Canonical Hilbertian realization of Harish Chandra modules .

Joseph Bernstein, Tel Aviv University

Abstract

Let G be a real reductive group. K ⊂ G a maximal compact subgroup, g -the Lie algebra of G.

It is well known that a topological admissible representation (π, V) of G can be well described by its space of K-finite vectors M which is a Harish Chandra module, i.e. a (g, K) - module satisfying some finiteness conditions.

Conversely, suppose we are given a Harish Chandra module M. It was shown by W. Casselman and N. Wallach that M can be uniquely realized by a smooth representation (π, V).

In my talk I will explain how to construct a canonical realization of M as a Hilbertian representation R(M). We will also see why this realization functor R is exact.

I will also discuss other possible Hilbertian structures (Sobolev norms) on the module M.