Proper affine actions in non-swinging representations

TL;DR

This paper establishes criteria for certain affine actions of semisimple Lie groups on vector spaces, ensuring the actions are free, properly discontinuous, and have Zariski-dense linear parts, expanding understanding of affine group actions.

Contribution

It provides a new sufficient criterion for the existence of free, properly discontinuous affine actions with Zariski-dense linear parts for semisimple Lie groups in non-swinging representations.

Findings

Criteria for affine actions with Zariski-dense linear parts

Existence of free, nonabelian, properly discontinuous actions

Application to non-swinging representations of semisimple groups

Abstract

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $\rho(G)$ and that is free, nonabelian and acts properly discontinuously on $V$.