Proper affine actions: a sufficient criterion

For a semisimple real Lie group G with representation \(\rho \) on finite-dimensional vector space V, we give sufficient criterion for existence of affine transformations V whose linear part is Zariski-dense in (G)\) and that free, nonabelian acts properly discontinuously V. This new more general than the one given Smilga (Groups Geom Dyn 12(2):449–528, 2018), insofar as it also deals “swinging” representations. When split, almost all irreducible representations have 0 weight satisfy this criterion. We conjecture actually necessary