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A proof of the first Kac-Weisfeiler conjecture in large characteristics


Authors: Benjamin Martin, David Stewart and Lewis Topley; with an appendix by Akaki Tikaradze
Journal: Represent. Theory 23 (2019), 278-293
MSC (2010): Primary 17B50; Secondary 17B10, 17B35, 03C60
DOI: https://doi.org/10.1090/ert/529
Published electronically: September 16, 2019
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Abstract: In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $ \mathfrak{g}$. The first predicts the maximal dimension of simple $ \mathfrak{g}$-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $ \mathfrak{gl}_n(k)$ whenever $ k$ is an algebraically closed field of sufficiently large characteristic $ p$ (depending on $ n$). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic.

In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac-Weisfeiler conjecture is given for the Lie algebra of a group scheme over a finitely generated ring $ R \subseteq \mathbb{C}$, after base change to a field of large positive characteristic.


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Additional Information

Benjamin Martin
Affiliation: Department of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, Aberdeen AB24 3UE, United Kingdom
Email: B.Martin@abdn.ac.uk

David Stewart
Affiliation: School of Mathematics and Statistics, University of Newcastle, Herschel Building, Newcastle, NE1 7RU, United Kingdom
Email: david.stewart@ncl.ac.uk

Lewis Topley
Affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, United Kingdom
Email: L.Topley@kent.ac.uk

Akaki Tikaradze
Affiliation: Department of Mathematics, Mail Stop 942, University of Toledo, 2801 W. Bancroft Street, Toledo, Ohio 43606-3390
Email: Akaki.Tikaradze@utoledo.edu, Akaki.Tikaradze@utoledo.edu

DOI: https://doi.org/10.1090/ert/529
Received by editor(s): November 16, 2018
Received by editor(s) in revised form: November 18, 2018, December 21, 2018, and July 30, 2019
Published electronically: September 16, 2019
Additional Notes: The second author is the corresponding author
The third author gratefully acknowledges the support of EPSRC grant number EP/N034449/1
Article copyright: © Copyright 2019 American Mathematical Society