April 19, 2014

Workshop on the Representation Theory of Reductive Algebraic Groups
January 18-21, 2007 (back to home page)


Jeffrey Adler, University of Akron
Multiplicity one upon restriction

Suppose $G$ is a quasisplit $p$-adic group, and $G'$ is its derived group. Conjecturally, the restriction to $G'$ of an irreducible, admissible representation of $G$ should decompose without multiplicity. I will place this conjecture into context, and present a simple proof (joint work with Dipendra Prasad) when $G$ is the group of symplectic or orthogonal similitudes.

Anne-Marie Aubert, Institut de Mathematiques de Jussieu
Springer correspondence for complex reflection groups

Recent work by a number of people has shown that complex reflection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called "spetses', of which reductive algebraic groups ought to be a special case.

In this talk (based on a joint work with Pramod Achar), we will propose a new algebraic construction associating to each complex reflection group (equipped with a root lattice) a set which plays the role of the set of unipotent classes. In the special case of dihedral groups, we will explain how to carry out the Lusztig-Shoji algorithm for calculating corresponding Green functions. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the ``unipotent variety'' associated to a spets. From this, we will determine the possible "Springer correspondences" and show that, as is true for algebraic groups, each special piece is rationnally smooth, as is the full unipotent variety.

Cristina Ballantine, College of the Holy Cross
Combinatorics and representation theory of p-adic groups

We use the Hecke algebra of $GL_n(\mathbb{Q}_p)$ with respect to $GL_n(\mathbb{Z}_p)$ to explore the relationship between representations of $GL_n(\mathbb{Q}_p)$ and the combinatorial properties of its Bruhat-Tits building. On the one hand, for $n=2$ or $3$, using the classification of the unramified representations of $GL_2(\mathbb{Q}_p)$, resp. $U_3(\mathbb{Q}_p)$, we show that
quotients of the corresponding building are Ramanujan graphs. On the other hand, we use combinatorial properties of the building to produce new p-adic representations of $GL_n(\mathbb{Q}_p), n=2,3$.

Lassina Dembele, University of Calgary
Explicit Jacquet-Langlands for GSp(4)

In this talk we consider the Langlands Correspondence as it pertains to a class of automorphic representations corresponding to families of Hilbert-Siegel modular forms. Assuming the relevant form of the Jacquet-Langlands Correspondence, we find geometric objects which produce global galois representations whose partial L-functions match the partial L-functions of these so-called Hilbert-Siegel representations. For GSp(4) we are able to provide new explicit examples of both the Langlands Correspondence and the Jacquet-Langlands Correspondence which do not rely on any conjectures.
(Joint with C. Cunningham)

Julia Gordon, The University of British Columbia
Motivic Integration and its Applications ot p-adic Groups

Motivic integration was first introduced by M. Kontsevich in the context of algebraic geometry. Later J. Denef and F. Loeser developed a
theory of arithmetic motivic integration, which specializes to ntegration over the $p$-adics. Denef and Loeser stated the general principle that ``all natural $p$-adic integrals are motivic''. The value of a motivic integral is a geometric object defined over the field of rational numbers. The value of the initial $p$-adic integral can be recovered from it by a procedure that generalizes counting points on the reduction of this object over the residue field.

Replacing $p$-adic integrals with motivic integrals has three main advantages: it clarifies exactly in what way the $p$-adic integral depends on $p$; motivic integration is an algorithmic procedure (based on elimination of quantifiers), so motivic integrals are computable; finally, motivic calculations allow to switch freely between function fields and extensions of ${\mathbb Q}_p$. Naturally, $p$-adic integrals appear prominently in the representation theory of $p$-adic groups (e.g. orbital integrals, and Harish-Chandra distribution characters). It is, however, a largely open question whether they are ``natural'' in the sense of Denef and Loeser.

Recently, R. Cluckers and F. Loeser introduced a new theory of motivic integration, which encompasses all the previous ones, has a large class of integrable functions, and is complete with all the tools expected of a theory of integration, such as Fubini theorem, and Fourier transform. The goal of these talks is to give an exposition of the theory from the perspective of applying it to the problems that arise in the context of representation theory of $p$-adic groups.

Ju-Lee Kim, University of Illinois at Chicago, M.I.T. Visiting Professor
Recent progress in the classification of supercuspidal representations

We discuss a recent progress in the classification of supercuspidal representations of reductive p-adic groups. In the first lecture, we give an introduction to the construction of tame supercuspidal representations (due to Yu) and their equivalence relation (due to Hakim and Murnaghan). In the second lecture, we sketch the proof of an exhaustion theorem of Yu's supercuspidal representations.

Fiona Murnaghan, University of Toronto
Ordinary characters and spherical characters: the supercuspidal case

We will recall some properties of characters of tame supercuspidal representations. Preliminary investigations indicate that spherical characters of many distinguished tame supercuspidal representations appear to have analogous properties. In particular, we will give a formula expressing such a spherical character in terms of integrals of a certain matrix coefficient of the representation. We will also
discuss possible relations between spherical characters and Fourier transforms of orbital integrals.

A. Raghuram, Oklahoma State University

The University of Ottawa Colloquium
Arithmetic of L-functions.

An L-function is a function of one complex variable that is attached to some interesting arithmetic or geometric data. The values of such an L-function, at interesting points, give structural information about the data to which it is attached. This talk will be an introduction, via examples, to the subject of special values of L-functions. I will begin by recalling some classical formulae which one usually encounters in an advanced course in Calculus. These formulae, when recast in modern language, are the prototypes of special values of L-functions. Starting at an elementary level, I will build up toward the conjectures of Deligne, which has guided a lot of research over the last thirty years on this theme.

Conference Talk
Special values of certain automorphic L-functions.

In this talk I will present some results obtained in collaboration with Freydoon Shahidi. Using some recent results on special values of automorphic L-functions due to Mahnkopf, and while using certain new cases of Langlands functoriality, we have made some progress on the special values of symmetric power L-functions associated to modular forms.

These results are in accordance with Deligne's conjectures on the special values of motivic L-functions. Although the results can be stated in a classical language, the technical inputs to our work mostly involve the representation theory of p-adic, real and adelic groups.

Michael Schein, Hebrew University
Weights in Serre-type conjectures and the mod p Langlands correspondence

Serre's conjecture specifies when a mod p representation of the absolute Galois group of Q is modular, and, if so, of what weights. We will explain a generalization of this conjecture to representations of absolute Galois groups of totally real fields (extending work of Fred Diamond et al. in the case when p is unramified in the totally real field) and present theoretical and computational evidence for the conjecture.

We will mention a connection with the mod p Langlands correspondence and discuss an intriguing correspondence between mod p Galois representations and characteristic zero representations of linear groups over rings.

Teruyoshi Yoshida, Harvard University
Non-abelian Lubin-Tate theory and Deligne-Lusztig theory revisited

We review some concrete local methods to show that the etale cohomology of Lubin-Tate spaces realizes the local Langlands correspondence for GL(n) (non-abelain Lubin-Tate theory). Especially we compare the level 0 supercuspidals (as automorphic induction from a tame character) and the Deligne-Lusztig trace formula; we will mention local computations for the Iwahori level case if time permits.

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