Functional calculus and coadjoint orbits.

Abstract

Let G be a compact Lie group and let π be an irreducible representation of G of highest weight λ. We study the operator-valued Fourier transform of the product of the j-function and the pull-back of ð by the exponential mapping. We show that the set of extremal points of the convex hull of the support of this distribution is the coadjoint orbit through ë + ä. The singular support is furthermore the union of the coadjoint orbits through ë + wä, as w runs through the Weyl group.

Our methods involve the Weyl functional calculus for noncommuting operators, the Nelson algebra of operants and the geometry of the moment set for a Lie group representation. In particular, we re-obtain the Kirillov-Duflo correspondence for compact Lie groups, independently of character formulae. We also develop a "noncommutative" version of the Kirillov character formula, valid for noncentral trigonometric polynomials. This generalises work of Cazzaniga, 1992.