  The goal of this article is to study the branching problem for a holomorphic
discrete series representation of the conformal group of a simple Euclidean
Jordan algebra $V$ restricted to the subgroup
$\operatorname{PSL}_2(\mathbb{R})\times\operatorname{Aut}(V)$ where
$\operatorname{Aut}(V)$ denotes the compact group of automorphisms of $V$. We
use a realization of the holomorphic discrete series on a space of
vector-values $L^2$-functions as well as the stratified model developed by the
second author to relate the branching problem to the decomposition of certain
representations of the compact group $\operatorname{Aut}(V)$ and to
vector-valued orthogonal polynomials.
