Computing Global Characters

Let π be an irreducible representation of a real reductive group G. The global character θπ of π may be considered as a function on the regular semisimple elements of G. The global character determines π, and it is of great interest to compute it. For example Harish-Chandra found the discrete series representations of G by computing their characters. Fix a Cartan subgroup H of G and let D be the Weyl denominator. Let λ be the infinitesimal character of π. The function θπ restricted to the regular elements of H is roughly of the form