College of Science and Mathematics
Department of Mathematics
Mathematics Seminar Series - Spring 2007
Monday,
April 30
2:30pm-3:55pm, Science 3-065
2:30pm-3:55pm, Science 3-065
Steven Jackson
UMass BostonComputing generators for the centralizer of a maximal compact subgroup in the universal enveloping algebra
Abstract:
Let G be a real reductive Lie group with Lie algebra
g, and let K be a maximal compact subgroup of G.
Denote the centralizer of K in the universal enveloping algebra of
g by U(g)^K. Harish-Chandra's original approach to representation
theory centered on the observation that an irreducible admissible
(g,K)-module is determined up to infinitesimal equivalence by the action
of U(g)^K on any K-primary component. Despite its conceptual
simplicity, Harish-Chandra abandoned this approach due to the difficulty
of computing generators for U(g)^K. In this talk we discuss how to use algebraic
geometry, prehomogeneous varieties, finite group invariants, integer programming,
and Grobner bases to compute generators and relations for this "hideously complicated"
object. This is joint work with Alfred Noel.
The presentations cover a large variety of topics and are intended for a general math audience. The seminar is organized by Prof. Alfred Noël and we usually meet Monday afternoons, from 2:30 pm to 4:00 pm.
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