Discrete series of simple Lie groups with admissible restriction to a $SL_2(\mathbb R)$

TL;DR

This paper classifies certain irreducible square integrable representations of simple Lie groups that, when restricted to a subgroup isomorphic to SL_2(R), remain admissible, identifying the unique holomorphic case and providing multiplicity formulas.

Contribution

It characterizes the irreducible square integrable representations with admissible restriction to SL_2(R) subgroups and determines the unique holomorphic case with explicit multiplicity formulas.

Findings

Representation is holomorphic when admissible

Identifies the unique subgroup with this property

Provides multiplicity formulas for the representations

Abstract

In this note we determine the irreducible square integrable representations of a simple group which admits an admissible restriction to a subgroup $H$ locally isomorphic to $SL_2(\mathbb R).$ We show such representation is holomorphic and we determine the essentially unique $H$ with this property as well as multiplicity formulae.