  Deformation quantization and geometric quantization on K\"ahler manifolds
give the mathematical description of the algebra of quantum observables and the
Hilbert spaces respectively, where the later forms a representation of quantum
observables asymptotically via Toeplitz operators. When there is a Hamiltonian
$G$-action on a K\"ahler manifold, there are associated symmetries on both the
quantum algebra and representation aspects. We show that in nice cases of
coadjoint orbits and K\"ahler-Einstein manifolds, these symmetries are strictly
compatible (not only asymptotically).
