Geometry and smooth representations of reductive Lie groups
by
Dragan Milicic
University of Utah, Salt Lake City, Utah, USA
Coauthors: Bill Casselman and Henryk Hecht

The standard way of studying representations of reductive Lie groups is to study the corresponding category of Harish-Chandra modules. This method reduces the problem to an essentially algebraic problem. In my lecture at the last Congress, I discussed how this problem can be transferred to a problem in D-module theory, where powerful methods from algebraic geometry can be used. A feature of this approach, which is sometimes unpleasant, is that Harish-Chandra modules do not admit an action of the reductive Lie group G itself (in fact, they are modules for the action of the enveloping algebra of the Lie algebra of the group G and the action of its maximal compact subgroup K).

Casselman and Wallach defined a category of ``smooth representations of moderate growth and finite length'' of G and constructed a ``completion'' functor from the category of Harish-Chandra modules into this category. They also established that this functor is an equivalence of categories. Its quasi-inverse is the functor which attaches to each smooth representation the dense Harish-Chandra module of its K-finite vectors. Therefore, smooth representations are just ``Harish-Chandra modules in disguise'', but with an action of the group G itself.

This allows us to apply to smooth representations some ideas inspired by the arguments used in the representation theory of p-adic reductive groups. In this lecture, I'll discuss the construction of the Bruhat filtration on smooth principal series representations. This is a natural filtration attached to the Bruhat cell stratification of the real flag variety. Using some powerful results from analytic geometry, one can describe completely the graded object attached to this filtration. Using standard techniques from homological algebra this leads to a number of interesting results on smooth representations. Among others, I'll discuss Kostant's results on existence and uniqueness of Whittaker vectors and Wallach's results on holomorphic continuation of Jacquet integrals.

Date received: May 17, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-85.