Abstract
Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p ≥ 0 and let ϕ be a nontrivial p-restricted irreducible representation of G. Let T be a maximal torus of G and s ϵ T. We say that s is Ad-regular if α(s) ≠ β(s) for all distinct T-roots α and β of G. Our main result states that if all but one of the eigenvalues of ϕ(s) are of multiplicity 1 then, with a few specified exceptions, s is Ad-regular. This can be viewed as an extension of our earlier work, in which we show that, under the same hypotheses, either s is regular or G is a classical group and ϕ is “essentially” (a twist of) the natural representation of G.
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References
N. Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5 et 6, Masson, Paris, 1981.
N. Bourbaki, Groupes et Algebres de Lie, Chapitres 7 et 8, Hermann, Paris, 1975.
Humphreys, J.: Modular Representations of Finite Groups of Lie Type. Cambridge Univ. Press, Cambridge (2006)
Lübeck, F.: Small degree representations of Finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4, 135–169 (2001)
F. Lübeck, Tables of weight multiplicities, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html.
Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Univ. Press, Cambridge (2011)
A. A. Πремет, Веса инфинимезимально неприводитых представлений групп Шевалле над полем простой характеристики, Матем. сб. 133(175) (1987), ном. 2(6), 167-183. Engl. transl.: A. A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSR Sb. 61 (1988), no. 1, 167- 183.
G. M. Seitz, The Maximal Subgroups of Classical Algebraic Groups, Memoirs Amer. Math. Soc. 365, Amer. Math. Soc., Providence, 1987.
G. M. Seitz, Bounds for dimensions of weight spaces of maximal tori, in: Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 157- 161.
T. Springer, R. Steinberg, Conjugacy classes, in: Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes in Math., Vol. 131, Springer Verlag, Berlin, 1970, pp. 167-266.
Testerman, D., Zalesski, A.: Subgroups of simple algebraic groups containing maximal tori, and representations with multiplicity 1 nonzero weights. Transform. Groups. 20, 831–861 (2015)
Testerman, D., Zalesski, A.E.: Spectra of nonregular elements in irreducible representations of simple algebraic groups. North-Western European J. Math. 7, 185–212 (2021)
Zalesski, A.: On eigenvalues of group elements in representations of simple algebraic groups and infinite Chevalley groups. Acta Appl. Math. 108, 175–195 (2009)
Zalesski, A.: Unisingular representations of finite symplectic groups. Comm. Algebra. 50, 1697–1719 (2022)
Zalesski, A.: Singer cycles in 2-modular representations of GLn+1(2). Archiv der Math. 110, 433–446 (2018)
A. E. Залесский, И. Супруненко, Представления размерностей (pn ∓1)/2 симплектической группы степени 2n над полет характеристики p , Вести AN БССР, Сеp. физ.- мат. навук (1987), нom. 6, 9- 15. Engl. transl.: A. Zalesski, I. Suprunenko, Representations of dimensions (pn ∓ 1)/2 of the symplectic group of degree 2n over a field of characteristic p, arXiv:2108.10650v2 (2021).
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D. M. Testerman acknowledges the support of the Swiss National Science Foundation, research grant number 200020 175571. In addition, this work was initiated while the second author visited the EPFL; he gratefully acknowledges this institute's support.
We dedicate this paper to the memory of James Humphreys. His books, research articles, and mathematical correspondence had a great inuence on our own mathematics and will continue to train generations of young mathematicians. We are proud to be able to contribute to this issue.
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TESTERMAN, D.M., ZALESSKI, A.E. ALMOST CYCLIC REGULAR SEMISIMPLE ELEMENTS IN IRREDUCIBLE REPRESENTATIONS OF SIMPLE ALGEBRAIC GROUPS. Transformation Groups 28, 1299–1324 (2023). https://doi.org/10.1007/s00031-023-09793-5
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DOI: https://doi.org/10.1007/s00031-023-09793-5