December  5, 2023

Workshop on the Representation Theory of p-adic Groups
May 5-9, 2004

to be held at the University of Ottawa


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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