Branching laws for discrete Wallach points

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.