For every semisimple coadjoint orbit O^ of a complex connected semisimple Lie group G^, we obtain a family of G^-invariant products ∗^ℏ on the space of holomorphic functions on O^. For every semisimple coadjoint orbit O of a real connected semisimple Lie group G, we obtain a family of G-invariant products ∗ℏ on a space A(O) of certain analytic functions on O by restriction. A(O), endowed with one of the products ∗ℏ, is a G-Fréchet algebra, and the formal expansion of the products around ℏ=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.