Computing Kazhdan-Lusztig polynomials for real Lie groups

where k (p) is the +1 (−1) eigenspace of θ on g. Let Z(g) be the center of the universal enveloping algebra of g, and fix a character χ of Z(g). Let HC be the category of Harish-Chandra modules for G (i.e. finitely generated (g,K)–modules), and let HCχ be the full subcategory of HC of the modules with generalized central character χ. We are interested in the distribution characters, say, of the irreducible modules in HCχ. These are of course a basis of the Grothendieck group of HCχ. Now there is another basis D of this Grothendieck group, which we may consider known, made up by the characters of standard representations (parabolically induced from discrete series on cuspidal parabolic subgroups.) To each γ ∈ D, Langlands assigns a specific irreducible subquotient γ; this establishes a bijection from D to the irreducibles in HCχ. If we denote m(γ, δ) the multiplicity of γ in δ, we can write