Double quantization on coadjoint representations of simple Lie groups and its orbits

Let M be a manifold with an action of a Lie group G, A the function algebra on M . The first problem we consider is to construct a Uh(g) invariant quantization, Ah, of A, where Uh(g) is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M . The second problem we consider is to construct a Uh(g) invariant two parameter (double) quantization, At,h, of A such that At,0 is a G invariant quantization of s. We call At,h a Uh(g) invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple Lie group and M is the coadjoint representation g∗ of G or a semisimple orbit in this representation. First of all, we describe Poisson brackets and pairs of Poisson brackets related to Uh(g) invariant quantizations for arbitrary algebras. After that we construct a two parameter quantization on g∗ for g = sl(n) and s the Lie bracket and show that such a quantization does not exist for other simple Lie algebras. As the function algebra on g∗ we take the symmetric algebra Sg. In sl(n) case, we also consider the problem of restriction of the family (Sg)t,h on orbits. In particular, we describe explicitly the Poisson bracket along the parameter h of this family, which turns out to be quadratic, and prove that it can be restricted on each orbit in g∗. We prove also that the family (Sg)t,h can be restricted on the maximal semisimple orbits. For M a manifold isomorphic to a semisimple orbit in g∗, we describe the variety of all brackets related to the one parameter quantization. Actually, it is a variety making M into a Poisson manifold with a Poisson action of G. It turns out that not all such brackets and not all orbits admit a double quantization with s the KirillovKostant-Souriau bracket. We classify the orbits and pairs of brackets admitting a double quantization and construct such a quantization for almost all admissible paires.