  We consider the problem of existence of the diagonal representation for
operators in the space of a family of generalized coherent states associated
with an unitary irreducible representation of a (compact) Lie group. We show
that necessary and sufficient conditions for the possibility of such a
representation can be obtained by combining Clebsch-Gordan theory and the
reciprocity theorems associated with induced unitary group representation.
  Applications to several examples involving $SU(2),$ $SU(3),$ and the
Heisenberg-Weyl group are presented, showing that there are simple examples of
generalized coherent states which do not meet these conditions. Our results are
relevant for phase-space description of quantum mechanics and quantum state
reconstruction problems.
