On Rossmann's character formula for discrete series

Abstract

In a recent article in this journal, W. Rossmann [3] has proved a formula conjectured by Kirillov connecting characters of the representations of semisimple Lie groups occurring in the Plancherel formula with Fourier transforms of the measures on orbits in the coadjoint representation. Rossmann's formula has considerable conceptual importance, as it established, via the orbit method, the connection between the representation theory of general Lie groups and the cabalistic study of representations of semi-simple Lie groups. I give here a simple proof of Rossmann's basic theorem, which relates the Fourier transforms on g and on a Cartan subalgebra of compact type. 1. Let V be a real vector space with a non-degenerate symmetric form B. I consider the multiplication operator m 8 given by (mB.f)(x)=B(x,x)f(x), and AB the c~176 Laplace ~176 (AB" f)(x)=(B (~--x' ~x)" f )(x)" i ing the harmonic oscillator i(mB--AB) as JB, [if V is one dimensional, JB=i [.X 2