  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 $U_h(\g)$ invariant
quantization, $\A_h$, of $\A$, where $U_h(\g)$ 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 $U_h(\g)$ invariant two parameter (double) quantization,
$\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$.
We call $\A_{t,h}$ a $U_h(\g)$ 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.
  The paper is founded on the papers: J.Donin, Double quantization on the
coadjoint representation of $sl(n)^*$, 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.
