# GEOMETRIC STRUCTURE IN THE REPRESENTATION THEORY OF REDUCTIVE P-ADIC GROUPS

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Mathieu Stiénon, Nigel Higson, Ping Xu.

**Speaker:** Paul Baum, Penn State

**Abstract:** Let G be a reductive p-adic group. Examples are GL(n, F) SL(n, F) where n can be any positive integer
and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual
of G is the set of equivalence classes of smooth irreducible representations of G. The representations
are on vector spaces over the complex numbers. The smooth dual has one point for each distinct smooth
irreducible representation of G. Within the smooth dual there are subsets known as the Bernstein components,
and the smooth dual is the disjoint union of the Bernstein components. This talk will explain a conjecture
due to Aubert-Baum-Plymen (ABP) which says that each Bernstein component is a complex affine variety.
These affine varieties are explicitly identified as certain extended quotients. The infinitesimal character of Bernstein
and the L-packets which appear in the local Langlands conjecture are then described from this point of view.
Recent results by a number of mathematicians (e.g. V. Heiermann, M. Solleveld) provide positive evidence for ABP.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 04/12/2011

**Time:** 2:30pm - 3:30pm