Branching laws for discrete Wallach points

TL;DR

This paper studies how certain holomorphic discrete series representations of automorphism groups of symmetric tube domains decompose when restricted to a subgroup, using Riesz distributions and explicit integral operators.

Contribution

It provides an explicit decomposition formula for these representations at Wallach points, employing analytic continuation and special functions.

Findings

Explicit decomposition formulas for Wallach points

Construction of intertwining operators via Riesz distributions

Description of Plancherel measure involving gamma functions

Abstract

We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain $V\oplus i\Omega$ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of $GL(\Omega)$. Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density for the Plancherel measure involves quotients of $\Gamma$-functions and the $c$-function for a symmetric cone of smaller rank.