Structure of prehomogeneous spaces associated with certain graded Lie algebras
type: Item
instance of: scholarly article
title: Structure of prehomogeneous spaces associated with certain graded Lie algebras
zbMATH Open document ID: 0568.17007
DOI: 10.1007/BF01458019
author: Iris Muller
author: Hubert Rubenthaler
author: Gérard Schiffmann
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164089
review text: Let \(\mathfrak g\) be a complex semisimple Lie algebra. We consider \(\mathbb Z\)-gradations of the following type: \(\mathfrak g={\mathfrak g}_{-1}\oplus {\mathfrak g}_ 0\oplus {\mathfrak g}_ 1\). Let \(G\) be the adjoint group of \(\mathfrak g\), \(G_ 0\) be the connected subgroup corresponding to \({\mathfrak g}_ 0\) and let \(N_ 0\) be a maximal unipotent subgroup of \(G_ 0\). It is known that the natural action of \(G_ 0\) on \({\mathfrak g}_ 1\) is prehomogeneous (i.e. has a Zariski open orbit). We first describe, using the root system of \(\mathfrak g\), the orbital decomposition of the preceding action. If the parabolic subalgebra \({\mathfrak g}_ 0\oplus {\mathfrak g}_ 1\) is auto-opposed we study the \(N_ 0\)-action on \({\mathfrak g}_ 1\) and we prove a generalization of the well known ``Gauss lemma'' concerning reduction of quadratic forms. We also give the structure of the ring of \(N_ 0\)-invariant polynomials. Finally, the preceding results enable us to calculate (via the Sato-Kashiwara-Kimura-Oshima algorithm) the b-functions associated to these prehomogeneous spaces.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 11S90
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 3906688
zbMATH Keywords: complex semi-simple Lie algebra
zbMATH Keywords: gradations
zbMATH Keywords: maximal unipotent subgroup
zbMATH Keywords: natural action
zbMATH Keywords: orbital decomposition
zbMATH Keywords: b-functions
zbMATH Keywords: prehomogeneous spaces
zbMATH Keywords: ring of invariant polynomials
MaRDI profile type: Publication
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OpenAlex ID: W2911984536
description: scientific article; zbMATH DE number 3906688