Inventiones Mathematicae

Suppose that G is a Lie group. which for the purpose of this introduction, we take to be a real form of a simply connected complex semisimple group. Suppose that square integrable representations for G exist and that f is a matrix coefficient of a square integrable representation belonging to the unitary equivalence class co. Harish-Chandra has shown how to evaluate the integral off with respect to the G-invariant measure on any regular semisimple conjugacy class. In fact suppose that h is a regular semisimple element of G. The Cartan subgroup T which centralizes h may be assumed to be stable with respect to a fixed Cartan involution Q. In other words, there is a &stable decomposition