝑋 , 𝜎𝑊₁⊗ 𝑌 ⊗𝑊₂)andb⁰ ∈Hom𝐺ˆ(𝑌 ⊗𝑊⁰

2⊗𝑊⁰

1, 𝑌). We omit choosing appropriate integers (𝑚_𝑖, 𝑛_𝑖)for simplicity. Then we have

C (b⁰◦ (id⊗𝑓₂⊗𝑓₁)) ◦DΓ^∗

𝐹⁻¹◦C (b) =C (b⁰) ◦DΓ^∗

𝐹⁻¹◦C ( (𝜎 𝑓₁◦id⊗𝑓₂) ◦b). (11.11) In particular, the cohomological correspondenceS𝑉₁,𝑉₂(a)equals to the composition of the following cohomological correspondences:

C (𝛿_𝜎𝑊 ⊗id𝑉₁): 𝑆(𝑉e₁) −→𝑆(𝜎𝑊f^∗(𝜎𝑊e ⊗𝑉e₁)), DΓ^∗

𝐹⁻¹

(𝑊⊗𝑉1)𝑊∗ :𝑆(𝜎𝑊f^∗(𝜎𝑊e ⊗𝑉e₁)) −→𝑆( (𝜎𝑊e ⊗𝑉e₁)𝑊f^∗) C (a⁰) :𝑆( (𝜎𝑊e ⊗𝑉e₁)𝑊f^∗) −→𝑆(𝑉e₂).

Proof. Consider the following diagram

𝑆(𝑋e) 𝑆(𝜎𝑊₁⊗𝑌e⊗𝑊f₂) 𝑆(e𝑌 ⊗𝑊f₂⊗𝑊f₁)

𝑆(𝜎𝑊⁰

1⊗𝑌e⊗𝑊f⁰

2) 𝑆(e𝑌 ⊗𝑊f⁰

2⊗𝑊f⁰

1) 𝑆(e𝑌)

C (b)

C ( (𝜎 𝑓₂◦id⊗𝑓₁)◦b)

DΓ_𝐹−1

C (𝜎 𝑓₁⊗id⊗𝑓₂)

C (b⁰◦(id⊗𝑓₂⊗𝑓₁)) C (id⊗𝑓₂⊗𝑓₁)

DΓ_𝐹−1

C (b⁰)

.

(11.12)

The bent triangles on the left and right are clearly commutative by Corollary 8.0.15.

It suffices to prove that the rectangle in the middle is commutative. But this is a direct consequence of [XZ17, Lemma 6.1.13].

Let 𝑋 = 𝑉₁, 𝑌 = 1, 𝑊₁ = 𝑊⁰

1 = 𝑊^∗, 𝑊₂ = 𝜎𝑊 ⊗𝑉₁, 𝑊⁰

2 = 𝑊 ⊗ 𝑉₂. Write a⁰⁰ for the image ofa under the canonical isomorphism Hom(𝜎𝑊 ⊗𝑉₁ ⊗𝑊^∗, 𝑉₂) Hom(𝜎𝑊⊗𝑉₁, 𝑊⊗𝑉₂). Takeb=𝛿_𝜎𝑊⊗id, 𝑓₁=id, and 𝑓₂ =a⁰⁰. Then the second assertion follows from the above commutative diagram.

Lemma 11.2.3. For any𝛼 ∈ Hom_𝐺ˆ(𝑉e₁,𝑉e₂), the construction ofS𝑉₁,𝑉₂ is indepen- dent from the choice of

(1) projectiveΛ-modules𝑊 ∈Rep_Λ(𝐺ˆ_Λ),

(2) a∈Hom_𝐺ˆ(𝑉₁, 𝜎𝑊^∗⊗𝑉₂⊗𝑊), such thatΘ𝑊(a) =𝛼, (3) (𝑉₁⊗𝑉₂) ⊗𝑊𝑊^∗-acceptable integers(𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂).

Proof. The proof is completely similar to that of [XZ17, Lemma 6.2.5], and we briefly discuss it here.

We start by proving the independence of (3). Choose another quadruple of (𝑉₁⊗ 𝑉₂) ⊗𝑊𝑊^∗-acceptable integers (𝑚⁰

1, 𝑛⁰

1, 𝑚⁰

2, 𝑛⁰

2) ≥ (𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂). We have the following diagram of Hecke correspondences

Sht^loc(𝑚

0 1,𝑛⁰

1)

𝑉₁ Sht^{𝜆,loc(𝑚}

0 1,𝑛⁰

1)

𝑉₁|𝑉₂ Sht^loc(𝑚

0 2,𝑛⁰

2) 𝑉₂

Sht_𝑉^{loc(𝑚1,𝑛1)}

1 Sht_𝑉^{𝜆,loc(𝑚1,𝑛1)}

1|𝑉₂ Sht_𝑉^{loc(𝑚2,𝑛2)}

2

.

res

𝑚0 1𝑛0

1 𝑚1, 𝑛

1 res

𝑚0 1𝑛0

1 𝑚1, 𝑛

1 res

𝑚0 2𝑛0

2 𝑚2, 𝑛 2

This is the upper face of diagram (10.5). As we discussed in §10, all the vertical maps are smooth, the two squares are commutative, and the left square is Cartesian.

ThenC^loc(𝑚

0 1,𝑛⁰

1)

𝑊 (a)equals the pullback of C^{loc(𝑚1,𝑛1)}

𝑊 (a)along the vertical maps.

Next, we prove the independence of (1) and (2) simultaneously. Consider that ˆ𝐺 acts on the filtration of O𝐺 by right regular representation. Then O𝐺 is realized as an ind-object in Rep_Λ(𝐺ˆ). Let𝑋 ∈ Rep_Λ(𝐺ˆ) be a projective object and we denote by𝑋 the underline𝐸-module of𝑋 equipped with the trivial ˆ𝐺-action. Consider the following ˆ𝐺-equivariant maps

a𝑋 : 𝑋 → O𝐺 ⊗ 𝑋 , 𝑥 ↦→ a𝑋(𝑥) (𝑔) :=𝑔𝑥 ,

𝑚_𝑋 : 𝑋^∗ ⊗ 𝑋 → O𝐺, (𝑥^∗, 𝑥) ↦→ 𝑚_𝑋(𝑥^∗, 𝑥) (𝑔) :=𝑥^∗(𝑔𝑥),

where we identifyO𝐺 ⊗𝑋as the space of𝑋-valued functions on ˆ𝐺in the definition of a𝑋 and𝑚_𝑋. Taking 𝑋 =𝑊, we have the following ˆ𝐺×𝐺ˆ-module maps

𝜎𝑊^∗⊗𝑉₂⊗𝑊

a𝜎 𝑊∗

−−−−→𝑊^∗⊗𝜎O𝐺 ⊗𝑉₂⊗𝑊

𝑚𝑊

−−−→ 𝜎O𝐺 ⊗𝑉₂⊗ O𝐺.

The map ˆ𝐺×𝐺ˆ → 𝐺 𝜎,ˆ (𝑔₁, 𝑔₂) ↦→ 𝜎(𝑔₁)⁻¹𝜎(𝑔₂)𝜎 induces a natural map 𝑑_𝜎 : 𝐸[𝐺 𝜎ˆ ] → 𝜎O𝐺 ⊗ O𝐺 which intertwines the 𝜎-twisted conjugation action on 𝐸[𝐺 𝜎ˆ ]and the diagonal action of ˆ𝐺on𝜎O𝐺⊗O𝐺. For any𝛼∈Hom_𝐺ˆ(𝑉₁,O𝐺⊗𝑉₂), denote by𝛼⁰the image of𝛼under the following map

Hom_𝐺ˆ(𝑉₁,O𝐺 ⊗𝑉₂) −−→^𝑑𝜎 Hom_𝐺ˆ(𝑉₁, 𝜎O𝐺 ⊗𝑉₂⊗ O𝐺). Direct computation yields the followings

(𝑚_𝑊 ◦a𝜎𝑊^∗) ◦a⁰=𝑑_𝜎(𝛼⁰):𝑉₁→ 𝜎O𝐺 ⊗𝑉₂⊗ O𝐺, and

id𝑉₂ ⊗𝑒_𝑊 =ev₍₁,1) ◦ (𝑚_𝑊 ◦𝑎_𝑊∗) :𝑉₂⊗𝑊 ⊗𝑊^∗ →𝑉₂,

where ev₍₁,1) denotes the evaluation at (1,1) ∈ 𝐺ˆ × 𝐺ˆ. In Lemma 11.2.2, let 𝑊₁ ⊗𝑊₂ := 𝑊 ⊗𝑊^∗, 𝑊⁰

1 ⊗𝑊⁰

2 := O𝐺 ⊗ O𝐺, 𝑓₁⊗ 𝑓₂ := 𝑚_𝑊 ◦𝑎_𝑊∗, b := a⁰, and b⁰:=ev₍₁,1). Then we have

C𝑊(a) =C (id𝑉₂ ⊗ 𝑒_𝑊) ◦DΓ^∗

𝐹⁻¹

(𝑉

2⊗𝑊)𝑊∗

◦ C (a⁰)

=C (ev₍₁,1)) ◦DΓ^∗

𝐹⁻¹

(𝑉

2⊗ O𝐺)O𝐺

◦ C (𝑑_𝜎(𝛼⁰)).

We see from the last equality in the above that C𝑊(a) depends only on 𝛼 and the

lemma is thus proved.

We claim that our construction of S𝑉₁,𝑉₂ is compatible with the composition of morphisms. More precisely, we have the following lemma.

Lemma 11.2.4. For any representations𝑉₁, 𝑉₂, 𝑉₃, let𝑆₁, 𝑆₂, 𝑆₃ ∈ Rep_Λ(𝐺ˆ_Λ) be projectiveΛ-modules, and we have the following commutative diagram

HomCoh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₂) ⊗Hom

Coh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₂,𝑉e₃) Hom

Coh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₃) Hom_𝐺ˆ(𝜎 𝑆₁⊗𝑉₁⊗ 𝑆^∗

2, 𝑉₂) ⊗Hom_𝐺ˆ(𝜎 𝑆₂⊗𝑉₂⊗ 𝑆^∗

2, 𝑉₃) Hom_𝐺ˆ(𝜎 𝑆₂⊗𝜎 𝑆₁⊗𝑉₁⊗ 𝑆^∗

1⊗𝑆^∗

2, 𝑉₃) Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₂)) ⊗Corr_Shtloc(𝑆(𝑉e₂), 𝑆(𝑉e₃)) Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₃)).

𝜙

C𝑆 1⊗C𝑆

2

𝜙⁰

C𝑆 1⊗𝑆

2

𝜙⁰⁰

(11.13)

Here

• the unlabelled vertical arrows are given by the Peter-Weyl theorem

• 𝜙is the compositions of morphisms inCoh^𝐺ˆ(𝐺 𝜎ˆ )

• 𝜙⁰⁰ is the composition described in §10.2

• 𝜙⁰(a1⊗a2)is defined to be the homomorphism

𝜎 𝑆₂⊗𝜎 𝑆₁⊗𝑉₁⊗𝑆^∗

1⊗ 𝑆^∗

2

id𝜎 𝑆2⊗a₁⊗id𝑆∗

−−−−−−−−−−−→2 𝜎 𝑆₂⊗𝑉₂⊗ 𝑆^∗

2 a₂

−→𝑉₃. Proof. The lemma can be proved following the same idea in the proof of [XZ17,

Lemma 6.2.7].

We study the endomorphism ring of the unit object in P(Sht^loc_¯

𝑘

,Λ). This will be used to prove the "𝑆=𝑇" theorem for Shimura sets in §12.3.

Let 𝛿₁ denote the intersection cohomology sheaf IC₀ on Sht^loc(₀ ^𝑚,𝑛). The group theoretic description of the moduli of restricted local Shtukas (cf. [XZ17, §5.3.2]) implies that Sht^loc₀ ^(𝑚,𝑛) is perfectly smooth. Thus𝛿₁may be realized as

𝛿

𝑚,𝑛

1 := Λh(𝑚−𝑛)dim𝐺i ∈ P(Sht^{loc(𝑚,𝑛)}₀ ,Λ) for every𝑚 ≥ 𝑛. Fix a square root𝑞^1/2.

Corollary 11.2.5. (1) There is a natural isomorphism Corr_Shtloc(𝛿₁, 𝛿₁) ' H𝐺 ,𝐸

whereH𝐺 ,𝐸 denotes the Hecke algebra𝐶^∞

𝑐 (𝐺(O)\𝐺(𝐹)/𝐺(O), 𝐸). (2) We denote the map

S_O

[𝐺 𝜎ˆ /𝐺ˆ],O_{[𝐺 𝜎ˆ /𝐺ˆ]} : End

Coh^𝐺ˆ(𝐺 𝜎ˆ )(O_{[𝐺 𝜎ˆ /𝐺ˆ]}) →Corr_Shtloc(𝛿₁, 𝛿₁)

byS_Ofor simplicity. Under the isomorphism in(1), the mapS_O⊗id𝐸[𝑞^−1/2,𝑞^1/2]

coincides with the classical Satake isomorphism.

Proof. Recall the definition of the Borel-Moore homology H^BM𝑖 (𝑋) for a perfect pfp algebraic space which is defined over an algebraically closed field (cf. [XZ17, A.1.3]). Assume 𝑋₁and𝑋₂to be perfectly smooth algebraic spaces of pure dimen- sion. Let𝑋₁← 𝐶 → 𝑋₂be a correspondence. Then

Corr𝐶 (𝑋₁, 𝐸h𝑑₁i),(𝑋₂, 𝐸h𝑑₂i)

(11.14)

=Hom𝐷^𝑐

𝑏(𝐶 ,𝐸) 𝐸h𝑑₁i, 𝜔_𝐶h𝑑₂−2 dim𝑋₂

=H^BM_{2 dim𝑋}

2+𝑑₁−𝑑₂(𝐶).

Then if 2 dim𝐶 = 2 dim𝑋₂+ 𝑑₁ −𝑑₂, the cohomological correspondences from (𝑋₁, 𝐸h𝑑₁i)to(𝑋₂, 𝐸h𝑑₂i)can be identified as the set of irreducible components of 𝐶of maximal dimension.

For a perfect pfp algebraic space 𝑋 of dimension 𝑑, define 𝐼 to be the set of top- dimensional irreducible components of 𝑋. Then H^BM_𝑑 (𝐼) is the free 𝐸-module generated by the 𝑑-dimensional irreducible components of 𝑋, and thus can be identified with the space 𝐶(𝐼 , 𝐸) of 𝐸-valued functions on 𝐼. The map 𝑓 ↦→

Í

𝐶𝑖∈𝐼 𝑓(𝐶_𝑖) [𝐶_𝑖] establishes a bijection

𝐶(𝐼 , 𝐸) =H^BM_𝑑 (𝑋). (11.15) With the above preparations, we get an isomorphism

H𝐺 ,𝐸 'Corr_Shtloc(𝛿₁, 𝛿₁), (11.16) via a similar argument as for [XZ17, Proposition 5.4.4], and we finish the proof of (1).

To prove part (2), we first note that the statement holds for 𝐸 = Q^ℓ by [XZ17, Theorem 6.0.1(2)]. We sketch the proof here. Let 𝜇 be a central minuscule dominant coweight, and 𝜈 be a dominant coweight such that 𝜎(𝜈) = 𝜈. Choose (𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂) to be(𝜈+𝜇, 𝜈)-acceptable. Takea∈ Hom_𝐺ˆ(𝑉_𝜈 ⊗𝑉_𝜇 ⊗𝑉_𝜈∗, 𝑉_𝜇)to be the map induced by the evaluation mape𝜈 :𝑉_𝜈⊗𝑉_𝜈∗ →1. Consider the following diagram

pt 𝐺 𝑟_≤𝜈∗ ^Δ 𝐺 𝑟_≤𝜈∗ ×𝐺 𝑟_≤𝜈∗ ^𝜎×id 𝐺 𝑟_𝜇∗×𝐺 𝑟_≤𝜇∗ ^Δ 𝐺 𝑟_≤𝜈∗ pt. Recall the cohomological correspondences𝛿_Ic𝜈∗and𝑒_Ic𝜈∗defined in [XZ17, §A.2.3.4].

Then C^{loc(𝑚1,𝑛1)}

𝑉𝜈

(a) = 𝛿_IC𝜈∗ ◦Γ_𝜎×id^∗ ◦𝑒_Ic𝜈∗ ∈ H^BM₀ (𝐺 𝑟_𝜈∗(𝑘)), and the cohomologi- cal correspondenceC^{loc(𝑚1,𝑛1)}

𝑉𝜈

(a) can be identified with the function 𝑓 on𝐺 𝑟_𝜈∗(𝑘)

whose value at 𝑥 ∈𝐺 𝑟_𝜈∗(𝑘) is given by tr(𝜙_𝑥 | Sat(𝑉_𝜈∗)𝑥_¯). Then up to a choice of 𝑞^1/2, the map 𝑆_O,O ⊗_Qℓ id_Qℓ[𝑞1/2,𝑞^−1/2] coincides with the classical Satake isomor- phism.

Now we come back to the case 𝐸 = Z^ℓ. Write 𝑄 for Q^ℓ[𝑞^1/2, 𝑞^−1/2]. The above argument shows that

S_O ⊗𝑄 : End

Coh^𝐺ˆ(𝐺 𝜎)ˆ (O_{[𝐺 𝜎/ˆ 𝐺]ˆ} ) ⊗_Z

ℓ

𝑄→ Corr_Shtloc(𝛿₁, 𝛿₁) ⊗_Z

ℓ

𝑄

coincide with the classical Satake isomorphism. Note that EndCoh^𝐺ˆ(𝐺 𝜎)ˆ (O_{[𝐺 𝜎/ˆ ˆ}

𝐺]) ⊗_Zℓ 𝑄 'Z^ℓ[𝐺ˆ]^𝐺ˆ ⊗_Zℓ 𝑄 ,

where ˆ𝐺 acts on ˆ𝐺 by the𝜎-twisted conjugation. Considering the Satake transfer of the image ofZ^ℓ-basis of Z^ℓ[𝐺ˆ]^(𝐺ˆ) inZ^ℓ[𝐺ˆ]^(𝐺ˆ) ⊗_Zℓ𝑄, we conclude the proof of

(2).

C h a p t e r 12

COHOMOLOGICAL CORRESPONDENCES BETWEEN SHIMURA VARIETIES

In this section, we adapt the machinery developed in previous sections and apply it to the study of the cohomological correspondences between different Hodge type Shimura varieties following the idea of [XZ17].

12.1 Preliminaries

Let (𝐺 , 𝑋) be a Shimura datum and 𝐸 be its reflex field (cf. [Mil05]). Let 𝐾 ⊂ 𝐺(A^𝑓) be a (sufficiently small) open compact subgroup and denote by Sh𝐾(𝐺 , 𝑋) the corresponding Shimura variety defined over𝐸. Fix a prime 𝑝 >2 such that𝐾_𝑝 is a hyperspecial subgroup of𝐺(Q^𝑝). We write 𝐺 for the reductive group which extends 𝐺 to Z(𝑝) and such that𝐺(Z^𝑝) = 𝐾_𝑝. Choose 𝜈 to be a place of 𝐸 lying over 𝑝. We write O_{𝐸 ,(𝜈)} for the localization of O𝐸 at 𝜈. Results of Kisin [Kis10]

and Vasiu [Vas07] state that for any Hodge type Shimura datum (𝐺 , 𝑋), there is a smooth integral canonical modelS𝐾(𝐺 , 𝑋) of Sh𝐾(𝐺 , 𝑋), which is defined over O_{𝐸 ,(𝜈)}. Let 𝑘_𝜈 denote the residue field ofO𝐸 ,𝜈 and fix an algebraic closure ¯𝑘_𝜈 of 𝑘_𝜈. We denote by Sh𝜇,𝐾 :=(S𝐾(𝐺 , 𝑋) ⊗𝑘_𝜈)^pfthe perfection of the special fiber of S𝐾(𝐺 , 𝑋). The perfection of mod 𝑝fibre of Shimura varieties and moduli of local Shtukas are related by a map loc𝑝 : Sh𝜇,𝐾 → Sht^loc𝜇 . The construction of loc𝑝is via a𝐺-torsor over the crystalline cite (S𝐾 , 𝑘𝜈/O𝐸 ,𝜈)_CRISand we refer to [XZ17, §7.2.1]

for a detailed discussion. In the Siegel case, it may be understood as the perfection of the morphism sending an abelian variety to its underlying 𝑝-divisible group. We need the following result of Xiao-Zhu [XZ17, Proposition 7.2.4] for our proof of the main theorem.

Proposition 12.1.1. Let(𝑚, 𝑛) be a pair of𝜇-large integers. The morphism loc𝑝(𝑚, 𝑛) :=res𝑚,𝑛◦loc𝑝 : Sh𝜇 →Sht^loc(𝜇 ^𝑚,𝑛)

is perfectly smooth.

Étale Local Systems onSh𝜇,𝐾

Let ℓ ≠ 𝑝 be a prime number. Assume that 𝜌 : 𝐺 → 𝐺 𝐿

Q^ℓ(𝑊) is a Q^ℓ- representation of 𝐺. If 𝐾 ⊂ 𝐺(A^𝑓) is sufficiently small, we associate an étale

local system Lℓ,𝑊 on Sh𝜇,𝐾 to 𝑊 following the idea of [LZ17, §4] and [Mil90,

§III.6] as follows.

Write𝐾 =𝐾_ℓ𝐾^ℓ with𝐾_ℓ ⊂ 𝐺(Q^ℓ)and𝐾^ℓ ⊂ 𝐺(A^ℓ𝑓). The representation𝜌restricts to a representation

𝜌_𝐾

ℓ : 𝐾(Q^ℓ) →𝐺(Q^ℓ) →𝐺 𝐿(𝑊

Q^ℓ). Note that𝐾(Q^ℓ)is compact, and there exists a latticeΛ𝑊 ,ℓ ⊂ 𝑊

Q^ℓ fixed by𝐾(Q^ℓ). Now we vary the levels atℓ. Define

𝐾⁽

𝑛)

ℓ :=𝐾_ℓ∩𝜌⁻¹

𝐾(Q^ℓ)({𝑔 ∈𝐺 𝐿(Λ𝑊 ,ℓ) | 𝑔≡ 1 mod ℓ^𝑛}).

Then we get a system of open neighborhoods of 1 ∈ 𝐺(Q^ℓ). For each 𝑛, the construction of𝐾^(𝑛)

ℓ gives rise to a representation 𝜌^𝑛

𝐾ℓ :𝐾_ℓ/𝐾^(𝑛)

ℓ →𝐺 𝐿(Λ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓ). The natural projection map Sh

𝜇,𝐾⁽

𝑛)

ℓ 𝐾^ℓ → Sh𝜇,𝐾ℓ𝐾^ℓ is a finite étale cover with the group of deck transformations being 𝐾_ℓ/𝐾⁽

𝑛)

ℓ . Then the trivial étaleZ/ℓ^𝑛Z-local system Sh_𝜇,𝐾^(𝑛)

ℓ

𝐾^ℓ ×Λ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓ on Sh_𝜇,𝐾^(𝑛)

ℓ

𝐾^ℓ gives rise to the étaleZ/ℓ^𝑛Z-local system

L𝑊 ,ℓ,𝑛 :=Sh_𝜇,𝐾^(𝑛)

ℓ

𝐾^ℓ ×^{𝐾ℓ/𝐾ℓ(𝑛)} Λ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓ. Let

L𝑊 ,Z^ℓ :=lim

←−−

𝑛

L𝑊 ,ℓ,𝑛. (12.1)

This is an étaleZ^ℓ-local system on Sh𝜇,𝐾. It can be checked thatL𝑊 ,Q^ℓ :=L𝑊 ,Z^ℓ⊗Q is an étaleQ^ℓ-local system on Sh𝜇,𝐾 which is independent of the choice ofΛℓ. 12.2 Main Theorem

Let(𝐺₁, 𝑋₁) and(𝐺₂, 𝑋₂)be two Hodge type Shimura data (cf. [Mil05]) equipped with an isomorphism 𝜃 : 𝐺_1,

A^𝑓 ' 𝐺_2,

A^𝑓. Let {𝜇_𝑖} denote the conjugacy class of Hodge cocharacters determined by 𝑋_𝑖 and consider them as dominant characters of ˆ𝑇. In particular, 𝜇₁ and 𝜇₂ are both minuscule. Then [XZ17, Corollary 2.1.5]

implies that there is a canonical inner twistΨ_R:𝐺₁→ 𝐺₂overC. Recall notations in §1.3. We define𝜇_𝑖,ad to be the composition of𝜇_𝑖with the quotient𝐺 → 𝐺_adand consider it as a character of ˆ𝑇_sc. We assume that

𝜇_1,ad |

𝑍(𝐺ˆ

ΓQ

sc )= 𝜇_2,ad |

𝑍(𝐺ˆ

ΓQ sc ) .

It follows from [XZ17, Corollary 2.1.6] thatΨ_Rcomes from a unique global inner twistΨ : 𝐺

1 ¯Q → 𝐺

2 ¯Q such thatΨ = Int(ℎ) ◦𝜃, for some 𝜃 : 𝐺_1,

A^𝑓 ' 𝐺_2,

A^𝑓 and ℎ∈𝐺_2,ad(A¯^𝑓).

We assume that𝐾_𝑖 ⊂ 𝐺(A^𝑓) to be sufficiently small such that𝜃 𝐾₁=𝐾₂. Choose a prime 𝑝such that𝐾_{1, 𝑝} (and therefore 𝐾_{2, 𝑝}) is hyperspecial. Let𝐺_𝑖 be the integral model of𝐺_𝑖,

Q^𝑝overZ^𝑝determined by𝐾_{𝑖, 𝑝}. Then𝐺₁ '𝐺₂, and we can thus identify their Langlands dual groups (𝐺 ,ˆ 𝐵,ˆ 𝑇ˆ). Choose an isomorphism 𝜄 : C ' Q¯^𝑝. Let 𝜈 | 𝑝 be a place of the compositum of reflex fields of (𝐺_𝑖, 𝑋_𝑖) determined by our choice of isomorphism𝜄. We write Sh𝜇𝑖for the mod 𝑝fibre of the canonical integral model of Sh𝐾𝑖(𝐺_𝑖, 𝑋_𝑖) base change to𝑘_𝜈. We make the following assumption

𝜇₁ |

𝑍(𝐺ˆ

ΓQ𝑝)= 𝜇₂ |

𝑍(𝐺ˆ

ΓQ𝑝) . (12.2)

The assumption guarantees the existence of the ind-scheme Sh𝜇₁|𝜇₂ which fits into the following commutative diagram

Sh𝜇₁,𝐾₁ Sh𝜇₁|𝜇₂ Sh𝜇₂,𝐾₂

Sht^loc𝜇₁ Sht^loc_𝜇

1|𝜇₂ Sht^loc𝜇₂

←− ℎ𝜇

1

loc𝑝

−

→ ℎ𝜇

2

loc𝑝

←− ℎ^loc

𝜇1

−

→ ℎ^loc

𝜇2

, (12.3)

and makes both squares to be Cartesian.

Remark 12.2.1. In the case that(𝐺₁, 𝑋₁)= (𝐺₂, 𝑋₂),Sh𝜇₁|𝜇₂is the perfection of the mod p fibre of a natural integral model of some Hecke correspondence. If(𝐺₁, 𝑋₁) ≠ (𝐺₂, 𝑋₂), thenSh𝜇₁|𝜇₂can be regarded as “exotic Hecke correspondences” between mod p fibres of different Shimura varieties. We refer to [XZ17, §7.3.3, §7.3.4] for a detailed discussion.

Let (𝐺_𝑖, 𝑋_𝑖) 𝑖 = 1,2,3 be three Hodge type Shimura data, together with the iso- morphisms𝜃_{𝑖, 𝑗} :𝐺_𝑖,

A^𝑓 '𝐺_{𝑗 ,}

A^𝑓 satisfying the natural cocycle condition. Choose a common level 𝐾 using the isomorphism𝜃_{𝑖, 𝑗}. Let 𝑝 be an unramified prime, such that the assumption(12.2)holds for each pair of( (𝐺_𝑖, 𝑋_𝑖),(𝐺_𝑗, 𝑋_𝑗)). Choose a half Tate twistQ^ℓ(1/2).

Let 𝑉_𝑖 := 𝑉_𝜇

𝑖 be the highest weight representation of ˆ𝐺

Q^ℓ of highest weight 𝜇_𝑖. Write 𝑉e_𝑖 ∈ Coh^𝐺ˆQℓ(𝐺ˆ

Q^ℓ𝜎) for the vector bundle associated to 𝑉_𝑖 analogous to

§11.4. Recall from§12.1 that, to each representation𝑊 of𝐺

Q^ℓ, we can attach the étale local systemL𝑊 ,Q^ℓ on Sh𝜇𝑖. Let𝑑_𝑖 =h2𝜌, 𝜇_𝑖i=dim Sh𝐾(𝐺_𝑖, 𝑋_𝑖). Denote the global section of the structure sheaf on the quotient stack [𝐺 𝜎ˆ /𝐺ˆ] by J, and the prime-to-𝑝 Hecke algebra byH^𝑝.

Theorem 12.2.2. There exists a map Spc : Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) →HomH^𝑝⊗J(H^∗𝑐(Sh𝜇₁,L𝑊 ,Q^ℓh𝑑₁i),H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i), (12.4)

which is compatible with compositions on the source and target.

Proof. Choose a latticeΛ𝑖 ∈Rep_Z

ℓ(𝐺ˆ

Z^ℓ)in𝑉_𝑖. We denote byΛe𝑖 ∈Coh^𝐺ˆZℓ(𝐺ˆ

Z^ℓ𝜎) the coherent sheaf which corresponds toΛ𝑖 as in §11.1. Then

HomCoh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) 'Hom_𝐺ˆ

Qℓ

(𝑉₁, 𝑉₂⊗Q^ℓ[𝐺ˆ]) (12.5) 'Hom_𝐺ˆ

Qℓ

(Λ₁⊗_Z

ℓQ^ℓ,(Λ₂⊗_Z

ℓ Z^ℓ[𝐺ˆ]) ⊗_Z

ℓQ^ℓ) 'Hom𝐺ˆ

Zℓ

(Λ₁,Λ₂⊗_Zℓ Z^ℓ[𝐺ˆ]) ⊗_Zℓ Q^ℓ 'Hom

Coh^𝐺ˆZℓ(𝐺ˆ

Zℓ𝜎)(fΛ₁,Λf₂) ⊗_ZℓQ^ℓ. By Theorem 11.2.1, we get a map

S_Λ1,Λ₂ : Hom

Coh^𝐺ˆZℓ(𝐺ˆ

Zℓ𝜎)(Λf₁,Λf₂) →Corr_Shtloc(𝑆(Λf₁), 𝑆(Λf₂)). (12.6) Combining(12.5)with(12.6), we get the following map

HomCoh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) →Corr_Shtloc(𝑆(Λf₁), 𝑆(Λf₂)) ⊗_Zℓ Q^ℓ. (12.7) Choose a dominant coweight𝜈and a quadruple (𝑚₁, 𝑛₁, 𝑚₂, 𝑛₂)that is (𝜇₁+𝜈, 𝜈)- acceptable and(𝜇₂+𝜈, 𝜈)-acceptable. We have the following diagram

Sh𝜇₁ Sh^𝜈_𝜇

1|𝜇₂ Sh𝜇₂

Sht^loc𝜇₁ Sht^𝜈,loc

𝜇₁|𝜇₂ Sht^loc𝜇₂

Sht^loc(𝜇₁ ^𝑚1,𝑛1) Sht^𝜈,_𝜇^{loc(𝑚1,𝑛1)}

1|𝜇₂ Sht^loc(𝜇₂ ^𝑚2,𝑛2)

←− ℎ_𝜇

1

loc^𝑝 loc^𝜈𝑝

−

→ ℎ_𝜇

2

loc^𝑝

←− ℎ^loc

𝜇1

res^𝑚₁^{, 𝑛}₁

−

→ ℎ^loc

𝜇2

res^𝜈𝑚1, 𝑛1 res^𝑚₂^{, 𝑛}₂

←− ℎ^loc(𝑚1

, 𝑛1) 𝜇1

−

→ ℎ^loc(𝑚2

, 𝑛2) 𝜇2

, (12.8)

where

• all squares are commutative (discussions on diagram (10.5) and diagram (12.3),

• except for the square at the down right corner, and the other three squares are Cartesian (discussions on diagram(12.3)and diagram (12.5),

• the morphism←− ℎ_𝜇

1 is perfectly proper ([XZ17, Lemma 5.2.12]),

• the morphisms loc𝑝(𝑚_𝑖, 𝑛_𝑖) are perfectly smooth (Proposition 12.1.1).

Then the morphism loc^𝜈𝑝(𝑚₁, 𝑛₁) := res^𝜈𝑚₁,𝑛₁ ◦loc^𝜈𝑝 is also perfectly proper. Thus we can pullback the cohomological correspondences (cf. [XZ17, A.2.11)]) on the right hand side of (12.6) along loc^𝜈𝑝(𝑚₁, 𝑛₁)to obtain a map

loc^𝜈𝑝(𝑚₁, 𝑛₁)^★: Corr_Shtloc(𝑆(Λf₁), 𝑆(Λf₂)) → Corr_Sh^𝜈

𝜇|𝜇(loc𝑝(𝑚₁, 𝑛₁)^★𝑆(fΛ₁),loc𝑝(𝑚₂, 𝑛₂)^★(𝑆(Λf₂)). Note that 𝜇_𝑖 are minuscule, then the★-pullback of𝑆(Λe𝑖)along loc𝑝(𝑚_𝑖, 𝑛_𝑖)equals

Z^ℓh𝑑_𝑖i. Next, we construct a natural map ℭ𝑊 : Corr_Sh^𝜈

𝜇1|𝜇 2

(Sh𝜇₁,Z^ℓh𝑑₁i),(Sh𝜇₂,Z^ℓh𝑑₂i)

→Corr_Sh^𝜈

𝜇1|𝜇 2

(Sh𝜇₁,L𝑊 ,Z^ℓh𝑑₁i),(Sh𝜇₂,L𝑊 ,Z^ℓh𝑑₂i) . (12.9)

For each𝑛 ∈Z⁺, we note that there exists an ind-scheme Sh^(𝑛)

𝜇₁|𝜇₂ which fits into the following commutative diagram such that both squares are Cartesian

Sh𝜇₁,𝐾^(𝑛)

ℓ 𝐾^ℓ Sh^𝜈,(𝑛)

𝜇₁|𝜇₂ Sh

𝜇₂,𝐾^(𝑛)

ℓ 𝐾^ℓ

Sh𝜇₁ Sh^𝜈_𝜇

1|𝜇₂ Sh𝜇₂.

←− ℎ_𝜇^(𝑛)

1

𝑝^𝑛

1

−

→ ℎ^(𝑛)_𝜇

2

𝑝^𝑛 𝑝^𝑛

2

←− ℎ𝜇

1

−

→ ℎ𝜇

2

Here the three vertical maps are the natural quotients by the finite group𝐾_ℓ/𝐾^𝑛

ℓ and are thus étale.

Let (𝑓_𝑛)𝑛 : (←− ℎ_𝜇

1)^∗(Z/ℓ^𝑛Zh𝑑₁i)𝑛 → (→− ℎ_𝜇

2)^!(Z/ℓ^𝑛Zh𝑑₂i)𝑛 be a cohomological cor- respondence in Corr_Sh^𝜈

𝜇1|𝜇2

(Sh𝜇₁,Z^ℓh𝑑₁i),(Sh𝜇₂,Z^ℓh𝑑₂i)

. For each 𝑛 ∈ Z⁺, the shifted pullback (cf. [XZ17, A.2.12]) of 𝑓_𝑛gives rise to a cohomological correspon- dence

˜ 𝑓_𝑛: (←−

ℎ⁽

𝑛)

𝜇₁ )^∗(Z/ℓ^𝑛Zh𝑑₁i) → (→− ℎ⁽

𝑛)

𝜇₂)^!(Z/ℓ^𝑛Zh𝑑₂i)