The $L_p$-dual space of a semisimple Lie group

TL;DR

This paper classifies irreducible representations of semisimple Lie groups on Lp-spaces for p not equal to 2, linking them to natural actions on quotient spaces and providing a complete classification in the real rank one case.

Contribution

It characterizes all irreducible Lp-space representations of semisimple Lie groups, connecting them to parabolic subgroups and natural quotient actions, with a full classification for real rank one groups.

Findings

Representations correspond to actions on G/Q spaces twisted by characters.

Complete classification of irreducible representations for real rank one groups.

Identification of the structure of all such representations for p ≠ 2.

Abstract

Let $G$ be a semisimple Lie group. We describe the irreducible representations of $G$ by linear isometries on $L_p$-spaces for $p\in (1,+\infty)$ with $p\neq 2.$ More precisely, we show that, for every such representation $\pi,$ there exists a parabolic subgroup $Q$ of $G$ such that $\pi$ is equivalent to the natural representation of $G$ on $L_p(G/Q)$ twisted by a unitary character of $Q.$ When $G$ is of real rank one, we give a complete classification of the possible irreducible representations of $G$ on an $L_p$-space for $p\neq 2,$ up to equivalence.