Abstract
Let G be a semisimple Lie group. Geometric quantization is a machinery which transforms a symplectic G-manifold X to a unitary G-representation . Let C be a Cartan subgroup of G, and L the stabilizer of an element in the Lie algebra of C. Let , where L ss is the commutator subgroup of L, and ) is the Lie algebra of the centralizer of L in C. When G is split, we perform geometric quantization to G × H-invariant symplectic forms on X. As a result, we construct a regular principal model in the sense that every regular principal series representation of G occurs once in . We also perform symplectic reduction to X and show that "quantization commutes with reduction". © Walter de Gruyter.