Summary: Inventiones math. 32, 205 - 261 (1976) Inventiones
t:'~-by Springer-Verlag 1976
The Characters of Discrete Series as Orbital Integrals
James Arthur* (Bures-sur-Yvette)
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 central-
izes h may be assumed to be stable with respect to a fixed Cartan involution Q.
In other words, there is a &stable decomposition
where TI is compact and TRis a vector group. Then according to Harish-Chandra,
where 0, is the character of ca and e(T)equals 1 if T is compact and is 0 otherwise.
Implicit in this formula is the absolute convergence of the integral on the left. The
vanishing statement (the case that T is noncompact) is sometimes known as the
Selberg principle. The purpose of this paper is to establish a formula which