Spherical unitary representations for reductive groups

A classical problem in representation theory, motivated by abstract harmonic analysis and number theory, is the study of unitary representations of reductive algebraic groups (for example the general linear, symplectic, or orthogonal groups) defined over real and p-adic fields.

A unitary representation of a group G is a continuous homomorphism \pi from G to the group of unitary operators on a complex Hilbert space. One defines the irreducible unitary representations to be those without proper closed invariant subspaces. Of particular interest is the identification of the spherical irreducible unitary representations of G, that is, those which have nontrivial fixed vectors under the action of a maximal compact subgroup K. The main motivation for their study comes from the theory of automorphic forms.

In this talk, I will present the background for this problem, and report on joint work with D.~Barbasch.

Refreshments will be served at 12:15 p.m. (Math Lounge, MATX 1115).