GEOMETRIC-QUANTIZATION AND MULTIPLICITIES

Let G be a compact Lie group acting on a compact symplectic manifold M in a Hamiltonian way. We suppose M prequantized and let C be a Kostant-Souriau line bundle on M. Let Q (M, L) be the quantized space of M. This representation of G can be realized as the equivariant index of the Dirac operator D(L). If G is a torus, I prove a conjecture of Guillemin-Sternberg on the multiplicities Of Q (M, L).