Key Theorem

Chapter XI: Key Theorem for Constructing the Jacquet-Langlands Transfer . 64

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)

in Corr_Sh^{𝜈 ,(𝑛)}

𝜇1|𝜇2

(Sh_𝜇

1,𝐾⁽

𝑛) ℓ

𝐾^ℓ

,Z/ℓ^𝑛Zh𝑑₁i),(Sh_𝜇

2,𝐾⁽

𝑛) ℓ

𝐾^ℓ

,Z/ℓ^𝑛Zh𝑑₂i)

. For any repre- sentation𝑊of𝐺

Q^ℓ, recall theZ/ℓ^𝑛ZmoduleΛ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓconstructed in §12.2. The cohomological correspondence ˜𝑓_𝑛gives rise to a cohomological correspondence

˜

𝑔_𝑛∈Corr

Sh^{𝜈 ,(𝑛)}_𝜇

1|𝜇2

(Sh𝜇₁,𝐾^(𝑛)

ℓ

𝐾^ℓ ×Λ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓh𝑑₁i,Sh

𝜇₂,𝐾^(𝑛)

ℓ

𝐾^ℓ ×Λ𝑊 ,ℓ/ℓ^𝑛Λ𝑊 ,ℓh𝑑₂i). In addition, the cohomological correspondence ˜𝑓_𝑛 is𝐾_ℓ/𝐾^(𝑛)

ℓ -equivariant. Then it follows that the cohomological correspondence ˜𝑔_𝑛 is also𝐾_ℓ/𝐾⁽

𝑛)

ℓ -equivariant and descends to a cohomological correspondence

𝑔_𝑛 ∈Corr_Sh^𝜈

𝜇1|𝜇2

( (Sh𝜇₁,L𝑊 ,ℓ,𝑛h𝑑₁i),(Sh𝜇₂,L𝑊 ,ℓ,𝑛h𝑑₂i)). Definingℭ𝑊( (𝑓_𝑛)𝑛) :=(𝑔_𝑛)𝑛completes the construction ofℭ𝑊. Compose the maps we previously construct,

CorrSht^{𝜈 ,}_𝜇^{loc(𝑚1, 𝑛1)}

1|𝜇2

(Sht^loc𝜇₁^{(𝑚1,𝑛1)}, 𝑆(fΛ₁)^{loc(𝑚1,𝑛1)}),(Sht^loc𝜇₁^{(𝑚2,𝑛2)}, 𝑆(Λf₂)^{loc(𝑚2,𝑛2)}) (12.10)

loc^𝜈𝑝(𝑚₁,𝑛₁)^★

−−−−−−−−−−→Corr_Sh^𝜈

𝜇1|𝜇2

( (Sh𝜇₁,Z^ℓh𝑑₁i),(Sh𝜇₂,Z^ℓh𝑑₂i))

ℭ𝑊

−−−→Corr_Sh^𝜈

𝜇1|𝜇 2

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

H^∗𝑐

−−→HomH^𝑝(H^∗𝑐(Sh𝜇₁,L𝑊 ,Z^ℓh𝑑₁i),H^∗𝑐(Sh𝜇₂,L𝑊 ,Z^ℓh𝑑₂i)).

We justify that the composition of maps in(12.11)factors through Corr_Shtloc(𝑆(𝑉e₁), 𝑆(𝑉e₂)).

Note that the proof of Lemma 11.2.3.(3) and the definition of loc^𝜈𝑝(𝑚₁, 𝑛₁)^★imply that for a quadruple (𝑚⁰

1, 𝑛⁰

1, 𝑚⁰

2, 𝑛⁰

2) of (𝜇₁ + 𝜈, 𝜈)-acceptable and (𝜇₂ + 𝜈, 𝜈)- acceptable integers, the functor loc^𝜈𝑝(𝑚₁, 𝑛₁)^★commutes with the connecting mor- phism in(10.12)(with𝜇₁,𝜇₂,𝜆fixed). Let𝜈 ≤ 𝜈⁰and(𝑚⁰

1, 𝑛⁰

1, 𝑚⁰

2, 𝑛⁰

2)be a quadruple of non-negative integers satisfying appropriate acceptance conditions. The proper smooth base change shows that loc^𝜈𝑝(𝑚⁰

1, 𝑛⁰

1)^★commutes with enlarging𝜈 to𝜈⁰. In addition, the proper smooth base change together with the construction ofℭ𝑊 show that the following diagram commutes:

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))

Corr_Sh^𝜈0

𝜇1|𝜇2

( (Sh𝜇₁,Z^ℓh𝑑₁i),(Sh𝜇₂,Z^ℓh𝑑₂i)) Corr_Sh^𝜈0

𝜇1|𝜇2

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

ℭ𝑊

𝑖_∗ 𝑖_∗

ℭ𝑊

Thus the map ℭ𝑊 is compatible with the enlargement of 𝜈. Finally, by [XZ17, Lemma A.2.8], the composition of maps H^∗𝑐◦ℭ𝑊 commutes with enlarging𝜈to𝜈⁰. We complete the proof of the statement at the beginning of this paragraph.

Composing(12.7)with(12.11), we get a canonical map HomCoh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) →HomH^𝑝(H^∗𝑐(Sh𝜇₁,L𝑊 ,Q^ℓh𝑑₁i),H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i)). (12.11) The fact that (12.9) is compatible with the compositions of the source and target can be proved in an analogous way as [XZ17, Lemma 7.3.12], and we omit the details. Then the action of J naturally translates to the right hand side of (12.9) and upgrades it to our desired map

Spc : Hom

Coh^𝐺ˆ(𝐺 𝜎ˆ )(𝑉e₁,𝑉e₂) →HomH^𝑝⊗J(H^∗𝑐(Sh𝜇₁,L𝑊 ,Q^ℓh𝑑₁i),H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i)). As discussed in loc.cit, the action of J on H^∗𝑐(Sh𝜇_𝑖,L𝑊 ,Q^ℓh𝑑_𝑖i) is expected to coincide with the usual Hecke algebra action, which may be understood as the Shimura variety analogue of V. Lafforgue’s "𝑆 = 𝑇" theorem (cf. [Laf18]). We prove this in the case of Shimura sets.

Proposition 12.2.3. LetSh𝐾(𝐺 , 𝑋) be a zero-dimensional Shimura variety. Then the action ofJonH^∗𝑐(Sh𝜇𝑖,L𝑊 ,Q^ℓh𝑑_𝑖i)is given by the classical Satake isomorphism.

Proof. Let 𝑓 ∈ J. Since the Shimura variety we consider is zero-dimensional, it follows from [XZ17, A.2.3(5)] that the cohomological correspondence loc^★𝑝(S_O(𝑓)) can be identified with a Z^ℓ-valued function on Sh𝜇|𝜇. By our construction of the map Spc, this function is given by the pullback of a function 𝑓⁰ on Sht^loc_𝜇|𝜇 = 𝐺(Z^𝑝)\𝐺(Q^𝑝)/𝐺(Z^𝑝). Corollary 11.2.5(2) thus implies that the function 𝑓⁰ is exactly the function S_O(𝑓) ∈ 𝐻

𝐺 ,𝐸[𝑝^−1/2, 𝑝^1/2] which is the image of 𝑓 under the classical Satake isomorphism.

For any 𝑛 ∈ Z⁺, take𝑊 = Z^𝑛ℓ. Recall our construction ofℭ𝑊, the cohomological correspondence ˜𝑓_𝑛is given by a finite direct sum of the function loc^★𝑝(S_O(𝑓))since the Shimura variety we consider is a set of discrete points. Then the action of Spc(𝑓) on H^∗𝑐(Sh𝜇_𝑖,L𝑊 ,Q^ℓh𝑑_𝑖i) is given by the classical Satake isomorphism. For general𝑊, we take resolutions of it as in (12.10), and the statement follows from

the case𝑊 =Z^𝑛ℓ.

12.3 Non-Vanishing of the Geometric Jacquet-Langlands Transfer In Theorem 12.2.2, we constructed the geometric Jacquet-Langlands transfer Spc : Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) →HomH^𝑝⊗J(H^∗𝑐(Sh𝜇₁,L𝑊 ,Q^ℓh𝑑₁i),H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i). It is natural to ask when this transfer map is nonzero. We discuss this issue in this

section. The idea essentially follows from the discussion in [XZ17, §7.4], and we briefly sketch it here.

Assume that Sh𝜇₁,𝐾₁(𝐺₁, 𝑋₁) is a zero dimensional Shimura variety. The Jacquet- Langlands transfer map induces the following map

JL₁,2(a): H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ) →H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i), fora ∈Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎𝑝)(𝑉e₁,𝑉e₂). Leta⁰ ∈Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎𝑝)(𝑉e₂,𝑉e₁)be the mor- phism such that the induced map

JL₂,1(a⁰) : H^∗𝑐(Sh𝜇₂,L𝑊 ,Q^ℓh𝑑₂i) →H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ)

is dual to JL₁,2(a) when viewing it as a cohomological correspondence (cf. [XZ17,

§A.2.18]).

The composition map JL₂,1(a⁰) ◦JL₁,2(a) gives rise to an endomorphism of𝑉g_𝜇

1 ∈ Coh^𝐺ˆQℓ(𝐺ˆ

Q^ℓ𝜎_𝑝). By [XZ17, Theorem 1.4.1], the hom spaces Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₁,𝑉e₂) and Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎)(𝑉e₂,𝑉e₁)are both finite projective J-modules. Thus it makes sense to consider the determinant of the pairing

HomCoh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎𝑝)(𝑉e₂,𝑉e₁) ⊗Hom

Coh^𝐺ˆQℓ(𝐺ˆ

Qℓ𝜎𝑝)(𝑉e₁,𝑉e₂) → J. (12.12) In particular, this determinant can be regarded as a regular function on the stack [𝐺ˆ

Q^ℓ𝜎_𝑝/𝐺ˆ

Q^ℓ]; for a detailed discussion on the pairing(12.12), see [XZ19].

By Theorem 6.1.2 inloc.cit, we conclude the following result:

Theorem 12.3.1. Let𝜋_𝑓 be an irreducibleH𝐾-module, and let

H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ) [𝜋_𝑓] :=HomH𝐾(𝜋_𝑓,H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ)) ⊗𝜋_𝑓 denote the𝜋_𝑓-isotypical component. Then, the map

JL₁,2(a) : H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ) →H^𝑑𝑐(Sh𝜇₂,L𝑊 ,Q^ℓ)

restricted to H⁰𝑐(Sh𝜇₁,L𝑊 ,Q^ℓ) [𝜋_𝑓] is injective if the Satake parameters of 𝜋_𝑓 is general with respect to𝑉_𝜇

2 in the sense of [XZ17].

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