Strict quantization of coadjoint orbits

Abstract

For every semisimple coadjoint orbit Oy of a complex connected semisimple Lie group Gy, we obtain a family of G-invariant products *h„ on the space of holomorphic functions on Oy. For every semisimple coadjoint orbit O of a real connected semisimple Lie group G, we obtain a family of G-invariant products *h on a space A.O/of certain analytic functions on O by restriction. A.O/, endowed with one of the products *h„, is a G-Fréchet algebra, and the formal expansion of the products around h = 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.