Invariant quantization in one and two parameters on semisimple coadjoint orbits 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. We study one and two parameter quantizations of A, Ah, At,h, such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, Uh(g). In particular, the algebra At,h specializes at h = 0 to a U(g), or G, invariant quantization, At,0. We prove that the Poisson bracket corresponding to Ah 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 dimH2(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, At,h, corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the KirillovKostant-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.