Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture
type: Item
instance of: scholarly article
title: Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture
zbMATH Open document ID: 0828.17008
DOI: 10.1007/BF01884291
author: Alexander Premet
published in: Inventiones Mathematicae
publication date: 1996-01-04T00:00:00Z
full work available at URL: https://eudml.org/doc/144292
review text: Let \({\mathfrak g}\) be the Lie algebra of a connected reductive group \(G\) over an algebraically closed field of characteristic \(p>0\). Suppose that the commutator subgroup of \(G\) is simply connected and that \(p\) is greater than any coefficient of any positive root of the root system \(R= R(G)\) relative to a basis of simple roots in \(R\). If \(p=2\), suppose also that \({\mathfrak g}\) admits a nondegenerate \(G\)-invariant trace form. Let \(V\) be a faithful irreducible \({\mathfrak g}\)-module with \(p\)-character \(\chi\in {\mathfrak g}^*\). The author proves that \(\dim V\) is divisible by \(p^{1/2}\dim \Omega (\chi)}\) where \(\Omega (\chi)\) is the order of \(\chi\) under the coadjoint action of \(G\). A conjecture mentioned by \textit{V. Kac} and \textit{B. Weisfeiler} [Funct. Anal. Appl. 5, 111--117 (1971); translation from Funkts. Anal. Prilozh. 5, No. 2, 28--36 (1971; Zbl 0237.17003)] is thus confirmed. The necessity of \({\mathfrak g}\) admitting a nondegenerate trace form in order for the conjecture to be true is shown by an example. The paper concludes by listing some related open questions.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 784020
zbMATH Keywords: irreducible representations
zbMATH Keywords: Kac-Weisfeiler conjecture
zbMATH Keywords: Lie algebra of a connected reductive group
zbMATH Keywords: trace form
Wikidata QID: Q115393101
MaRDI profile type: Publication
cites work: The Lie algebras with a quotient trace form
cites work: Q5607044
cites work: Q3682663
cites work: On Injective Modules for Infinitesimal Algebraic Groups, I
cites work: Noncommutative Extensions of Hilbert Rings
cites work: Q3264252
cites work: Q3972941
cites work: Quantum Coadjoint Action
cites work: Q3251484
cites work: Geometry of p-unipotent Lie algebras
cites work: Support varieties for restricted Lie algebras
cites work: Modular Representation Theory of Lie Algebras
cites work: Deformations of Lie Algebra Representations
cites work: Induction, deformation, and specialization of Lie algebra representations
cites work: Q4750828
cites work: Q4089001
cites work: Q5732712
cites work: Kohomologie von p-Lie-Algebren und nilpotente Elemente. (Cohomology of p- Lie-algebras and nilpotent elements)
cites work: Q3800267
cites work: Conjugacy Classes in Parabolic Subgroups of Semisimple Algebraic Groups, II
cites work: Q4087381
cites work: Q3998928
cites work: THE ZASSENHAUS VARIETY OF A CLASSICAL SEMISIMPLE LIE ALGEBRA IN FINITE CHARACTERISTIC
cites work: An Associative Orthogonal Bilinear Form for Hopf Algebras
cites work: IRREDUCIBLE REPRESENTATIONS OF THE LIE ALGEBRA $ \mathrm{sl}(n)$ OVER A FIELD OF POSITIVE CHARACTERISTIC
cites work: An Analogue of the Jacobson-Morozov Theorem for Lie Algebras of Reductive Groups of Good Characteristics
cites work: On the Endomorphism Ring of a Simple Module Over an Enveloping Algebra
cites work: Irreducible representations of a simple three-dimensional lie algebra over a field of finite characteristic
cites work: Simple singularities and simple algebraic groups
cites work: Q3977493
cites work: Representations of the Witt algebra
cites work: The Representations of Lie Algebras of Prime Characteristic
Recommended article: An Analogue of the Jacobson-Morozov Theorem for Lie Algebras of Reductive Groups of Good Characteristics
Recommended article: Faithful completely reducible representations of modular Lie algebras
Recommended article: G -complete reducibility and semisimple modules
Recommended article: Modular representations of Lie algebras of reductive groups and Humphreys' conjecture
Recommended article: 模李代数的Ext-转移(英)
Recommended article: Modules with Involution for Algebraic Groups
Recommended article: Q4485762
Recommended article: Q4664366
Recommended article: Cohomology of reductive modular Lie algebras
Recommended article: A Character formula for B-modules
description: scientific article; zbMATH DE number 784020