2008-04-25

Steve Jackson, University of Massachusetts (Boston)

A Molien series for the centralizer of a maximal compact subgroup

Let (G,K) be the complex symmetric pair associated with a real reductive Lie group G_0, and let U(g)^K denote the centralizer of K in the universal enveloping algebra U(g). By a theorem of Harish-Chandra, an irreducible (g,K)-module is determined up to infinitesimal equivalence by the action of U(g)^K on any K-primary component. For this reason, the problem of determining generators for U(g)^K has been considered by several authors, but complete results have been obtained only for G_0 = SU(2,2) and the families G_0 = SU(n,1) and G_0 = SO(n,1). In 2006, Kostant proved that U(g)^K is generated by elements of filtration degree (2 dim g \choose 2) dim p, reducing the general problem to a finite but computationally difficult algorithm. In this talk I will discuss a method by which Kostant's algorithm can be significantly accelerated by exploiting the Kostant-Rallis theorem via a certain homomorphism from U(g)^K to the ring of regular functions on the nilpotent cone in p. The situation is analogous to that in the invariant theory of finite groups, where the Molien series is used to accelerate the algorithm suggested by Noether's degree bound.