Kirillov's character formula for reductive Lie groups

Abstract

Inventiones math. 48, 2007-220 (1978); on-line version. Kirillov’s famous formula says that the characters X of the irreducible unitary representations of a Lie group G should be given by an equation of the form (Φ) χ(exp x )= p(x) −1 Ω e i(λ,x) dµΩ(λ) where ω =Ω (X )i s aG-orbit in the dual g ∗ ft he Lie algebrag of G, µΩ is Kirillov’s canonical measure on Ω, and p is a certain function on g ,n amely p(x )= det 1/2 {sinh(ad(x/2)) /ad(x/2)} at least for generic orbits Ω [10]. This formula cannot be taken too literally, of course (the integral in (Φ) is usually divergent), but has to be interpreted as an equation of distributions on a certain space of test functions on g. To make this precise, denote by g o an open neighborhoodod of zero in g so that exp : g → G restricts to an invertible analytic map of g o onto an open subset of G. For our purposes, the formula (Φ) should be interpreted as saying that (Φ � )t r g ϕ(x)π(exp(x)) dx = Ω g e i(λ,x) ϕ(x) p(x) −1 } dµΩ(λ) for all C ∞ functions ϕ with compact support in g o . (Here π is the representation of G with character χ.) Of course, Kirillov’s formula does not hold in this generality. It is in fact a major problem in representation theory to determine its exact domain of validity. In this paper we shall show that Kirillov’s formula holds for the characters of a reductive real Lie group which occur in the Plancherel formula. Actually, we shall deal in detail only with the discrete series characters. The formula for the other characters can then be reduced to the formula for the discrete series characters by familiar methods. (Duflo [3]). Kirillov’s formula for the discrete series is a consequence of a formula relating the Fourier transform on g with the Fourier transform on Cartan subalgebras of compact type by means of the invariant integral. This is the form in which Kirillov’s formula will be proved.