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 $$\rho $$ ρ on a finite-dimensional real vector space V , that does not satisfy the criterion from the previous paper. Assuming that $$\rho $$ ρ is irreducible and under some additional assumptions on G and $$\rho $$ ρ , 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 $$\rho (G)$$ ρ ( G ) .

中文翻译：

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米尔诺式表征的构建

我们证明了作者之前论文正确仿射动作的主要定理的部分相反：一个充分的标准（已提交；可在 arXiv:1612.08942 获得）。更准确地说，设 G 是一个半单实李群，在有限维实向量空间 V 上具有 $$\rho $$ ρ 表示，它不满足前一篇论文中的标准。假设 $$\rho $$ ρ 是不可约的，并且在 G 和 $$\rho $$ ρ 的一些额外假设下，我们然后证明不存在一组在 V 上正确不连续作用的仿射变换，其线性部分是 Zariski -dense in $$\rho (G)$$ ρ ( G ) 。