Branching laws, some results and new examples

TL;DR

This paper investigates the restriction behavior of square integrable representations of certain noncompact Lie groups to specific subgroups, providing formulas for multiplicities and conditions for admissibility in the context of Hermitian manifolds.

Contribution

It offers new formulas for multiplicities and criteria for admissibility of representations when restricted to subgroups containing a three-dimensional ideal.

Findings

Derived a formula for the multiplicity of irreducible factors.

Established necessary and sufficient conditions for admissibility over semisimple factors.

Analyzed the restriction of representations in the context of Hermitian G-manifolds.

Abstract

For a connected, noncompact matrix simple Lie group $G$ so that a maximal compact subgroup $K$ has three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contains the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that a square integrable representations of the ambient group is admissible over the semisimple factor of $K.$