Let H n be the weighted Bergman space on a bounded symmetric domain D ¼ G=K: It has analytic continuation in the weight n and for n in the so-called Wallach set H n still forms unitary irreducible (projective) representations of G: We give the irreducible decomposition of the tensor product H n1 #H n2 of the representations for any two unitary weights n and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of SLð2; CÞ: As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product H n1 #H n2 modulo the subspace of functions vanishing of certain degree on the diagonal. r

Let D ¼ G=K be a complex bounded symmetric domain of tube type in a Jordan algebra V C ; and letThe analytic continuation of the holomorphic discrete series on D forms a family of interesting representations of G: We consider the restriction on D and the branching rule under H of the scalar holomorphic representations. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group L is a spherical subgroup of K and we find a canonical basis of L-invariant polynomials in the components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transform of those L-invariant polynomials are, under the spherical transform on D; multi-variable Wilson-type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on D; when extended to a holomorphic function in a neighborhood of 0 2 D; in terms of the L-spherical holomorphic polynomials on D; the coefficients being the Wilson polynomials. # 2002 Elsevier Science (USA)

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form [Formula: see text] indexed by a complex multiparameter [Formula: see text] and defined on the space of smooth functions on the product of three spheres in 𝔽n, where 𝔽 is either ℝ, ℂ, ℍ, or 𝕆 with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν1, ν2, ν3), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein–Reznikov integral [Formula: see text] in terms of hypergeometric functions.

Let Ω be a bounded symmetric domain of non-tube type in C n with rank r and S its Shilov boundary. We consider the Poisson transform P s f (z) for a hyperfunction f on S defined by the Poisson kernel P s (z, u) = (h(z, z) n/r /|h(z, u) n/r | 2 ) s , (z, u) × Ω × S, s ∈ C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in M n,m (C) (n m), we prove that an eigenvalue equation for the second order M n,n -valued Hua operator characterizes the image.

LetD be the covariant Cauchy-Riemann operator and D the covariant holomorphic differential operator on a line bundle over a Hermitian symmetric space G/K. We study the Shimura invariant differential operators defined viaD and D. We find the eigenvalues of a family of the Shimura operators and of the generators. (1991): 32C17, 32M15, 58G30, 53A45, 05E05, 17C50 Mathematics Subject ClassificationNotation. Here is a list of main notation used in this paper.