Résumé
In the 70s, W. Schmid has shown that the representations of the discrete series of a real semi-simple Lie group G could be realized as the quantization of elliptic coadjoint orbits. In this paper we show that such orbits, equipped with the Hamiltonian action of a maximal compact subgroup K of G, are non-compact examples where the philosophy of Guillemin-Sternberg "Quantization commutes with reduction" applies. If H(O) is a representation of the discrete series of G associated to a coadjoint orbit O, we express the K-multiplicities of H(O) in terms of Spinc-index on symplectic reductions of O.