 
Summary: Inventiones math. 32, 205  261 (1976) Inventiones
mathematicue
t:'~by SpringerVerlag 1976
The Characters of Discrete Series as Orbital Integrals
James Arthur* (BuressurYvette)
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.
HarishChandra has shown how to evaluate the integral off with respect to the
Ginvariant 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 HarishChandra,
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
generalizes (1).
