The geometry of generalized Steinberg varieties.
type: Item
instance of: scholarly article
title: The geometry of generalized Steinberg varieties.
zbMATH Open document ID: 1059.20044
author: J. Matthew Douglass
author: Gerhard Röhrle
published in: Advances in Mathematics
publication date: 2004-10-01T00:00:00Z
review text: Let \(G\) be a reductive algebraic group over an algebraically closed field. Assume the characteristic is zero or good, so that the variety \(\mathcal N\) of nilpotent elements of the Lie algebra \(\mathfrak g\) may be identified with the unipotent variety. The authors introduce a variety \(X^{\mathcal{P,Q}}_{c,d}\) that is designed to cover many earlier constructions. Here \(\mathcal{P,Q}\) are conjugacy classes of parabolic subgroups, \(c(P)\), \(d(Q)\) are closed subvarieties of \(\mathcal N\), depending \(G\)-equivariantly on \(P\in\mathcal P\), \(Q\in\mathcal Q\). One requires \({\mathfrak u}_P\subseteq c(P)\subseteq{\mathcal N}\cap\mathfrak p\) and \({\mathfrak u}_Q\subseteq d(Q)\subseteq{\mathcal N}\cap\mathfrak q\). One defines \(X^{\mathcal{P,Q}}_{c,d}\) to be the subset of \(\mathcal N\times\mathcal P\times\mathcal Q\) consisting of the \((x,P,Q)\) with \(x\in c(P)\cap d(Q)\). The cases where \(c\), \(d\) are both minimal (maximal) are best understood. For those it is shown that \(X^{\mathcal{P,Q}}_{c,d}\) is equidimensional and that the irreducible components are indexed by \((W_{\mathcal P},W_{\mathcal Q})\) double cosets in the Weyl group \(W\). One also gets a map that associates to one of these double cosets a nilpotent orbit in \(G{\mathfrak u}_P\cap G{\mathfrak u}_Q\).
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B45
zbMATH DE Number: 2105048
zbMATH Keywords: Steinberg varieties
zbMATH Keywords: Springer representations
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: unipotent elements
zbMATH Keywords: Borel subgroups
zbMATH Keywords: double coset representatives
zbMATH Keywords: Weyl groups
zbMATH Keywords: nilpotent orbits
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: j.aim.2003.09.002
OpenAlex ID: W1967986548
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DOI: 10.1016/J.AIM.2003.09.002
description: scientific article; zbMATH DE number 2105048