Branching laws for square integrable representations

TL;DR

The paper surveys branching laws for square-integrable representations of a semisimple group $G$ when restricted to a closed reductive subgroup $H$, using direct-integral and reproducing-kernel models. It shows that discrete-series representations realized as $H^2(G,\tau)_\lambda$ decompose into $H$-factors that are tempered, with $H$-discrete factors again square-integrable, and that intertwiners between these spaces can be realized as integral operators or differential operators under natural conditions. It provides practical criteria for discrete decomposability: a restriction is $H$-discretely decomposable if there exists a discrete-series $H^2(H,\sigma)_\mu$ and a nonzero differential-operator intertwiner from $V$ to $H^2(H,\sigma)_\mu$, with finiteness of multiplicities when all intertwiners are differential operators; a spherical-function criterion via the lowest $K$-type, with $\Phi$ being $\mathfrak z_{\mathfrak h}$-finite, completes the picture. In symmetric-pair settings, the paper aligns with complete classifications by Kobayashi–Oshima–Vargas and extends Harish-Chandra-type analysis to restriction problems, offering tools for explicit branching laws in representation theory and harmonic analysis.

Abstract

We present an overview of results on branching laws for square integrable representations of a semisimple Lie group, restricted to a closed reductive subgroup. The overview is partial and it is based on joint work with Bent Ørsted and the deep work of Toshiyuki Kobayashi.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Theorem 3.4