SCIENTIFIC PROGRAMS AND ACTIVITIES

April 18, 2014

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.

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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.

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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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