Monodromy groups of certain Kloosterman and hypergeometric sheaves

Katz, Nicholas M.; Tiep, Pham Huu

doi:10.48550/arxiv.2005.09051

Cited by 1 publication

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References 32 publications

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“…We note as well that the consideration of Problem 1.1 is an extension of the analogous question for finite quasi-simple groups of Lie type and their representations in defining characteristic (see [18,19]), as well as the classification (in [11,17]) of irreducible representations of simple algebraic groups for which a maximal torus acts with 1-dimensional weight spaces [11,17]. A similar problem for irreducible representations of finite simple groups occurring as subgroups of GL n (C) has been studied in [7].…”

Section: Introductionmentioning

confidence: 99%

Spectra of non-regular elements in irreducible representations of simple algebraic groups

Testerman,

Zalesski

2021

Preprint

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We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and φ : G → GL(V ) is a non-trivial irreducible representation for which there exists a non-regular non-central semisimple element s ∈ G such that φ(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.

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