Monday, April 30th, 2007
2:30pm - 3:55pm, in Science 2-065

Steven Jackson

UMass Boston

Computing 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.