Ladder representation norms for hermitian symmetric groups

Let G be a connected noncompact simple Hermitian symmetric group with finite center. Let H(lambda) denote the geometric realization of an irreducible unitary highest weight representation with highest weight lambda. Then H(lambda) consists of vector-valued holomorphic functions on G/K and the action of G on H(lambda) is given in terms of a factor of automorphy. For highest weights lambda corresponding to ladder representations, we obtain the G-invariant inner product on H(lambda). This inner product arises as the pullback of an isometry Phi(lambda) : H(lambda) --> H(lambda) x Y-lambda, where Y-lambda is finite dimensional and the weight <(lambda)over tilde> corresponds to a scalar valued representation. In all but finitely many cases the G-invariant inner product on H(<(lambda)over tilde>) is known and is used to express the G-invariant inner product on H(lambda). Explicit examples are given for families of ladder representations of SU(p, q) and SO*(2n). Finally, inversion formulas for unitary intertwining operators between H(lambda) and any equivalent realization are exhibited.