Construction of Milnorian representations

TL;DR

This paper establishes conditions under which certain affine actions cannot be properly discontinuous, providing a partial converse to previous criteria for proper affine actions of semisimple Lie groups.

Contribution

It proves a partial converse to a prior criterion, showing non-existence of proper affine actions under specific irreducibility and density assumptions.

Findings

No properly discontinuous affine actions with Zariski-dense linear parts under given conditions.

Provides new constraints on affine actions of semisimple Lie groups.

Extends understanding of the relationship between group representations and affine actions.

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 $\rho(G)$.