For a semisimple real Lie group G with an irreducible representation ρ on a finite-dimensional real vector space V , we give a sufficient criterion on ρ for existence of a group of affine transformations of V whose linear part is Zariskidense in ρ(G) and that is free, nonabelian and acts properly discontinuously on V . (2010): 20G20, 20G05, 22E40, 20H15. Mathematics Subject Classification

For any noncompact semisimple real Lie group G, we construct a group of affine transformations of its Lie algebra g whose linear part is Zariski-dense in Ad G and which is free, nonabelian and acts properly discontinuously on g.

For any semisimple real Lie algebra g R , we classify the representations of g R that have at least one nonzero vector on which the centralizer of a Cartan subspace, also known as the centralizer of a maximal split torus, acts trivially. In the process, we revisit the notion of g-standard Young tableaux, introduced by Lakshmibai and studied by Littelmann, that provides a combinatorial model for the characters of the irreducible representations of any classical semisimple Lie algebra g. We construct a new version of these objects, which differs from the old one for g = so(2r) and seems, in some sense, simpler and more natural.

For a semisimple real Lie group G with a representation ρ on a finitedimensional real vector space V , we give a sufficient criterion on ρ for existence of a group of affine transformations of V whose linear part is Zariski-dense in ρ(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.Proof. This follows from the general Theorem 7.2 in [Tit71]. This is also stated as Lemma 2.5.1 in [Ben97].Example 2.13.1. If G = SL n (R), we may take V i = Λ i R n , so that ρ i is the i-th exterior power of the standard representation of G on R n .2. More generally, if G is split, then all restricted weight spaces correspond to ordinary weight spaces, hence have dimension 1. So we may simply take every n i to be 1, so that the ρ i 's are precisely the fundamental representations.Lemma 2.14. Fix an index i such that 1 ≤ i ≤ r. Then all restricted weights of ρ i other than n i ̟ i have the formwith c j ≥ 0 for every j.

We construct a fundamental region for the action on the 2d + 1-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature (d + 1, d), where d is any odd positive integer.