On the Helgason-Johnson bound

TL;DR

This paper develops a framework to refine the Helgason-Johnson bound for infinite-dimensional representations of simple non-compact Lie groups, providing explicit results for exceptional groups using advanced representation theory tools.

Contribution

It introduces a new framework to sharpen the Helgason-Johnson bound for infinite-dimensional representations, with explicit results for exceptional Lie groups.

Findings

Refined bounds for infinite-dimensional representations.

Explicit results for exceptional Lie groups.

Application of Dirac operator inequality and Vogan pencil.

Abstract

Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any irreducible unitary representation of $G$ with infinitesimal character $\Lambda$ whose continuous part is $\nu$. The beautiful Helgason-Jonson bound in 1969 says that the norm of $\nu$ is upper bounded by the norm of $\rho(G)$, which stands for the half sum of the positive roots of $G$. The current paper aims to give a framework to sharpen the Helgason-Johnson bound when $\pi$ is infinite-dimensional. We have explicit results for exceptional Lie groups. Ingredients of the proof include Parathasarathy's Dirac operator inequality, Vogan pencil, and the unitarily small convex hull introduced by Salamanca-Riba and Vogan.