A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

Abstract

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group [Formula: see text] of a tube domain [Formula: see text] over a symmetric cone [Formula: see text] restricted to the conformal group [Formula: see text] of a tube domain [Formula: see text] holomorphically embedded in [Formula: see text]. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the [Formula: see text]-fold tensor product of holomorphic discrete series of the universal covering of [Formula: see text]. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group [Formula: see text] to the subgroup [Formula: see text], and finally to the subgroup [Formula: see text].