Back to Departmental Colloquium: Spring 2009
Departmental Colloquium
Date: Thursday, Apr 23, 2009
Time: 4:15PM
Location: JWB 335
Dan Barbasch
Cornell
Title |
Theunitarydualoftherationalpointsofarealorp-adiclinearreductivegroup |
Abstract |
While maybe as hard to parse, sadly it is not as catchy as Mark Twain’s “Constantinopolitanischerdudelsackspfeifenmachersgesellschafft”. In the 1930’s I.M. Gelfand outlined a program of abstract harmonic analysis, which offered a paradigm for the use of symmetry to study a very wide class of mathematical problems. A key technical step is the following: Problem: For every locally compact group G, determine the set G^_u of irreducible unitary representations G. The group G is usually the symmetry group of a problem. In mathematical physics, differential geometry, or differential equations it is a real Lie group. In number theory, the group may be an algebraic group over a local fields. In combinatorics, it is often a finite group. In Gelfand’s program, G is acting (as a symmetry group) on a measure space X, preserving the measure. In this setting there is a Hilbert space H = L^2(X) of (complex-valued) square-integrable functions on X. Then G acts linearly on H by p(g)f:=f(g^{-1}v), and the fact that the action preserves the measure amounts to the fact that p(g) is unitary. The first step in Gelfand’s program is to express questions about X (related to the symmetry group G) as questions about L^2(X) (and the linear operators p(g)). Knowledge of the unitary dual G^_u is a crucial ingredient. In this talk I will explain the nature of the answer of the unitary dual of a reductive real or p-adic group, and give some examples of its uses. |
Number Theory
Representation Theory
Combinatorics