On the Helgason-Johnson bound

Abstract

Let $G$ be a simple non-compact linear Lie group. Let $π$ be any irreducible unitary representation of $G$ with infinitesimal character $Λ$ whose continuous part is $ν$. The beautiful Helgason-Jonson bound in 1969 says that the norm of $ν$ is upper bounded by the norm of $ρ(G)$, 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 $π$ 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.