Abstract
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.
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17 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00208-021-02294-4
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Acknowledgements
I am very grateful to my Ph.D. advisor, Yves Benoist, who introduced me to this exciting and fruitful subject, and gave invaluable help and guidance in my initial work on this project. I would also like to thank Bruno Le Floch for some interesting discussions, which in particular helped me gain more insight about weights of representations. Thanks to the Yale University (and my colleagues from there), where almost all of the work on this paper has been conducted, for the wonderful working conditions.
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Communicated by Thomas Schick.
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Smilga, I. Proper affine actions: a sufficient criterion. Math. Ann. 382, 513–605 (2022). https://doi.org/10.1007/s00208-020-02100-7
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DOI: https://doi.org/10.1007/s00208-020-02100-7