### Abstracts

**Clifton Cunningham**, University of Calgary

*Some Remarks on Geometry, Orbital Integrals and Characters*

In the first part of this talk we describe recent work with Hales proving
that (good) orbital integrals are motivic, in the sense of Denef-Loeser,
and that the Fundamental Lemma over characteristic-p p-adic fields implies
the Fundamental Lemma over characteristic-0 p-adic fields. In the second
part of this talk we consider related issues for characters of p-adic
groups as related to recent work with Aubert; specifically, we introduce
`depth-zero character sheaves' and use these to describe Waldspurger's
basis for nilpotent distributions.

**Stephen DeBacker**, University of Michigan

*Nilpotent orbits and Bruhat-Tits theory: a second look*

In this talk we shall briefly recall the (depth zero) classification
of nilpotent orbits in terms of Bruhat-Tits theory. We shall then discuss
two consequences of this classification: (1) the determination of the
Levi subgroup attached to a nilpotent orbit (joint with Jeff Adler)
and

(2) the determination of the component group of the centralizer of a
nilpotent element (joint with David Kazhdan). Both of these consequences
were motivated by harmonic analysis considerations which we shall also
discuss.

**William J. Haboush**, University of Illinois
at Urbana-Champaign

*Spaces of Lattices over a Local Hilbert Class Field*

I will summarize a number of recent results concerning the construction
of certain schemes of lattices. Let $k$ denote the algebraic closure
of the finite field, $\mathbb F_p,$ let $\V $ denote the ring of Witt
vectors over $k$ and let $K$ denote its field of fractions. Let $K^n$
be the $n$-dimensional space over $K$ and let $F$ denote a fixed maximal
$\V$-lattice in $K^n$ which we view as generated by the standard basis.
A lattice is called special if its top exterior power is the top exterior
power of $F.$ I will show that the special lattices contained in $p^{-r}F$
can be viewed as the rational points of a finite dimensional projective
variety, $\Lrn ,$ and that there is a universal family of sublattices
over this scheme. I will describe how one can compute the tangent space
to a lattice on this scheme and use it to describe the singular locus
of $\Lrn .$ I will use this information on the tangent space to describe
a canonical projective embedding of this scheme. Finally I will discuss
some results of A. Sano who has shown that these lattice varieties are
normal.

**Volker Heiermann**, Purdue University

*Unipotent Orbits and poles of Harish-Chandra's $\mu $-function*

In a previous work, we have shown that a representation of a $p$-adic
group obtained by (normalized) parabolic induction from an irreducible
supercuspidal representation $\sigma $ of a Levi subgroup $M$ contains
a subquotient which is square integrable, if and only if Harish-Chandra's
$\mu $-function has a pole in $\sigma $ of order equal to the parabolic
rank of $M$. The aim of the present talk is to interpret this result
in terms of Langlands' functoriality principle.

**Phil Kutzko**, University of Iowa

*A sign which arises in the representation theory of classical groups
over finite fields and its relation to local $L$-functions*

Joint work with Lawrence Morris. Let $\pi $ be a self-contragredient
irreducible unitarizable supercuspidal representation of $GL(n,F)$ where
$F$ is a $p$-adic field, $p\neq 2,$ and where $n=2m$ is a (necessarily)
even integer. \ Let $G_{1}=SO(2n,F),$ $% G_{2}=SO(2n+1,F).$ \ Then $G_{i},$
$i=1,2$ has a parabolic subgroup $P_{i}$ whose Levi subgroup $L_{i}$
is isomorphic to $GL(n,F).$ \ Using this we may identify $\pi $ with
a representation of $L_{i}$ and so consider the representations $\iota
_{P_{i}}^{G_{i}}\pi $ where $\iota _{P_{i}}^{G_{i}}$ is the functor
of normalized parabolic induction. \ It is then a theorem of Shahidi
that, with an appropriate choice of the isomorphisms $L_{i}\cong GL(n,F)$,
one has that $\iota _{P_{1}}^{G_{1}}\pi $ is irreducible if and only
if $\iota _{P_{2}}^{G_{2}}\pi $ is reducible. \ Shahidi's proof relies
on a analysis of the relevent local $L-$functions. \ On the other hand,
at least in the case that $\pi $ is a representation of level (depth)
zero, this result may, using the theory of types and covers, be made
to follow from the calculation of a certain sign that occurs naturally
in the study of classical groups over finite fields. \ This talk will
be devoted to a relatively self-contained exposition of this approach
to Shahidi's result.

**Fiona Murnaghan**, University of Toronto

*Distinguished tame supercuspidal representations*

Let H be the fixed points of an involution of a group G. A representation
of G is said to be H-distinguished if there exists a nonzero H-invariant
linear functional on the space of the representation. In this talk,
we describe recent results (joint with Jeff Hakim) concerning distinguishedness
of tame supercuspidal representations of reductive p-adic groups. We
discuss necessary conditions for distinguishedness, dimensions of spaces
of invariant linear functionals, and equivalence of supercuspidal representations
arising from different G-data.

**Vytautas Paskunas**, Universitaet Bielefeld

*A construction of a good number of supersingular representations
of GL(2,F)*

Let $F$ be a non-Archimedean local field with the residual characteristic
$p$. We construct a "good" number of smooth irreducible $\overline{\mathbf{F}}_p$-representations
of $GL_2(F)$, which are supersingular in the sense of Barthel and Livn\'e.
If $F=\mathbf{Q}_p$ then results of Breuil imply that our construction
gives all the supersingular representations up to the twist by an unramified
quasi-character. We conjecture this is true for arbitrary $F$.

**A. Raghuram**, University of Iowa

*Reciprocity, GL(2) and Division algebras*

For a local field $F$, a result of Waldspurger says that an irreducible
representation $\pi$ of $GL(2,F)$ contains a character $\chi_1 \otimes
\chi_2$ of the diagonal torus $F^* x F^*$ if and only if the central
character of $\pi$ is $\chi_1\chi_2$. In this talk I will consider some
variations and generalizations of this phenomenon.

**Paul Sally** (University of Chicago)

*Characters for Reductive p-adic Groups*

I will talk about the past, present, and future of admissible characters
for reductive p-adic groups with emphasis on discrete series and applications
to harmonic analysis.

**Allan J. Silberger **(Cleveland State
University)

*On Explicit Matching in the Level Zero Case*

The "Abstract Matching Theorem", proved by Badulescu in char
p and DKV in char 0, implies the existence of a match-up of discrete
series characters which preserves character values on the regular elliptic
set, up to known sign, between the unit groups of any two central simple
algebras of the same reduced degree over a p-adic field. "Explicit
matching" means that we give the predicted match-up by first finding
for each unit group a set of invariants for its discrete series (e.g.
a set of K types) and then mapping the set of invariants to a parameter
set which is independent of the algebra. In this talk we will discuss
the main results of the joint research of the speaker and E.-W. Zink.
These results treat only the depth zero case.

**Yakov Varshavsky**, Hebrew University
of Jerusalem

*On endoscopic decomposition of characters of certain cuspidal representations*

This a joint work with David Kazhdan. Langlands conjectured that characters
of irreducible representations of p-adic groups have endoscopic decomposition.
In this talk I will explain this result in the case of representations
induced from inflations of irreducible cuspidal Deligne-Lusztig representations
of finite fields. Moreover, the obtained decomposition is compatible
with inner twistings.

For the proof we reduce the question (using topological Jordan decomposition)
to analogous question about Lie algebras. Now the statement follows
from a generalization (to inner forms) of a theorem of Waldspurger (which
asserts that endoscopic pieces are preserved by Fourier transform) and
a Springer hypothesis (which describes traces of Deligne-Lusztig representations
in terms of Fourier transform of an orbit).

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