Kirillov's formula and Guillemin-Sternberg conjecture

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