Unitary representations of simple Lie groups

Unitary representations of simple Lie groups

-
David Vogan, MIT
Fine Hall 314

By 1950, work of Gelfand and others had led to a general program for "non-commutative harmonic analysis": understanding very general mathematical problems (particularly of geometry or analysis) in the presence of a (non-commutative) symmetry group G. A first step in that program is classification of unitary representations - that is, the realizations of G as automorphisms of a Hilbert space. Despite tremendous advances from the work of Harish-Chandra, Langlands, and others, completing this first step is still some distance away. Since functional analysis is not as fashionable now as it was in 1950, I'll explain some of the ways that Gelfand's problem can be related to algebraic geometry (particularly to equivariant K-theory). I'll also discuss the (closely related) question of whether computers may be able to help solve these problems.