For a semisimple real Lie group G with a 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 Smilga (Groups Geom Dyn 12(2):449–528, 2018), insofar as it also deals with “swinging” representations. When G is split, almost all the irreducible representations of G that have 0 as a weight satisfy this criterion. We conjecture that it is actually a necessary and sufficient criterion.

对于在有限维实向量空间V上具有表示\(\rho \)的半单实李群G，我们给出了\(\rho \)上的一组V的仿射变换的充分准则， 其线性部分在\(\rho (G)\) 中是 Zariski-dense 的，它是自由的、非 abelian 并且在V上正确地不连续地作用 。这个新标准比 Smilga (Groups Geom Dyn 12(2):449–528, 2018) 中给出的标准更通用，因为它还处理“摆动”表示。当摹分裂，几乎所有的不可约表示 g ^以 0 作为权重的那些满足这个标准。我们推测它实际上是一个充分必要的标准。