Spinc-quantization and the K-multiplicities of the discrete series

Paul-Emile Paradan (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

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.

Domaines: Géométrie symplectique [math.SG]; Théorie des représentations [math.RT]

Paul-Emile Paradan. Spinc-quantization and the K-multiplicities of the discrete series. Annales Scientifiques de l'École Normale Supérieure, 2003, 36 (5), pp.805-845. ⟨10.1016/j.ansens.2003.03.001⟩. ⟨hal-00773251⟩