Double quantization on some orbits in the coadjoint representations of simple Lie groups

Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g-fraktur sign. We study one and two parameter quantizations A_h and A_t,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, U_h(g-fraktur sign). In particular, the algebra A_t,h specializes at h = 0 to a U(g-fraktur sign)-invariant (G-invariant) quantization, A_t,0. We prove that the Poisson bracket corresponding to A_h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, A_t,h, corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.