PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 8, August 2007, Pages 2567–2571 S 0002-9939(07)08781-3 Article electronically published on March 22, 2007 MATRIX COEFFICIENTS AND COADJOINT ORBITS OF COMPACT LIE GROUPS A. H. DOOLEY AND R. W. RAFFOUL (Communicated by Jane M. Hawkins) Abstract. Let G be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the j -function and the pull-back of an arbitrary unitary irreducible repre- sentation of G to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups. 1. Introduction Let G be a tame unimodular Lie group, g its Lie algebra and g ∗ the vector space dual of g. The group G operates on g ∗ by the coadjoint action yielding, according to the Kirillov-Kostant philosophy, a consequent parametrization of the unitary irreducible representations of G by orbits satisfying a certain integrality condition, equivalent to the following character formula: let π be a finite-dimensional unitary irreducible representation of G related to the coadjoint orbit O – for almost all π in the reduced dual of G, as an equality of distributions, (1.1) j (X)Tr π(exp(X)) =  O e iη(X) dμ O (η) for all X ∈ g in a sufficiently small neighbourhood of 0, where μ O is a Liouville measure on O and the G-invariant function j is the analytic square root of the jacobian of the exponential map, j (0) = 1. When G is compact, (1.1) follows from the Weyl character formula and a well- known result [7, Theorem 2] of Harish-Chandra (see [15] or [12] for an exposition). It holds as an identity over all of g, and a unitary irreducible representation of highest weight λ corresponds to the orbit though λ + δ, where δ is half the sum of the positive roots. This was originally proved by Kirillov in [10], where the validityof formula (1.1) for nilpotent groups was also demonstrated and its universality conjectured. It has since been verified for many other classes of Lie groups, notable contributions being [16] and [9]. Received by the editors April 18, 2006. 2000 Mathematics Subject Classification. Primary 43A77, 22E99; Secondary 47Nxx. Key words and phrases. Coadjoint orbits, Lie groups, matrix coefficients, moment map, Weyl functional calculus. The authors gratefully acknowledge the support of the Australian Research Council. c 2007 American Mathematical Society 2567 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use