  We emphasize some properties of coherent state groups, i.e. groups whose
quotient with the stationary groups, are manifolds which admit a holomorphic
embedding in a projective Hilbert space. We determine the differential action
of the generators of the representation of coherent state groups on the
symmetric Fock space attached to the dual of the Hilbert space of the
representation. This permits a realization by first-order differential
operators with holomorphic polynomial coefficients on K\"ahler coherent state
orbits.

Differential Geometry