Ann. Scient. Éc. Norm. Sup.

4

e

série, t. 48, 2015, p. 409 à 451

THE GEOMETRIC SATAKE CORRESPONDENCE

FOR RAMIFIED GROUPS

X ZHU

A. – We prove the geometric Satake isomorphism for a reductive group deﬁned over

F = k((t)), and split over a tamely ramiﬁed extension. As an application, we give a description of the

nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.

R. – Nous démontrons l’isomorphisme de Satake géométrique pour un groupe réductif dé-

ﬁni sur F = k((t)) et déployé sur une extension modérément ramiﬁée. Nous donnons comme applica-

tion une description des cycles évanescents sur certaines variétés de Shimura via les modèles locaux de

Rapoport-Zink-Pappas.

Introduction

The Satake isomorphism (for unramiﬁed groups) is the starting point of the Langlands

duality. Let us ﬁrst recall its statement. Let F be a non-Archimedean local ﬁeld with ring

of integers O and residue ﬁeld k, and let G be a connected unramiﬁed reductive group

over F (e.g., G = GL

n

). Let A ⊂ G be a maximal split torus of G, and W

0

be the

Weyl group of (G, A). Let K be a hyperspecial subgroup of G(F ) containing A( O) (e.g.,

K = GL

n

( O)). Then the classical Satake isomorphism describes the spherical Hecke algebra

Sph = C

c

(K \ G(F )/K), the algebra of compactly supported bi-K-invariant functions

on G(F ) under convolution. Namely, there is an isomorphism of algebras

Sph ' C[X

•

(A)]

W

0

,

where X

•

(A) is the coweight lattice of A, and C[X

•

(A)]

W

0

denotes the W

0

-invariants of the

group algebra of X

•

(A).

If F has positive characteristic p > 0, then the classical Satake correspondence has a vast

enhancement. For simplicity, let us assume that G is split over F (for the general case, see

Theorem A.12). Let us write G = H ⊗

k

F for some split group H over k so that K = H( O).

Let Gr

H

= H(F )/H( O) be the aﬃne Grassmannian of H. Choose a prime diﬀerent from p,

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

0012-9593/02/© 2015 Société Mathématique de France. Tous droits réservés

410 X. ZHU

and let Sat

H

be the category of (K ⊗

¯

k)-equivariant perverse sheaves with Q

`

-coeﬃcients

on Gr

H

⊗

¯

k. Then this is a Tannakian category and there is an equivalence

Sat

H

' Rep(G

∨

Q

`

),

where G

∨

Q

`

is the dual group of G and Rep(G

∨

Q

`

) is the tensor category of algebraic represen-

tations of G

∨

Q

`

(cf. [10, 19]).

There is also a version of Satake isomorphism for an arbitrary reductive group over F ,

as recently proved by Haines and Rostami (cf. [12])

(1)

. Namely, let B(G) be the Bruhat-Tits

building of G and v ∈ B(G) be a special vertex. Let K

v

⊂ G(F ) be the special parahoric

subgroup of G(F ) corresponding to v. Let A be a maximal split F -torus of G such that

K

v

⊃ A( O), let M be the centralizer of A in G and W

0

= N

G

(A)/M be the Weyl group

as before. Let M

1

be the unique parahoric subgroup of M(F ), and Λ

M

= M(F )/M

1

, which

is a ﬁnitely generated Abelian group. Then

(0.1) C

c

(K

v

\G(F )/K

v

) ' C[Λ

M

]

W

0

.

More explicitly, suppose that G is quasi-split so that M = T is a maximal torus. Then

Λ

M

= (X

•

(T )

I

)

σ

,

where I is the inertial group and σ is the Frobenius, and (X

•

(T )

I

)

σ

denotes the σ-invariants

of the I-coinvariants of the group X

•

(T ).

The goal of this paper is to provide a geometric version of the above isomorphism when

F has positive characteristic p and the group G is quasi-split and splits over a tamely ramiﬁed

extension. More precisely, let k be an algebraically closed ﬁeld and let 6= char k be a prime.

Let G be a group over the local ﬁeld F = k((t)) (so that G is quasi-split automatically), which

is split over a tamely ramiﬁed extension. That is, there is a ﬁnite extension

˜

F /F such that

G

˜

F

is split and char k - [

˜

F : F ]. Let v ∈ B(G) be a special vertex in the building of G

and let G

v

be the parahoric group scheme over O = k[[t]] (in the sense of Bruhat-Tits),

determined by v. We write LG for the loop space of G and K

v

= L

+

G

v

for the jet space

of G

v

. By deﬁnition, for any k-algebra R, LG(R) = G(R

ˆ

⊗

k

F ) and K

v

(R) = G

v

(R

ˆ

⊗

k

O).

Let

F

v

= LG/K

v

be the (twisted) aﬃne ﬂag variety

(2)

, which is an ind-scheme over k. Let P

v

= P

K

v

( F

v

)

be the category of K

v

-equivariant perverse sheaf on F

v

, with coeﬃcients in Q

`

. Let H be a

split Chevalley group over Z such that G ⊗

F

F

s

' H ⊗ F

s

, where F

s

is a (ﬁxed) separable

closure of F . Then there is a natural action of I = Gal(F

s

/F ) on H

∨

:= H

∨

Q

`

(preserving a

ﬁxed pinning).

T 0.1. – The category P

v

has a natural tensor structure. In addition, as tensor

categories, there is an equivalence

R S : Rep((H

∨

)

I

) ' P

v

,

(1)

There is another version, known earlier, as in [6].

(2)

One would call F `

v

the aﬃne Grassmannian of G. However, we reserve the name “aﬃne Grassmannian” of G

for another object, as deﬁned in Deﬁnition A.2.

4

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SÉRIE – TOME 48 – 2015 – N

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THE GEOMETRIC SATAKE CORRESPONDENCE FOR RAMIFIED GROUPS 411

such that H

∗

◦ R S is isomorphic to the forgetful functor, where H

∗

is the hypercohomology

functor.

This theorem can be regarded as a categoriﬁcation of (0.1) in the case when k is alge-

braically closed and the group splits over a tamely ramiﬁed extension of k((t)). For the

description of (H

∨

)

I

when H is absolutely simple and simply-connected, see § 4.

Let us point out the following remarkable facts when the group is ramiﬁed. First, the

group (H

∨

)

I

is not necessarily connected as is shown in Remark (4.4). Second, it is well-

known that if G is unramiﬁed over F , then all the hyperspecial subgroups of G are conjugate

under G

ad

(F ) ([27, §2.5]), where G

ad

is the adjoint group of G. However, this is no longer

true for special parahoric of G if G is ramiﬁed. An example is given by the odd ramiﬁed

unitary similitude group GU

2m+1

. There are essentially two types of special parahorics

of GU

2m+1

, as given in (7.1). One of them has reductive quotient GO

2m+1

(denoted by G

v

0

),

and the other has reductive quotient GSp

2m

(denoted G

v

1

). Accordingly, the geometry

of the corresponding ﬂag varieties F

v

0

and F

v

1

are very diﬀerent, while P

v

0

' P

v

1

.

Indeed, their Schubert varieties (i.e., closures of K

v

i

-orbits) are both parameterized by

irreducible representations of GO

2m+1

. Let F

v

0

¯µ

2m,1

(resp. F

v

1

¯µ

2m,1

) be the Schubert

variety in F

v

0

(resp. F

v

1

) parameterized by the standard representation of GO

2m+1

.

Then it is shown in [31] that F

v

0

¯µ

2m,1

is not Gorenstein, while in [25] that F

v

1

¯µ

2m,1

is

smooth. On the other hand, the intersection cohomology of both varieties gives the standard

representation of GO

2m+1

. In addition, the stalk cohomologies of both sheaves are the

“same”. See Theorem 0.3 below.

R 0.1 . – Instead of considering a special parahoric K

v

of LG, one can begin with

the special maximal “compact” K

0

v

, (i.e., K

0

v

= L

+

G

0

v

, where G

0

v

is the stabilizer group

scheme of v as constructed by Bruhat-Tits), and consider the category of K

0

v

-equivariant

perverse sheaves on LG/K

0

v

. However, from a geometric point of view, this is less natural

since K

0

v

is not necessarily connected and the category of K

0

v

-equivariant perverse sheaves

is complicated. In fact, we do not know how to relate this category to the Langlands dual

group yet. In addition, when we discuss the Langlands parameters in Section 6, it is also more

“correct” to consider K

v

rather than K

0

v

.

The idea of the proof of the theorem is as follows. Using Gaitsgory’s nearby cycle functor

construction as in [8, 31], we construct a functor

Z : Sat

H

→ P

v

,

which is a central functor in the sense of [2]. By standard arguments in the theory of

Tannakian equivalence and the Mirkovic-Vilonen theorem, this already implies that

P

v

' Rep(

˜

G

∨

) for certain closed subgroup

˜

G

∨

⊂ H

∨

. Then we identify

˜

G

∨

with (H

∨

)

I

using the parametrization of the K

v

-orbits on F

v

.

R 0.2 . – (i) We believe that the same argument (maybe with small modiﬁcations)

should work for groups split over wild ramiﬁed extensions. However, we have not checked

this carefully.

(ii) Our approach is more inspired by [8] rather than [19]. However, it would be interesting

to know whether there is the similar theory of MV-cycles in the ramiﬁed case. It seems that

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

412 X. ZHU

the geometry of semi-inﬁnite orbits on F

v

is similar to the unramiﬁed case, except when

F

v

corresponds to one type of special parahorics for odd unitary groups (the one denoted

by G

v

1

as above). We do not know what happens in this last case.

(iii) Our theorem and the method also share the similar features with the results of Nadler

on geometric Satake for real groups [20].

When the group G is quasi-split over the non-Archimedean local ﬁeld F = F

q

((t)) and

v is a special vertex of B(G, F ), the aﬃne ﬂag variety F

v

is deﬁned over F

q

. We assume that

v is very special, i.e., it remains special when we base change G to F

q

((t)) (see § 6 for more dis-

cussions of this notion). Then we can consider the category of K

v

-equivariant semi-simple

perverse sheaves on F

v

, pure of weight zero, and denote it by P

0

v

. On the other hand, let I

be the inertial group of F and σ be the Frobenius of Gal(k/F

q

), where k = F

q

. Then the ac-

tion of Gal(F

s

/F ) on H

∨

(via the pinned automorphisms) induces a canonical action of σ

on (H

∨

)

I

, denoted by act

alg

. One can form the semidirect product (H

∨

)

I

o

act

alg

Gal(k/F

q

),

which can be regarded as a proalgebraic group over Q

`

, and consider the category of alge-

braic representations of (H

∨

)

I

o

act

alg

Gal(k/F

q

), denoted by Rep((H

∨

)

I

o

act

alg

Gal(k/F

q

)).

T 0.2. – In this case, the functor R S in Theorem 0.1 can be extended to an

equivalence

R S : Rep((H

∨

)

I

o

act

alg

Gal(k/F

q

)) ' P

0

v

,

whose composition with H

∗

is isomorphic to the forgetful functor.

Let us mention that under this equivalence, the restriction to Gal(k/F

q

) of the represen-

tation (H

∨

)

I

o

act

alg

Gal(k/F

q

) on H

∗

( F ) for F ∈ P

0

v

is NOT the natural Galois action

of Gal(k/F

q

) on H

∗

( F ). However, their diﬀerence can be described explicitly. See Section 4

and appendix for more details.

Our next result is to use the ramiﬁed geometric Satake isomorphism to obtain the stalk

cohomology of sheaves on F

v

(i.e., the corresponding Lusztig-Kato polynomial in ramiﬁed

case), following an idea of Ginzburg (cf. [10]). Let us state the result precisely. The centralizer

of A in our case is a maximal torus of G, denoted by T . Then the K-orbits on F

v

are labeled

by X

•

(T )

I

/W

0

, W

0

-orbits of the coinvariants of the cocharacter group of T . For ¯µ ∈ X

•

(T )

I

,

let

˚

F

v ¯µ

be the corresponding orbit. For a representation V of (H

∨

)

I

, let V (¯µ) be the weight

space of V for (T

∨

)

I

. Let X

∨

∈ Lie(H

∨

)

I

be a certain principal nilpotent element (see

Section 5 for the details), which induces a ﬁltration F

i

V (¯µ) = (ker X

∨

)

i+1

∩V (¯µ) on V (¯µ),

called the Brylinski-Kostant ﬁltration. Then we have

T 0.3. – For V ∈ Rep((H

∨

)

I

), let R S(V ) ∈ P

v

be the corresponding sheaf. Then

dim H

2i−(2ρ,¯µ)

R S(V )|

F `

v ¯µ

= dim gr

F

i

V (¯µ).

Here H

∗

denotes the cohomology sheaves, and 2ρ is the sum of positive roots of H, see

Section 1 for the meaning of (2ρ, ¯µ).

One of our main motivations of this work is to apply these results to the calculation of the

nearby cycles of certain ramiﬁed unitary Shimura varieties, via the Rapoport-Zink-Pappas

local models. For example, we obtain the following theorem (see Section 7 for details).

4

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THE GEOMETRIC SATAKE CORRESPONDENCE FOR RAMIFIED GROUPS 413

T 0.4. – Let G = GU(r, s) be a unitary similitude group associated to an imagi-

nary quadratic extension F/Q and a Hermitian space (W, φ) over F/Q. Let p > 2 be a prime

where F/Q is ramiﬁed and the Hermitian form is split. Let K

p

be a special parahoric subgroup

of G = G(Q

p

). Let K = K

p

K

p

⊂ G(Q

p

)G(A

p

f

) be a compact open subgroup with K

p

small

enough. Let Sh

K

be the associated Shimura variety over the reﬂex ﬁeld E and Sh

K

p

be the in-

tegral model of Sh

K

over O

E

p

(as deﬁned in [23]). Then for 6= p, the action of the inertial

subgroup I of Gal(Q

p

/F

p

) on the nearby cycle Ψ

Sh

K

p

⊗

O

E

p

O

F

p

(Q

`

) is trivial.

By applying Theorem 0.3, it will not be hard to determine the traces of Frobenius on these

sheaves explicitly, which will be the input of the Langlands-Kottwitz method to determine

the local Zeta factors of Sh

K

. Instead, we characterize these traces of Frobenius in terms of

Langlands parameters, which veriﬁes a conjecture of Haines and Kottwitz in this case (see

Proposition 7.4).

R 0.3 . – (i) While the deﬁnition of the integral model of a PEL-type Shimura

variety at an “unramiﬁed” prime p (i.e., the group is unramiﬁed at p and K

p

is hyperspe-

cial) is well-known (cf. [15]), the deﬁnition of such a model at the ramiﬁed prime p (even

for K

p

special) is a subtle issue. In [21, 23], the integral models Sh

K

p

are deﬁned as certain

closed subschemes of certain moduli problems of Abelian varieties. Except a few cases

(e.g., (r, s) = (n −1, 1) and n = r + s is small), there is no moduli description of Sh

K

p

so far. In general, Sh

K

p

are not smooth. Indeed, as shown in [21, 23], when n = r + s is

odd and (r, s) = (n −1, 1), for the special parahoric K

p

of G(Q

p

) with reductive quotient

GO

n

, Sh

K

p

is not even semi-stable.

(ii) If r 6= s, then we know that E = F and the above theorem gives a complete description

of the monodromy on the nearby cycles of Sh

K

p

. If r = s, then E = Q, and the complete

description of the monodromy is more complicated. See Section 7 for details. In any case, the

action of inertia on the nearby cycle is semi-simple.

(iii) We hope that there will be a “good” compactiﬁcation of such Shimura varieties Sh

K

p

.

Then the above theorem, together with the existence of such compactiﬁcation, would imply

that the monodromy of H

∗

c

(Sh

K

⊗

E

p

F

p

) is trivial.

(iv) The triviality of the monodromy as above would have the following surprising conse-

quence for the Albanese of Picard modular surfaces. Namely, in the case when (r, s) = (2, 1),

F/Q is ramiﬁed at p > 2 and K

p

= G(Q

p

) is a special parahoric, the Albanese Alb(Sh

K

p

)

of Sh

K

p

is trivial. It will be interesting to ﬁnd the “optimal” level structure at p so that

Alb(Sh

K

p

) can be possibly non-trivial. More detailed discussion will appear elsewhere.

Let us quickly describe the organization of the paper. We will prove Theorem 0.1 and

Theorem 0.2 in §1-4. Then we prove Theorem 0.3 in §5.

In § 6, we brieﬂy discuss the Langlands parameters associated to a smooth representation

of a quasi-split p-adic group, which has a vector ﬁxed by a special parahoric. We call them

“spherical” representations, and we will see that their Langlands parameters can be described

easily. Again, the correct point of view is to consider the special parahoric rather than the

special maximal compact. Then in § 7, we apply the previous results to study the nearby cycles

on certain unitary Shimura varieties.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE