Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II

Let (G,G₁) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D₁=G₁/K₁⊂ D=G/K, realized as bounded symmetric domains in complex vector spaces p⁺₁⁺ respectively. Then the universal covering group G of G acts unitarily on the weighted Bergman space H_λ(D)(D) on D. Its restriction to the subgroup G₁ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's formula in terms of the K₁-decomposition of the space P(p⁺₂) of polynomials on the orthogonal complement p⁺₂ of p⁺₁ in p⁺. The object of this article is to compute explicitly the inner product f(x₂),e^(x|z̄)p⁺_λ for f(x₂)ₖ(p⁺₂)(p⁺₂), x=(x₁,x₂),z⁺=p⁺₁⁺₂, and to construct explicitly G₁-intertwining operators (symmetry breaking operators) H_λ(D)|_G₁ε₁λ(D₁,Pₖ(p⁺₂)) from holomorphic discrete series representations of G to those of G₁, which are unique up to constant multiple for sufficiently large λ. In this article, we treat the case p⁺,p⁺₂ are both simple of tube type and rankp⁺=rankp⁺₂. When rankp⁺=3, we treat all partitions k, and when rankp⁺ is general, we treat partitions of the form k=(k,…,k,k-l).