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.