  Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair
of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded
symmetric domains in complex vector spaces
$\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal
covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman
space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to
the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely,
and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's
formula in terms of the $K_1$-decomposition of the space
$\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement
$\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of
this article is to compute explicitly the inner product $\bigl\langle
f(x_2),e^{(x|\overline{z})_{\mathfrak{p}^+}}\bigr\rangle_\lambda$ for
$f(x_2)\in\mathcal{P}_{\mathbf{k}}(\mathfrak{p}^+_2)\subset\mathcal{P}(\mathfrak{p}^+_2)$,
$x=(x_1,x_2),z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$, and to
construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry
breaking operators)
$\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_{\varepsilon_1\lambda}(D_1,\mathcal{P}_{\mathbf{k}}(\mathfrak{p}^+_2))$
from holomorphic discrete series representations of $\widetilde{G}$ to those of
$\widetilde{G}_1$, which are unique up to constant multiple for sufficiently
large $\lambda$. In this article, we treat the case
$\mathfrak{p}^+,\mathfrak{p}^+_2$ are both simple of tube type and
$\operatorname{rank}\mathfrak{p}^+=\operatorname{rank}\mathfrak{p}^+_2$. When
$\operatorname{rank}\mathfrak{p}^+=3$, we treat all partitions $\mathbf{k}$,
and when $\operatorname{rank}\mathfrak{p}^+$ is general, we treat partitions of
the form $\mathbf{k}=(k,\ldots,k,k-l)$.
