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Journal ArticleDOI

The Geometric Satake Correspondence for Ramified Groups

01 Mar 2015-Annales Scientifiques De L Ecole Normale Superieure (Elsevier)-Vol. 48, Iss: 2, pp 409-451
TL;DR: The geometrical Satake isomorphism for a reductive group defined over F=k((t)) and split over a tamely ramified extension was proved in this paper, and a description of nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
Abstract: We prove the geometrical 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.

Summary (6 min read)

1. Introduction

  • A distributed computing system consists of heterogeneous computing devices, communication networks, operating system services, and applications.
  • The authors approach to distributed applications management requires instrumentation; that is, code inserted into the application at strategic locations so a managed process can maintain management information, respond to management requests and generate event reports.
  • This research work is supported by the IBM Centre for Advanced Studies and the Natural Sciences and Engineering Research Council of Canada.
  • By developing an instrumentation architecture, the authors can make e orts toward automating some parts of the process, and provide guidance to facilitate the development of custom instrumentation in a controlled and structured manner.
  • Section 2 outlines some of the major objectives guiding the current work.

2. Objectives

  • A number of general objectives were established to guide their study of instrumentation and to in¯uence the development and re®nement of their instrumentation architecture and prototype.
  • For e ective management, the authors must be able to acquire information on the processes being managed.
  • Similarly, the architecture should accommodate a wide range of management functions including performance, accounting, fault tolerance, and security.
  • The instrumentation architecture should support the automation of the instrumentation process, so it can be done with little developer e ort.
  • In addition to reducing developer e ort, the notion of run-time transparency must be observed; users of managed applications should not be aware of substantial performance overhead caused by management.

3. Management environment

  • The authors de®ne the general framework of their distributed applications management system.
  • An agent receives management requests from managers and carries out the operations on the appropriate managed objects.
  • Fig. 2 illustrates the basic components of an agent architecture and how they interact with managed processes.
  • The managed object interface is used to coordinate requests on managed objects and to handle emitted noti®cations.
  • Process managed object classes are specialized through inheritance to re¯ect their unique characteristics including communication environment (sockets, DCE Remote Procedure Calls (RPCs), etc.), process role (client, server, or peer), and application type (database, etc.).

4. Instrumentation architecture

  • Having identi®ed the key objectives of instrumentation for managing distributed applications, the authors re®ne the management architecture developed in their previous work (Hong et al., 1995b, c; Katchabaw et al., 1996a) to re¯ect their current focus on instrumentation.
  • In Section 4, the authors describe the key components of the architecture and the interactions that occur between these components.

4.1. Instrumentation component overview

  • In Fig. 3, Instrumented Process 1 has been expanded to reveal a set of instrumentation components.
  • The coordinator is also responsible for creating and destroying sensors and actuators, as well as performing management initialization and termination activities within the managed process.
  • One can think of a sensor as a reusable software component that can be plugged into an application process via one or more probes for the purpose of capturing management data.
  • For other application-speci®c information, custom sensors can be built or derived from standard sensors.
  • As for sensors, it is anticipated that application developers will view actuators as reusable parts which can be plugged into an application process via one or more probes to provide speci®c control functions.

4.2. Service interfaces

  • The service interfaces represent points of interactions between the managed process (through the management coordinator) and the agent.
  • The authors will ®rst present the interface of the agent that is used by the managed process.
  • The authors will then present the interface of the managed process used by the agent.

4.2.1. Managed process to agent

  • Acquiring management information is critical to the management of distributed applications.
  • The authors must be able to support two types of ¯ow of information: (i) the periodic sending of management information to the agent, and (ii) the ability to send information when a signi®cant event occurs.
  • Id represent the process name and identi®er.
  • Noti®es the agent of the termination of the process in the context of the normal operation of the application, also known as Process_notifyTermination.
  • ServerType speci®es the type of server process that failed.

4.2.2. Agent to managed process

  • This service is used by the management agent to request information from managed processes in the system.
  • By specifying the set of attributes to retrieve, the appropriate information is returned to the agent.
  • This service instructs the process to terminate as quickly as possible.
  • Operations include the following: Process_controlManageProcess: Activates management in a process, informing the process which agent will be managing it.
  • For periodic reports, this operation can change the interval between reports.

4.3. Interactions

  • The interactions between the management system and managed processes include requests from the management system to the processes and reports ¯owing in the other direction.
  • Requests can be made to retrieve management information, exert control over a managed process, or to change the way in which a process is being managed.
  • Reports are generated at process initialization, at periodic intervals, upon the detection of alarm conditions, and at process termination.
  • The authors describe the roles of the instrumentation components presented in Fig. 3 as they relate to these interactions.

4.3.1. Management system to instrumentation interactions

  • Requests from the management system to an instrumented process are received by its management coordinator which, in turn, routes them to the appropriate instrumentation components as follows: · A request for management information is sent as a read request to the sensor (or sensors) responsible for the infor- mation requested.
  • The state for the sensors is updated by the sensors' probes, or through interactions with other sensors.
  • The results are returned through the coordinator to the management system.
  • · A request to execute a control operation is sent as an action request to the actuator (or actuators) responsible for that operation.
  • If the request calls for the collection of additional management information, new sensors are created or existing ones are enabled.

4.3.2. Instrumentation to management system interactions

  • Reports from process instrumentation to the management system go through the management coordinator.
  • There are a variety of circumstances in which such reports are sent from the coordinator to the management system.
  • This co- ordinator then creates and enables a default set of sensors and actuators.
  • To register with the management system (notifying the system of the existence of the instrumented process and of an interface through which the process can be managed), the coordinator issues read requests to the appropriate sensors to gather the information.
  • The sensors return this state information, which was originally collected by their probes.

5.1. Placing instrumentation

  • For e ective management of a distributed application process, instrumentation probes must be inserted at strategic locations (probe points) in the process source code.
  • Other instrumentation components (the management coordinator, sensors, and actuators) are located in the same address space as the process, but do not need to be placed within the original code of the process.
  • Every process entry point must be instrumented with probes to allow instrumentation to initialize and to allow process registration to occur.
  • At these points, performance metrics can be computed, faults can be detected, accounting can be performed, security policies can be validated, and application con®guration may be changed.
  • The above list of probe points is not exhaustive, but gives a strong sense of the level of instrumentation required for manageability.

5.2. Performing instrumentation

  • In the previous section, the authors described the process of instrumentation and identi®ed the strategic points in source code where probes should be placed.
  • The authors examine several instrumentation techniques and the relative merits of each.
  • Fig. 4 illustrates the alternative approaches the authors consider.

5.2.1. Hand-coded instrumentation

  • Most application process instrumentation is carried out by hand during application development.
  • In their work, the authors try to make use of standard components and apply them in a prescribed manner to achieve a degree of consistency across applications and minimize the risk of error.
  • A substantial amount of time, e ort, and resources can be expended instrumenting an application in this way.
  • Tools to automate the insertion of probes would greatly enhance this procedure.

5.2.2. Function or class wrapper instrumentation

  • One possible method of automating the instrumentation process is to provide application developers with management ``wrappers'' for functions, data structures, or objects.
  • A management wrapper mf around a function f might be implemented as p1 f p2, where p1 and p2 are standard probes.
  • To make use of the function f, mf is called instead (although its name is probably still the same), with the same result produced.
  • The same concept applies to wrapping data structures and objects.
  • Wrappers can help automate basic instrumentation and reduce somewhat the e ort and risk of human error.

5.2.3. IDL instrumentation

  • Many distributed application computing environments use a formal Interface De®nition Language (IDL) through which attributes and procedure or object method signatures are speci®ed.
  • IDL compilers could be modi®ed to insert probes automatically when generating communications stub code.
  • This approach would further reduce the e ort and risk of programmer error problems associated with probe insertion, but at a cost of some ¯exibility and customizability.
  • Since stub code is normally produced in the source language of the application, the developer does have opportunity to access and modify generated stubs, buying back some of the lost ¯exibility.
  • Editing stub code is usually considered a dangerous and messy practice.

5.2.4. System library or compiler instrumentation

  • A fourth approach hides most of the instrumentation procedure from the application developer.
  • By providing instrumented system libraries or a compiler that injects instrumentation at compile-time, most of the instrumentation task is handled automatically.
  • This could not otherwise be done by the developer alone.
  • Perhaps the best example of work on this type of internal instrumentation can be found in (OSF, 1993, 1995); however, this work is aimed mainly at performance measurement rather than general management.
  • Clearly, there is a need for automation to reduce the e ort and inherent risk of instrumentation by hand.

6.1. Management environment

  • The prototype builds upon University College London's OSI Management Information Service (Pavlou et al., 1993, 1991) which provides an object-oriented infrastructure for developing management applications and OSI agents.
  • OSIMIS includes a compiler (Cowan, 1993) to parse managed object class de®nitions (speci®ed in the Guidelines for the De®nition of Managed Objects (GDMO) (ISO, 1991c)) and generate the agent code required to access those managed objects.
  • Through extensions to OSIMIS, their management system is capable of managing DCE distributed applications for a variety of management tasks.

6.2. Instrumentation implementation

  • Instrumentation is provided to application developers through a C++ class library.
  • The library contains a variety of standard sensors, actuators, and management coordinators.
  • The library can also be specialized to develop new classes of instrumentation components to meet the unique needs of speci®c applications.
  • Figs. 5±7 show class hierarchies for management coordinators, sensors, and actuators, respectively, implemented to date.

6.2.1. Management coordinator

  • Interactions between the management system and the managed processes can be carried out using di erent communication protocols that include standard DCE RPCs and sockets.
  • Di erent management systems can be supported by replacing the management coordinator.
  • If the management coordinator uses DCE for communication between the instrumented processes and management agents and then changes to sockets then the authors would only have to replace the management coordinator; the sensors and actuators do not change.
  • Coordinator routines have also been developed to initialize the instrumentation, to route internal messages, and to manage sensors and actuators.
  • A scheduler routine in the coordinator handles time-based events; this has been implemented as a separate execution thread.

6.2.2. Sensors and actuators

  • In the current prototype, the instrumentation library includes the following categories of sensors and actuators.
  • Registration sensors allow processes and applications to be registered with the management system so that the system is aware of their existence.
  • These include remote procedure call timeouts, server congestion, middleware service failures, failed ®le or network operations, and other abnormal and undesirable conditions.
  • Resource utilization sensors collect information that characterizes the impact a process is having on system hardware resources, including memory, CPU, disk, and network usage measures.
  • Process control actuators control process termination, process suspension, process priority modi®cation and change the length of the interval between RPC statistics event reports.

6.2.3. Probes

  • The following probes were inserted into the source code of distributed applications by hand.
  • Since the authors are using DCE they retrieve the binding handle of the agent using the Cell Directory Service (CDS).
  • A management thread is created that becomes the management coordinator and the sensors.
  • This probe noti®es an agent of the termination of a process.
  • Hence, for any manageable process, the code for the process will exhibit a form similar to this: #include ``app_instrumentation.h'' Process_instrumentationInit; Process_rpcRequestBegin application remote procedure call; Process_rpcRequestEnd Process_instrumentationShutdown;.

6.3. Managers

  • To validate their instrumentation architecture the authors implemented, as part of the prototype system, several managers.
  • (Rolia et al., 1995; Rolia and Sevcik, 1995; Sun et al., 1997; Sun, 1997; Woodside et al., 1995) Delays caused by poor performance at the application level or network level can seriously a ect the usability and e ectiveness of a distributed application, or an entire distributed environment, also known as Performance management.
  • At the source of a failure is a fault (or possibly several faults).
  • (Lut®yya et al., 1997) Most manager applications, such as those described above, require access to information on computing devices, networks, system resources and services, and user applications in the environment they are managing, also known as Con®guration maintenance.
  • Registration sensors, such as those discussed in Section 6.2.2, are used by distributed applications to register, update, and remove their con®guration information with the Con®guration Maintenance Service.

6.4. Evaluation

  • Having designed and developed prototypes of their generic architecture in Section 4, it is important to evaluate this work in light of their objectives given in Section 2.
  • The authors have also provided a library of instrumentation components to support generic management of applications, and derived components speci®c to the needs of certain applications (DCE distributed applications).
  • As a result, the authors have met this objective with their generic architecture, and they have successfully validated it through their prototypes.
  • The authors work also supports a variety of management functions, with sensors and actuators to facilitate fault management, performance management, con®guration management, and accounting for system resources.
  • To assess the overhead of their management instrumentation, the authors have developed a Distributed Applications Management Testbed (Katchabaw et al., 1996b).

8. Concluding remarks

  • The work the authors describe here is part of an ongoing structured attack on the distributed applications management problem.
  • The authors have discussed several approaches to instrumentation based on this, and have concluded that, while automation and tools are required to facilitate the process, hand-coded instrumentation is still needed for custom management.
  • The authors see the following areas as potentially fruitful.
  • · Developing software engineering tools to enable building instrumented applications.
  • · Developing an information model and repository services for management components to support searching, browsing, and other queries.

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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 déplo 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 ane Grassmannian of H. Choose a prime dierent 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
`
-coecients
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
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) ane 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 coecients 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 ane Grassmannian of G. However, we reserve the name “ane Grassmannian of G
for another object, as defined in Definition A.2.
4
e
SÉRIE TOME 48 2015 N
o
2

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 dierent, 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 ane 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 dierence 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
e
SÉRIE TOME 48 2015 N
o
2

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

Citations
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Abstract: For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction \"from automorphic to Galois\". Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

150 citations

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41 citations

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TL;DR: In this article, the authors formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program.
Abstract: We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

37 citations

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TL;DR: In this article, the authors proved the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.
Abstract: We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.

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Journal ArticleDOI
25 Mar 2020
TL;DR: In this article, the authors study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type and determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.
Abstract: We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

27 citations

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TL;DR: In this article, the basic relationship between G and G is discussed, and a canonical construction of G, starting from G, is presented, which leads to a rather explicit construction of a Hopf algebra by Tannakian formalism.
Abstract: As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.

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511 citations

Frequently Asked Questions (10)
Q1. What are the contributions mentioned in the paper "The geometric satake correspondence for ramified groups " ?

For unramified groups, the classical Satake isomorphism is the starting point of the Langlands duality this paper. 

The only place where the complex topology is used, besides the issue of dealing with Z-coefficients as in [19], is to define the coproduct on H∗(GrH ,Z) by realizing GrH as being homotopic to the based loop space of a maximal compact subgroup ofHC. 

In other words, the group scheme Aut⊗H∗TH over tH of the tensor automorphism of this fiber functor, which a priori is an inner form of H∨, is canonically isomorphic to H∨ × tH . 

The authors write ρ(γ) = (ρ1(γ), γ) for γ ∈WF , where ρ1 is a map from WF to H∨.D 6.3. – A “spherical” parameter (or Langlands-Satake parameter) is a Langlands parameter ρ which can be conjugated to the form ρ(γ) = (1, γ) for γ in the inertial group I.Let (H∨)I be the I-fixed point subgroup ofH∨ (which could be non-connected according to Remark 4.4). 

Recall that under the (ramified) geometric Satake isomorphism, the cohomological grading corresponds to the grading by 2ρ : Gm → GSpn ⊂ GLn ×Gm. 

The authors define the set of dominant elements in X•(T )I to be(1.2) X•(T )+I = {µ̄|(µ̄, a) ≥ 0 for a ∈ Φ +}.Then the natural mapX•(T )+I ⊂ X•(T )I → X•(T )I/W0 is bijective. 

In this case, the functor R S in Theorem 0.1 can be extended to an equivalenceR S : Rep((H∨)I oactalg Gal(k/Fq)) ' P 0 v,whose composition with H∗ is isomorphic to the forgetful functor. 

By the assumption (i) and [7, Proposition 2.16], the tensor structures on RepG1 and RepG2 induce B ⊗B → B and A⊗ A→ A respectively. 

The proof of [29, Lemma 5.1] that cTH ( L) is primitive under this Hopf algebra structure can be replaced by the following argument: as is well-known (e.g., see [30, 1.1.9]), if L is ample on GrH , then there is an ample line bundle on Gr2, which away from the diagonal is L L and on the diagonal is L. 

As v is special, there is a bijection between the finite Weyl chambers for (GL, SL) and the affine Weyl chambers (or called alcove) with v as a vertex, and this bijection is compatible with the action of Gal(L/F ).