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, $\text{A}$ the function algebra on M. The first problem we consider is to construct a $U_h(\text{g})$ invariant quantization, ${\text{A}}_h$, of $\text{A}$, where $U_h(\text{g})$ 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 $U_h(\text{g})$ invariant two parameter (double) quantization, ${\text{A}}_{t,h}$, of $\text{A}$ such that ${\text{A}}_{t,0}$ is a G invariant quantization of $s$. We call ${\text{A}}_{t,h}$ a $U_h(\text{g})$ 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 ${\text{g}}^{\ast }$ of G or a semisimple orbit in this representation. The paper is founded on the papers: J.Donin, Double quantization on the coadjoint representation of $sl(n{)}^{\ast }$, Czechoslovak J. of Physics, 47 (1997), no 11, 1115-1122, q-alg/9707031, and J.Donin, D.Gurevich, and S.Shnider, Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups, Com. Math. Phys., 204 (1999), no. 1, 39-60, math/9807159, and contains some additional results.