Strict quantization of coadjoint orbits

For every semisimple coadjoint orbit (O) over cap of a complex connected semisimple Lie group (G) over cap we obtain a family of (G) over cap -invariant products (*) over cap ((h) over bar )on the space of holomorphic functions on (O) over cap. For every semisimple coadjoint orbit O of a real connected semisimple Lie group G, we obtain a family of G-invariant products *((h) over bar ) on a space A(O) of certain analytic functions on O by restriction. A(O), endowed with one of the products *((h) over bar ), is a G-Frechet algebra, and the formal expansion of the products around (h) over bar = 0 determines a formal deformation quantization of O, which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.