## 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.

-----

**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.

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.

-----

**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.)

-----

**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.

-----

**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.

-----

**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.)

-----

**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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