Nilpotent orbits in good characteristic and the Kempf-Rousseau theory
type: Item
instance of: scholarly article
title: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory
zbMATH Open document ID: 1020.20031
DOI: 10.1016/S0021-8693(02)00662-2
author: Alexander Premet
published in: Journal of Algebra
publication date: 2003-04-28T00:00:00Z
review text: Let \(G\) be an algebraic group over an algebraically closed field \(k\) and \(\mathfrak g\) be the Lie algebra of \(G\). The focus of this paper is nilpotent elements and orbits in \(\mathfrak g\) with the goal of extending two results known to hold over the complex numbers to a field of prime characteristic \(p>0\) where \(p\) is good for the root system of \(G\). First, suppose that \(G\) is connected and reductive. Over the complex numbers, the Bala-Carter theorem states that any nilpotent element in \(\mathfrak g\) is \(G\)-conjugate to a Richardson element in a distinguished parabolic subalgebra of the Lie algebra of a Levi subgroup of \(G\). \textit{K. Pommerening} [J. Algebra 49, 525-536 (1977; Zbl 0367.20046), ibid. 65, 373-398 (1980; Zbl 0437.20034)] showed that the same characterization holds when \(k\) has prime characteristic \(p>0\) as long as \(p\) is good for the root system of \(G\). In this work, the author provides a new proof of this theorem in prime characteristic which avoids computations and case-checking. This is done by replacing the traditional approach of using \({\mathfrak{sl}}(2)\)-theory (which is not as effective in prime characteristic) with Kempf-Rousseau theory or the theory of optimal tori for \(G\)-unstable vectors. This new approach also leads to an improved proof of the existence theorem for good transverse slices to nilpotent orbits in \(\mathfrak g\). For previous proofs, see the work of \textit{N. Kawanaka} [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 573-597 (1987; Zbl 0651.20046)], \textit{A. Premet} [Trans. Am. Math. Soc. 347, No. 8, 2961-2988 (1995; Zbl 0865.17015)], and \textit{N. Spaltenstein} [J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 283-286 (1984; Zbl 0581.17003)]. Secondly, suppose that \(G\) is a simple algebraic group of adjoint type. Over the complex numbers, \textit{E. Sommers} [Int. Math. Res. Not. 1998, No. 11, 539-562 (1998; Zbl 0909.22014)] found a bijection between \(G\)-conjugacy classes of pairs \((L,e)\) whe…
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B45
zbMATH DE Number: 1901828
zbMATH Keywords: algebraic groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: nilpotent elements
zbMATH Keywords: Bala-Carter theorem
zbMATH Keywords: Pommerening theorem
zbMATH Keywords: centralizers
zbMATH Keywords: component groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Kempf-Rousseau theory
zbMATH Keywords: optimal tori
zbMATH Keywords: transverse slices
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00662-2
OpenAlex ID: W1973914981
description: scientific article; zbMATH DE number 1901828