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Let G be a connected reductive real Lie group, and H a compact connectedsubgroup. Harish-Chandra associates to a regular coadjoint admissible orbit Mof G some unitary representations of G. Using the character formula for theserepresentations, we show that the multiplicities of the restriction to H can becomputed, under a suitable properness assumption, in terms of the fibers of therestriction map from M to the dual of the Lie algebra of H. In particular, thisgives an alternate proof of a result of Paradan.

Kirillov's formula and Guillemin-Sternberg conjecture