Abstract
Let H be a linear algebraic group whose connected component \(G\ne 1\) is simple, H/G is cyclic, and \(Z(H)\subseteq Z(G)\). We determine the irreducible representations \(\phi \) of H such that \(\phi (G)\) is irreducible and \(\phi (h)\) has simple spectrum for some \(h\in H\). The latter means that all irreducible constituents of the group \(\phi ( \langle h \rangle )\) are of multiplicity 1. (Here \( \langle h \rangle \) is the subgroup of H generated by h.) This extends an earlier known result for \(H=G\).
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References
Bourbaki, N.: Groupes et Algèbres de Lie, vol. IV–VI. Masson, Paris (1981)
Bourbaki, N.: Groupes et Algèbres de Lie, vol. VII–VIII. Springer, Berlin (2006)
Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups, vol. 3. American Mathematical Society, Providence (1998)
Katz, N.M., Tiep, P.H.: Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups. Princeton University, Princeton, Preprint (2023)
Lübeck, F.: Turning weight multipicities into Brauer characters. J. Algebra 558, 534–549 (2020)
Seitz, G.M.: The Maximal Subgroups of Classical Algebraic Groups. Memoirs of the American Mathematical Society, vol. 365. American Mathematical Society, Providence (1987)
Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of American Mathematical Society, vol. 80. American Mathematical Society, Providence (1968)
Steinberg, R.: Lectures on Chevalley Groups. American Mathematical Society University Lecture Series, vol. 66. American Mathematical Society, Providence (2016)
Suprunenko, I.D., Zalesski, A.E.: Irreducible representations of finite Chevalley groups containing a matrix with a simple spectrum. In: Groups, Combinatorics and Geometry. London Mathematical Society Lecture Note Series, vol. 165, pp. 327–332. Cambridge University Press, Cambridge (1992)
Suprunenko, I.D., Zalesski, A.E.: Irreducible representations of finite classical groups containing matrices with simple spectra. Comm. Algebra 26, 863–888 (1998)
Suprunenko, I.D., Zalesski, A.E.: Irreducible representations of finite groups of exceptional Lie type containing matrices with simple spectra. Comm. Algebra 28, 1789–1833 (2000)
Testerman, D., Zalesski, A.E.: Subgroups of simple algebraic groups containing maximal tori and representations with multiplicity \(1\) non-zero weights. Transform. Groups 20, 831–861 (2015)
Testerman, D., Zalesski, A.: Spectra of non-regular elements in irreducible representations of simple algebraic groups. North-Western Eur. J. Math. 7, 185–212 (2021)
Zalesskiĭ, A.E., Suprunenko, I.D.: Representations of dimensions \((p^n\pm 1)/2\) of the symplectic group of degree \(2n\) over a field of characteristic \(p\). Vesti Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 6, 9–15 (1987). (English translation: arXiv:2108.10650)
Acknowledgements
I am indebted to Robert Guralnick for rising the problem discussed in this paper and his encouraging comments and to Frank Lübeck for fixing the cases with \(q=4,8\) of Theorem 1.4.
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Dedicated to A.G. Elashvili on the occasion of his 80th birthday.
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Zalesski, A. Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups. Arch. Math. 121, 355–369 (2023). https://doi.org/10.1007/s00013-023-01892-9
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DOI: https://doi.org/10.1007/s00013-023-01892-9
Keywords
- Algebraic group representations
- Representations of finite simple groups of Lie type
- Matrices with simple spectrum