Abstract. The Kirillov-Duflo orbit correspondance for compact Lie groups is parametrisation of the unitary dual, associating to the irreducible representation of highest weight λ the coadjoint orbit through λ + δ, where δ is half the sum of the positive roots and justified by the character formulae of Weyl or Kirillov. In this paper we obtain this correspondence independently of character theory, showing that it arises from a convexity property of the Weyl functional calculus of the infinitesimal generators of the representation.

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 representation 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.