  We study a quantum moment map and propose an invariant for $G$-invariant star
products on a $G$-transitive symplectic manifold. We start by describing a new
method to construct a quantum moment map for $G$-invariant star products of
Fedosov type. We use it to obtain an invariant that is invariant under
$G$-equivalence. In the last section we give two simple examples of such
invariants, which involve non-classical terms and provide new insights into the
classification of $G$-invariant star products.
