Back to Departmental Colloquium: Spring 2000
Departmental Colloquium
Date: Thursday, Feb 24, 2000
Time: 4:15PM
Location: JWB 335
David Vogan
MIT
Title |
Three-dimensional subgroups and unitary representations |
Abstract |
The simplest noncommutative compact Lie group is SU(2), the group of unit quaternions. If G* is any compact Lie group, a natural problem is to understand the set D(G*) of conjugacy classes of homomorphisms of SU(2) into G*. Dynkin showed in the 1950s that D(G*) is a finite set, and calculated it in all cases. Suppose now that k is a local field. The structure theory of reductive groups attaches to G* and k a split reductive group G. The compact group G* is a “Langlands dual” of G. A fundamental unsolved problem in abstract harmonic analysis is to parametrize the “purely real” unramified unitary representations G. It’s known that such representations are parametrized by a compact polytope P(G,k). It turns out that the polytope depends very little on the field k ; for example, it’s the same for all p -adic fields k (for a fixed G* ). A conjecture of Arthur realizes D(G*) as a subset of P(G,k) ; these finitely many special points seem to control most of the geometry of the polytope P(G,k). I’ll discuss how the two problems illuminate each other. |
Number Theory
Representation Theory
Commutative Algebra