Closures of conjugacy classes in \(G_2\)
type: Item
instance of: scholarly article
title: Closures of conjugacy classes in \(G_2\)
zbMATH Open document ID: 0693.17004
DOI: 10.1016/0021-8693(89)90313-X
author: Hanspeter Kraft
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(\mathfrak g\) be the simple Lie algebra of type \(G_2\) and \(C_i\) be the nilpotent conjugacy class in \(\mathfrak g\) of dimension \(i = 6, 8, 10\) and \(12\). In this paper the author shows the following: (a) every conjugacy class of \(\mathfrak g\) except \(C_8\) has a normal closure with rational singularities, (b) [\textit{T. Levasseur} and \textit{S. P. Smith}, J. Algebra 114, 81--105 (1988; Zbl 0644.17005)] \(\bar C_8\) is not normal in \(\bar C_6=\bar C_8\setminus C_8\). The normalization \(\eta_8: \tilde C_8\to \bar C_8\) is bijective and \(\tilde C_8\) has an isolated rational singularity in \(\eta_8^{-1}(0)\), (c) \(\bar C_{12}\) has a singularity of type \(D_4\) in \(C_{10}\) and \(\bar C_{10}\) a singularity of type \(A_1\) in \(C_8\). The proof of these results is based on the same construction as in Levasseur-Smith (loc. cit.). Namely, he embeds \(\mathfrak g\) into \(\mathfrak{so}_7\) by the 7-dimensional standard representation and studies the \(\mathfrak g\)-equivariant projection \(p: \mathfrak{so}_7\to \mathfrak g\). At the end, the author gives a brief summary of what is known about normality of closures of conjugacy classes in reductive groups.
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 4134266
zbMATH Keywords: maximal torus
zbMATH Keywords: Borel subalgebra
zbMATH Keywords: simple Lie algebra
zbMATH Keywords: nilpotent conjugacy class
reviewed by: Eiichi Abe
MaRDI profile type: Publication
cites work: Classes of unipotent elements in simple algebraic groups. II
cites work: Green functions of finite Chevalley groups of type \(E_ n (n=6,7,8)\)
cites work: Über Schichten halbeinfacher Lie-Algebren
cites work: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen
cites work: Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups)
cites work: Q3682663
cites work: Cohomology and the resolution of the nilpotent variety
cites work: The normality of closures of orbits in a Lie algebra
cites work: On the collapsing of homogeneous bundles
cites work: Lie Group Representations on Polynomial Rings
cites work: Parametrisierung von Konjugationsklassen in \(\mathfrak {sl}_n\)
cites work: Q3993222
cites work: Closures of conjugacy classes of matrices are normal
cites work: Minimal singularities in \(GL_ n\).
cites work: On the geometry of conjugacy classes in classical groups
cites work: Primitive ideals and nilpotent orbits in type \(G_ 2\)
cites work: Q3778139
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: On the green polynomials of a chevalley group of type f4
cites work: Simple singularities and simple algebraic groups
cites work: ON A CLASS OF QUASIHOMOGENEOUS AFFINE VARIETIES
Recommended article: A note on the Kazhdan-Lusztig map for even orthogonal Lie algebras
Recommended article: Singularities of closures of \(B\)-conjugacy classes of nilpotent elements of height 2
Recommended article: Complexity and nilpotent orbits
Recommended article: The nullcone of the Lie algebra of \(G_2\)
Recommended article: The singularities of the closures of nilpotent orbits in certain symmetric pairs
Recommended article: Q4485760
Recommended article: Jet schemes of the closure of nilpotent orbits
Recommended article: On the dimension of the sheets of a reductive Lie algebra
Recommended article: Q4025332
Recommended article: Local geometry of Jordan classes in semisimple algebraic groups
description: scientific article; zbMATH DE number 4134266