Characters of irreducible Harish-Chandra modules and Goldie ranks of their annihilators
AUTOR(ES)
King, Donald R.
RESUMO
Assume that GC is a simply connected complex semi-simple Lie group with Lie algebra [unk]. Let G ⊂ GC be a real form, H ⊂ G be a maximally split Cartan subgroup with Lie algebra [unk]0, [unk] = [unk]0 [unk] C, and P ⊆ [unk]* be the weight lattice. If X is an irreducible Harish-Chandra module with infinitesimal character λ ∈ [unk]*, one can associate to X a family {θ(μ): μ ∈ λ + P} of Z-linear combinations of distribution characters of G, so that θ(λ) = X. θ(μ) is irreducible when μ lies in Cλ, a certain positive “Weyl” chamber containing λ. In this case let Ann θ(μ) be its annihilator in U([unk]) and set p(μ) = Goldie rank of U([unk])/ Ann θ(μ). Let d = Gelfand-Kirillov dimension of X. For most x ∈ [unk]0 if exp tx is regular for small t > 0 then (i) c(μ) = limt→0+td θ(μ) (exp tx) exists for all μ ∈ λ + P; (ii) c(μ) extends to a homogeneous Weyl group harmonic polynomial on [unk]* of degree ½(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.
