Semistable locus of a group compactification
type: Item
instance of: scholarly article
title: Semistable locus of a group compactification
zbMATH Open document ID: 1251.14034
DOI: 10.1090/S1088-4165-2011-00404-0
author: Xuhua He
author: Jason Michael Starr
publication date: 2012-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.0281
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 6043189
zbMATH Keywords: wonderful varieties
zbMATH Keywords: geometric quotients
zbMATH Keywords: Lusztig \(G\)-stable piece
MaRDI profile type: Publication
arXiv ID: 0907.0281
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OpenAlex ID: W2963759289
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published in: Representation Theory
review text: Let \(G\) be a connected, semi-simple algebraic group of adjoint type over an algebraically closed field. One can consider \(G\) as a \(G\times G\)-variety by left and right translation. Then there exists a canonical \(G\times G\)-equivariant embedding \(X\) of \(G\) which is called the wonderful compactification [\textit{C. De Concini} and \textit{C. Procesi}, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. The variety \(X\) is an irreducible, smooth projective \((G\times G)\)-variety with finitely many \(G \times G\)-orbits indexed by the subsets of the set of simple roots of \(G\). The boundary \(X\setminus G\) is a union of smooth divisors with normal crossing.NEWLINENEWLINEThe classification of the orbits by the action on \(X\) of the diagonal copy of \(G\) in \(G\times G\) was obtained by \textit{G. Lusztig} [Mosc. Math. J. 4, No. 4, 869--896 (2004; Zbl 1103.20041)] in terms of \(G\)-stable pieces.NEWLINENEWLINEThe authors give an explicit description of the semi-stable locus of \(X\) (for the diagonal \(G\)-action) using Lusztig's \(G\)-stable pieces and calculate the geometric quotient \(X/\!/G\). More generally, they consider the case where the action is twisted by a diagram automorphism. Let \(\sigma\) be a diagram automorphism of \(G\), in particular an automorphism on \(G\) such that \(\sigma(B) = B\) and \(\sigma(T) = T\). The author consider the variety \(X_\sigma\) isomorphic \(X\), but with the action twisted by \((g_1,g_2)\rightarrow(g_1,\sigma(g_2))\).NEWLINENEWLINEIn [\textit{C. De Concini, S. Kannan} and \textit{A. Maffei}, Mosc. Math. J. 8, No. 4, 667--696 (2008; Zbl 1161.14036)] is described the semi-stable locus and geometric quotient for complete symmetric varieties (which includes as a special case the nontwisted conjugation action of \(G\) on its wonderful compactification), in terms of the \(G\)-invariants of the Cox ring.NEWLINENEWLINEIn this work it is used a different approach based on the geometry of the nilpotent cone of \(X_\si…
reviewed by: Alessandro Ruzzi
description: scientific article; zbMATH DE number 6043189