Construction of Intertwining Operators between Holomorphic Discrete Series Representations

TL;DR

This paper explicitly constructs intertwining operators between holomorphic discrete series representations of a Lie group and its subgroup, using differential operators and series expansions, with applications to residue analysis at poles.

Contribution

It provides explicit constructions of intertwining operators for holomorphic discrete series representations in symmetric pairs, including differential and integral operators, advancing understanding of representation restrictions.

Findings

Constructed $G_1$-intertwining projection operators as differential operators.

Developed $G_1$-intertwining embedding operators as infinite-order differential operators.

Analyzed residues of intertwining operators at poles for subquotient maps.

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 $(G,G_1)=(G_0\times G_0,\Delta G_0)$ 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.