Construction of Milnorian representations

Abstract

We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let $G$ be a semisimple real Lie group with a representation $\rho$ on a finite-dimensional real vector space $V$, that does not satisfy the criterion from the previous paper. Assuming that $\rho$ is irreducible and under some additional assumptions on $G$ and $\rho$, we then prove that there does not exist a group of affine transformations acting properly discontinuously on $V$ whose linear part is Zariski-dense in $\rho (G)$.

This version differs from the previous one only by a reference that I added

References in corpus (2)

• Proper affine actions: a sufficient criterion
• Proper affine actions in non-swinging representations

Cited by in corpus (2)

• Representations having vectors fixed by a Levi subgroup
• Action of $w_0$ on $V^L$ for orthogonal and exceptional groups