## Workshop on Geometry Related to the Langlands Programme

May 27-31, 2009

to be held at the University of Ottawa

### Abstracts

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**Anne-Marie Aubert**, C.N.R.S.

*An introduction to perverse sheaves and character sheaves*

**Due to circumstances beyond control, Anne-Marie Aubert regrets she is unable
to attend. **

The mini-course will be given by David Treumann and Clifton Cunningham.

In the first half of the course, after a brief review on constructible sheaves,
derived categories, triangulated categories and t-structures, we shall introduce
perverse (with respect to an arbitrary perversity function p) sheaves on stratified
topological spaces and describe the link with intersection cohomology complexes.

Then we shall pass to the construction of Dcb(X,(Ql)--), the bounded "derived"
category of Ql-(constructible) sheaves on an algebraic variety over an algebraically
closed field in which the prime number l is invertible [BBD, 2.2.18] and describe
the corresponding subcategory M(X) of perverse sheaves on X.

In the second part of the course we shall define, and study some of the main
properties of character sheaves over G, a reductive algebraic group defined
over an algebraically closed field. Character sheaves on G, which were introduced
by Lusztig in [L] for a connected G (and extended by him more recently to
disconnected G), belong to the set of isomorphism classes of simple objects
of M(G).

References:

[BBD] A.A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Astérisque
100 (1982).

[L] G. Lusztig, Character sheaves I, Advances in Mathematics, 56, pp. 193-237
(1985).

Pre-requisites:

Some familiarity with sheaves should be useful: see for instance R. Hartshorne,
section 1 of chap. II in "Algebraic Geometry", Graduate Texts in
Mathematics, Springer-Verlag.

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**Ioan Badulescu**, University of Montpellier

*The Jacquet-Langlands correspondence*

This course is an introduction to local and global Jacquet-Langlands correspondence
in zero characteristic. If F is a local field and D is a central division
algebra of dimension d^2 over F, then there is a natural injection from the
set of regular semisimple conjugacy classes of the group GL(n,D) into the
set of regular semisimple conjugacy classes of the group GL(nd,F). This defines
via characters of representations a bijection between the set of irreducible
square integrable representations of the group GL(nd) over a local field F
and the set of irreducible square integrable representations of the group
GL(n) over a division algebra D (the local Jacquet-Langlands correspondence,
proved in full generality by Deligne, Kazhdan and Vignéras). There
is also a global version of this result.

After a short introduction to the representation theory of GL(n) over local
and global fields I will give the Deligne-Kazhdan-Vignéras proof of
the local Jacquet-Langlands correspondence using the simple trace formula.
I will state the global result which is based on the local one, and give the
idea of the proof, based on the simple (but not as simple as the other one)
trace formula of Arthur and Clozel.

-----

**Pierre-Henri Chaudouard**, CNRS, Paris

On the geometry of the Hitchin fibration (slides)

The stabilization of the Arthur-Selberg trace formula should give all Langlands'
functorialities of endoscopic types. But, to stabilize the trace formula,
one needs the fundamental lemma stated by Langlands-Shelstad: it is a family
of combinatorial identities between orbital integrals. One also needs a "weighted"
fundamental lemma, due to Arthur, which applies to the weighted orbital integrals.

In the equal characteristic case, the orbital integrals have a nice geometric
interpretation: they count rational points of projective varieties over finite
fields. These varieties are either quotients of affine Springer fibers (global
orbital integrals) or fibers of the Hitchin fibration (adelic orbital integrals).
Moreover, some quotients of Hitchin fibers are products of affine Springer
fibers. Ngô's recent proof of the fundamental lemma relies on a deep
geometric and cohomological study of the elliptic part of the Hitchin fibration.
With Laumon, we have extended Ngô's study to the hyperbolic part of
the Hitchin fibration and we have proved the "weighted" fundamental
lemma.

The aim of the course is to give an introduction to the geometry of the affine
Springer fibers and of the Hitchin fibration.

References:

[1] M. Goresky, R. Kottwitz, and R. Macpherson. Homology of affine Springer
fibers in the unramified case. Duke Math. J., 121(3):509--561, 2004.

[2] D. Kazhdan and G. Lusztig. Fixed point varieties on affine flag manifolds.
Israel J. Math., 62(2):129--168, 1988.

[3] B. C. Ngô. Le lemme fondamental pour les algèbres de Lie.
http://arxiv.org/abs/0801.0446.

[4] B. C. Ngô. Fibration de Hitchin et endoscopie. Invent. Math. ,
164(2):399--453, 2006.

[5] I. Biswas and S. Ramanan. An infinitesimal study of the moduli of Hitchin
pairs. J. London Math. Soc. (2), 49(2):219--231, 1994.

[6] A. Beauville, M. Narasimhan, and S. Ramanan. Spectral curves and the
generalised theta divisor. J. Reine Angew. Math., 398:169--179, 1989.

[7] Pierre-Henri Chaudouard, Gérard Laumon. Le lemme fondamental pondéré
I : constructions géométriques. http://fr.arxiv.org/abs/0902.2684v1

Pre-requisites:

Basic knowledge of algebraic geometry.

-----

**Jeffrey D. Adler**, American University

*Towards a lifting of representations of finite reductive groups*

The problem of understanding explicit base change for p-adic groups forces
one to consider certain liftings of representations of finite groups. I will
give an introduction to the former and a partial description of the latter.
This is joint work with Joshua Lansky.

**Atsushi Ichino**, Institute for Advanced Study

*Formal degrees and adjoint gamma-factors*

Harish-Chandra established the Plancherel formula for reductive groups over
local fields. Formal degrees and Plancherel measures are key ingredients in
the Plancherel formula and are important objects in harmonic analysis. In
this talk, we give an extension of Langlands' conjecture which relates these
objects with certain arithmetic invariants. Using twisted endoscopy, we prove
the conjecture for stable discrete series of U(3) over p-adic fields. We also
discuss its interaction with local theta correspondence.

This talk is based on a joint work with Kaoru Hiraga and Tamotsu Ikeda, and
that with Wee Teck Gan.

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**Syu Kato**, Kyoto University

*An exotic Deligne-Langlands correspondence for symplectic groups *
An affine Hecke algebra associated to a root datum is a q-analogue of its
affine Weyl group. In general, it admits several (up to three) parameters.
The classification of irreducible representations of affine Hecke algebras
with equal-parameters is given by Kazhdan-Lusztig (and Ginzburg) as a modification
of the Deligne-Langlands conjecture. Their approach is based on the geometry
of the nilpotent cone of the corresponding Lie algebra over complex numbers.
This approach is later deepened by Lusztig in order to deduce similar results
for (various) integrally-weighted one-parameter cases.

In this talk, we present a geometric realization of an affine Hecke algebra
H of type C with three parameters by replacing the nilpotent cone of the Lie
algebra with a certain Hilbert nilcone of a symplectic group in the Kazhdan-Lusztig
construction. This enables us to present a Deligne-Langlands type classification
of simple modules of H when the parameters are sufficiently good. (The title
Deligne-Langlands means that our classification looks unmodified when compared
with the Kazhdan-Lusztig theorem.)

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**Paul Mezo**, Carleton University

*Spectral transfer for real twisted endoscopy*

The theory of endoscopy attaches to a reductive algebraic group a collection
of structurally related groups, the so-called endoscopic groups. The harmonic
analysis of the reductive group is conjectured to be related to that of the
endoscopic groups. In the context of real groups, the standard conjectures
have been proven by Shelstad. In the more general real context, in which automorphisms
and characters twist the data, the transfer of twisted orbital integrals (i.e.
geometric transfer) has been partially established by Renard. We shall describe
the transfer of twisted characters under the assumption of geometric transfer.

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**Mark Reeder**, Boston College

*Explicit examples of the local Langlands correspondence*

I will discuss a family of supercuspidal representations of simple p-adic
groups and their Langlands parameters, both having minimal wild ramification.
The representations are constructed uniformly, without any restrictions on
p or the base field k, and they are new when p is small. On the other hand,
the corresponding parameters behave quite differently for small p. They arise
from Galois extensions of k whose existence was predicted by the existence
of the simple supercuspidal representations.

This is joint work with Benedict Gross.

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**Hadi Salmasian**, University of Windsor

*Character sheaves and representations of p-adic groups*

In this talk we give a geometric interpretation of characters of certain
representations of reductive p-adic groups, which are usually called depth
zero supercuspidal representations, in terms of characteristic functions of
certain character sheaves. Our work relies on Lusztig's theory of character
sheaves, and in some sense suggests a general framework for relating character
sheaves on groups over a p-adic field to smooth representations. (Joint work
with Clifton Cunningham.)

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**David Treumann**, University of Minnesota

*Localization and induction for Springer's representations*

The Springer correspondence is a relationship between the geometry of the
space of unipotent elements in a reductive algebraic group G and the representations
of the Weyl group of G. And "induction" result due to Alvis and
Lusztig relates the Springer correspondences for G and for a Levi subgroup
of G. I will discuss a proof and generalization of this result based on localization
techniques in equivariant cohomology.

###

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