Irreducible components of the nilpotent commuting variety of a symmetric semisimple Lie algebra
type: Item
instance of: scholarly article
title: Irreducible components of the nilpotent commuting variety of a symmetric semisimple Lie algebra
zbMATH Open document ID: 1189.17008
author: Michaël Bulois
publication date: 2009-04-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10396
review text: Given an involution \(\theta \) of a finite dimensional semisimple Lie algebra \(\mathfrak g\) and the associated Cartan decomposition \(\mathfrak g=\mathfrak k\oplus \mathfrak p\), the nilpotent commuting variety of \((\mathfrak g,\theta )\) consists of pairs of nilpotent elements \((x,y)\) of \(\mathfrak p\) satisfying the condition \([x,y]=0\). An interesting structural problem concerns the questions whether this variety is equidimensional and its irreducible components are indexed by the orbits of \(\mathfrak{p}\) distinguished elements (i.e., those elements whose centralizer in \(\mathfrak{p}\) is formed by nilpotent elements). The validity of this assertion was already established by \textit{A. Premet} in the case \((\mathfrak g\times \mathfrak g,\theta )\) where \(\theta (x,y)=(y,x)\) [Invent. Math. 154, No. 3, 653--683 (2003; Zbl 1068.17006)]. The main purpose of this to prove the conjecture for other types. In all, 18 cases are covered and proven to satisfy the assertion. The proof is based largely on two technical lemmas on the orbits and a reduction principle.
Mathematics Subject Classification ID: 17B08
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5541197
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: symmetric pair
zbMATH Keywords: commuting variety
zbMATH Keywords: nilpotent orbit
reviewed by: Rutwig Campoamor Stursberg
MaRDI profile type: Publication
OpenAlex ID: W2199848042
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DOI: 10.5802/AIF.2426
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5541197