Construction of Intertwining Operators between Holomorphic Discrete Series Representations

Abstract

In this paper we explicitly construct $G_1$-intertwining operators between holomorphic discrete series representations $\mathcal{H}$ of a Lie group $G$ and those $\mathcal{H}_1$ of a subgroup $G_1\subset G$ when $(G,G_1)$ is a symmetric pair of holomorphic type. More precisely, we construct $G_1$-intertwining projection operators from $\mathcal{H}$ onto $\mathcal{H}_1$ as differential operators, in the case $(G,G_1)=(G_0\times G_0,ΔG_0)$ and both $\mathcal{H}$, $\mathcal{H}_1$ are of scalar type, and also construct $G_1$-intertwining embedding operators from $\mathcal{H}_1$ into $\mathcal{H}$ as infinite-order differential operators, in the case $G$ is simple, $\mathcal{H}$ is of scalar type,and $\mathcal{H}_1$ is multiplicity-free under a maximal compact subgroup $K_1\subset K$. In the actual computation we make use of series expansions of integral kernels and the result of Faraut-Korányi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles.