Characters of irreducible Harish-Chandra modules and Goldie ranks of their annihilators

Abstract

Assume that G(C) is a simply connected complex semi-simple Lie group with Lie algebra [unk]. Let G subset G(C) be a real form, H subset G be a maximally split Cartan subgroup with Lie algebra [unk](0), [unk] = [unk](0) [unk] C, and P subset, dbl equals [unk](*) be the weight lattice. If X is an irreducible Harish-Chandra module with infinitesimal character lambda in [unk](*), one can associate to X a family {theta(mu): mu in lambda + P} of Z-linear combinations of distribution characters of G, so that theta(lambda) = X. theta(mu) is irreducible when mu lies in C(lambda), a certain positive "Weyl" chamber containing lambda. In this case let Ann theta(mu) be its annihilator in U([unk]) and set p(mu) = Goldie rank of U([unk])/ Ann theta(mu). Let d = Gelfand-Kirillov dimension of X. For most x in [unk](0) if exp tx is regular for small t > 0 then (i) c(mu) = lim(t-->0(+) )t(d) theta(mu) (exp tx) exists for all mu in lambda + P; (ii) c(mu) extends to a homogeneous Weyl group harmonic polynomial on [unk](*) of degree (1/2)(dim G - dim H) - d; (iii) up to a constant, c = the polynomial extending p to [unk](*). c is said to be the character polynomial of Ann X.