19 research outputs found

    Proper affine actions on semisimple Lie algebras

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    For any noncompact semisimple real Lie group GG, we construct a group of affine transformations of its Lie algebra g\mathfrak{g} whose linear part is Zariski-dense in AdG\operatorname{Ad} G and which is free, nonabelian and acts properly discontinuously on g\mathfrak{g}.Comment: Section 3 of this paper draws heavily on the section 3 from my earlier paper arXiv:1303.3766 This is the version that appeared in Geometriae Dedicata. I have corrected some mistakes, added a few examples and added a proof that the Margulis invariant is well-define

    Action of w0w_0 on VLV^L: the special case of so(1,n)\mathfrak{so}(1,n)

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    In this note, we present an algorithm that allows to answer any individual instance of the following question. Let GRG_{\mathbb{R}} be a semisimple real Lie group, and VV an irreducible representation of GRG_{\mathbb{R}}. How does the longest element w0w_0 of the restricted Weyl group WW act on the subspace VLV^L of VV formed by vectors that are invariant by LL, the centralizer of a maximal split torus of GRG_{\mathbb{R}}? This algorithm comprises two parts. First we describe a complete answer to this question in the particular case where GR=SO(1,n)G_{\mathbb{R}} = \operatorname{SO}(1,n) for any n2n \geq 2. Then, for an arbitrary GRG_{\mathbb{R}}, we show that it suffices to do the computation in a well-chosen subgroup SRGRS_{\mathbb{R}} \subset G_{\mathbb{R}} which is (up to isogeny) the product of several groups that are either compact, abelian or isomorphic to SO(1,n)\operatorname{SO}(1,n) for some n2n \geq 2.Comment: Main text: 9 pages, 1 figure. Appendix: 7 pages, of which 4.5 pages are occupied by 2 tables. The ancillary files contain an implementation of the algorithm in the LiE software package (with an explanation

    New sequences of non-free rational points

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    We exhibit some new infinite families of rational values of τ\tau, some of them squares of rationals, for which the group or even the semigroup generated by the matrices (1101)\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right) and (10τ1)\left( \begin{smallmatrix} 1 & 0 \\ \tau & 1 \end{smallmatrix} \right) is not free.Comment: Fixed a typo: I had written "free" instead of "non-free" in a few places; and made a few other minor edit

    New sequences of non-free rational points

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    We exhibit some new infinite families of rational values of τ\tau , some of them squares of rationals, for which the group or even the semigroup generated by the matrices (1101)({{\textstyle \begin{matrix} 1 & 1\\ 0 & 1 \end{matrix}}}) and  (10τ1)({{\textstyle \begin{matrix} 1 & 0 \\ \tau & 1 \end{matrix}}}) is not free
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