## Workshop on the Representation Theory of Reductive Algebraic Groups

January 18-21, 2007 (back
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## Abstracts

**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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