  Let $G$ be a reductive Lie group, $\g$ its Lie algebra, and $M$ a
$G$-manifold. Suppose $\A_h(M)$ is a $\U_h(\g)$-equivariant quantization of the
function algebra $\A(M)$ on $M$. We develop a method of building
$\U_h(\g)$-equivariant quantization on $G$-orbits in $M$ as quotients of
$\A_h(M)$. We are concerned with those quantizations that may be simultaneously
represented as subalgebras in $\U^*_h(\g)$ and quotients of $\A_h(M)$. It turns
out that they are in one-to-one correspondence with characters of the algebra
$\A_h(M)$. We specialize our approach to the situation $\g=gl(n,\C)$,
$M=\End(\C^n)$, and $\A_h(M)$ the so-called reflection equation algebra
associated with the representation of $\U_h(\g)$ on $\C^n$. For this particular
case, we present in an explicit form all possible quantizations of this type;
they cover symmetric and bisymmetric orbits. We build a two-parameter
deformation family and obtain, as a limit case, the $\U(\g)$-equivariant
quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits.
