Proper affine actions: a sufficient criterion

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 $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations.

This paper generalizes the author's previous papers arXiv:1406.5906 and arXiv:1605.03833 . The structure of the proof is similar; a few passages are borrowed from the earlier papers. In this version, I corrected a mathematical mistake in Definition 6.12.(iii). This mistake remains present in the published version; an erratum has been published to address it

References in corpus (3)

• Proper affine actions for right-angled Coxeter groups
• Proper affine actions in non-swinging representations
• Action of Weyl group on zero weight space

Cited by in corpus (6)

• Action of Weyl group on zero weight space
• Construction of Milnorian representations
• Finite step Rigidity of Hitchin representations and special Margulis-Smilga spacetimes
• Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups
• Representations having vectors fixed by a Levi subgroup
• Action of $w_0$ on $V^L$ for orthogonal and exceptional groups