Symmetric weights and \(s\)-representations
type: Item
instance of: scholarly article
title: Symmetric weights and \(s\)-representations
zbMATH Open document ID: 0994.53025
author: Sergio Console
author: Anna Fino
published in: Kodai Mathematical Journal
publication date: 2002-09-27T00:00:00Z
review text: In his classification of irreducible polar representations, \textit{J. Dadok} [Trans. Am. Math. Soc. 288, 125-137 (1985; Zbl 0565.22010)] assigned to an irreducible representation of a compact Lie group \(K\) with highest weight \(\lambda \) an integer \(k(\lambda)\). This can be characterized as the smallest positive integer \(l\) such that \(w\in U^l v_\lambda\) where \(w\) and \(v_\lambda\) are vectors of smallest respectively highest weight and \(U^l\) is the \(l\)-th filtration of the enveloping algebra of the Lie algebra of \(K\). Dadok proved that \(k(\lambda)\leq 4\) resp. \(k(\lambda)\leq 2\) for polar representations of real resp. complex type. Using results of \textit{M. Wang} and \textit{W. Ziller} [Symmetric spaces and strongly isotropy irreducible spaces, Math. Ann. 296, No. 2, 285-326 (1993; Zbl 0804.53075)] the authors show that a (not necessarily polar) faithful irreducible representation \(V\) of a compact connected Lie group \(K\) with symmetric highest weight \(\lambda \) is the isotropy representation of a symmetric space if \(k(\lambda)=4\) and \(V\) is of real type. If \(k(\lambda)=2\) and \(V\) is of complex type then \(V\) is the isotropy representation of a Hermitian symmetric space. The authors give a geometric interpretation of the upper bound of \(k(\lambda)\) in terms of the osculating spaces of the orbits. They show that the values \(k(\lambda)=4\) resp. \(k(\lambda)=2\) are maximal for representations of class \({\mathcal O}_2\) (such as polar representations). Generally, an irreducible orthogonal representation \(V\) is of class \({\mathcal O}_d\) if \(V\) is spanned by the derivatives \(c^{(l)}(0)\) of order \(l\leq d\) of curves \(c\) in an orbit \(Kp\subset V\) with \(p=c(0)\neq 0\).
reviewed by: Evangelia Samiou
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1547130
zbMATH Keywords: symmetric space
zbMATH Keywords: \(s\)-representation
zbMATH Keywords: strongly isotropy irreducible spaces
zbMATH Keywords: isotropy representation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044216
OpenAlex ID: W2039372450
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DOI: 10.2996/KMJ/1138044216
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description: scientific article; zbMATH DE number 1547130