Restriction of some representations of U(P,Q) to a symmetric subgroup

We consider the restriction of unitary representations A(q) of U(p, q) with nontrivial (g, K)-cohomology which are cohomologically induced from a theta-stable parabolic subalgebra q with Levi subgroup L = U(1)U-s(p - s, q) to the symmetric subgroups U(p, q - r)U(r) and U(r)U(p - r, q). By considering a generalized bottom layer map with respect to non-compact subgroups we show that the restriction of A(q) to a symmetric subgroup U(p, q - r)U(r) is a direct sum of irreducible representations each with finite multiplicity. For U(p, p) we find examples of representations A(q) and of subgroups H-1 and H-2 which are isomorphic but not conjugate, so that the restriction of A(q) to one subgroup is a direct sum, whereas the restriction to the the other subgroup has continuous spectrum. In the case r=1 we obtain information about the branching law in section 4. We also relate in section 5. harmonic forms with coefficients in A(q) to harmonic forms with coefficients in the U(p, q - 1)U(1) submodules generated by the minimal K-type.