On the separation property of orbits in representation spaces.
type: Item
instance of: scholarly article
title: On the separation property of orbits in representation spaces.
zbMATH Open document ID: 1055.20038
DOI: 10.1016/S0021-8693(02)00503-3
author: Hanspeter Kraft
author: Nolan Wallach
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The starting point of the paper under review was the following question asked by Jens-Carsten Jantzen in relation with some work of \textit{A. Premet} [see: J. Lond. Math. Soc., II. Ser. 55, No. 2, 236-250 (1997; Zbl 0889.17019)]: Question. Let \(K\) be an algebraically closed field. Is it true that the minimal nilpotent conjugacy class \(C_{\min}\) in the Lie algebra \(\mathfrak g\) of a simple affine algebraic \(K\)-group has the following ``Separation Property'': (SP) For any pair of linearly independent linear functions \((\alpha,\beta)\) on \(\mathfrak g\) there is an element \(x\in C_{\min}\) such that \(\alpha(x)=0\) and \(\beta(x)\neq 0\)? In the paper under review the authors show that the answer is ``yes'' except for \(\mathfrak{sl}_2\) and the symplectic Lie algebras \(\mathfrak{sp}_{2n}\) (\(n>2\)) where they give explicit counterexamples. As a consequence, some results in the above mentioned paper of Premet do not hold in the given generality. (In the mean-time the authors have been informed that the basic results of Premet are still correct and have been proved along different lines.) Nevertheless, the separation property is an interesting concept in itself and the authors study this property in the paper. In Section 2, they first discuss the case of matrices and show that the set of nilpotent rank 1 matrices in \(\mathfrak{sl}_n(K)\subset M_n(K)\) has the separation property (SP) in case \(n\geqslant 3\). In Section 3, the authors discuss the separation property in general and introduce in addition the ``strong'' and the ``weak'' separation property, (SSP) and (WSP). A number of examples illuminate these concepts. In Section 4 and 5, they study the separation property for orbits in representation spaces of algebraic groups and give a classification of those representations of reductive groups where the minimal orbit \(O_{\min}\) satisfies the various separation properties. For example, the authors discover the fact that in an irreducible representatio…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B45
zbMATH DE Number: 1868087
zbMATH Keywords: separation property
zbMATH Keywords: vector spaces
zbMATH Keywords: orbits
zbMATH Keywords: irreducible representations
zbMATH Keywords: semisimple groups
zbMATH Keywords: Lie algebras of simple affine algebraic groups
zbMATH Keywords: nilpotent conjugacy classes
zbMATH Keywords: root systems
zbMATH Keywords: highest weight vectors
zbMATH Keywords: Weyl modules
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1868087