Finite multiplicity theorems for induction and restriction

TL;DR

This paper establishes bounds and criteria for the multiplicities of irreducible representations in induced and restricted representations of semisimple Lie groups, linking geometric structures to representation theory properties.

Contribution

It provides new bounds and geometric criteria for the multiplicities of irreducible representations in induction and restriction, highlighting independence from real forms.

Findings

Bounds for multiplicities of irreducible representations

Geometric criteria for finiteness of Hom spaces

Uniform boundedness characterized by complex flag variety

Abstract

We find upper and lower bounds of the multiplicities of irreducible admissible representations $\pi$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^G\tau$ from irreducible representations $\tau$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Hom_G(\pi,Ind_H^G \tau)$ (induction) and of $Hom_H(\pi|_H,\tau)$ (restriction) by means of the real flag variety $G/P$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.