On the procedure of multidimensional quantization (Q750977)

Let G be a simply connected Lie group with Lie algebra L. Let P denote an orbit of G in \(L^*\) (dual of L) under the coadjoint action of G on \(L^*\). Kirillov showed that P has a natural structure of a G- homogeneous symplectic manifold and studied representations of G on the space of sections of complex line bundles over P. Kirillov's method has come to be known as the orbit method. This method has been extended in many directions and has led to the Kostant-Kirillov theory of geometric quantization. The paper under review discusses a new procedure of quantization, called multi-dimensional quantization. Its starting point is an irreducible G- bundle associated with a given Hamiltonian system with symmetry group G. The authors use the procedure to obtain representations of semi-simple and reductive Lie groups by a modification of the usual construction of holomorphically induced representations.