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

Let a be the function algebra on a semisimple orbit, hi, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g. 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 (8). In particular, the algebra A(t, h) specializes at h = 0 to a U(g)-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 H-2(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.