On M. Sato's classification of some reductive prehomogeneous vector spaces
type: Item
instance of: scholarly article
title: On M. Sato's classification of some reductive prehomogeneous vector spaces
zbMATH Open document ID: 1261.11078
DOI: 10.2977/PRIMS/40
author: Tatsuo Kimura
author: Yukimi Ishii
author: Inhyun Ryu
author: Michio Hamada
author: Yoshiteru Kurosawa
author: Masaya Ouchi
author: Tomohiro Kamiyoshi
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2011-07-22T00:00:00Z
review text: This paper under review is concerned with the classification theory of prehomogeneous vector spaces (abbreviated as PV). Let \((G, \rho, V)\) be a reductive PV over \(\mathbb{C}\), that is, \(G\) is a connected reductive \(\mathbb{C}\)-group and \(\rho: G \to \text{GL}(V)\) is a rational representation of \(G\) such that \(V\) admits a Zariski-dense \(G\)-orbit. Such a PV is called irreducible if \((\rho, V)\) is. The classification of irreducible reductive PV's is achieved by Tatsuo Kimura and Mikio Sato. Nonetheless, the complete classification in the non-irreducible case still seems to be unreachable, except in a few special cases. The authors consider the reductive PV's of the form \((G_0 \times G, \Lambda_1 \otimes \rho, V(n) \otimes V)\), where \(G_0\) is a connected semisimple subgroup of \(\text{SL}(n)\), \(\Lambda_1\) is the standard representation of \(\text{GL}(n)\), and \(\rho: G \to \text{GL}(V)\) is a rational representation of a connected reductive group \(G\). Under some conditions on the decomposition of \((\rho, V)\) into irreducibles, the authors classify all reductive PV's of this form. Roughly speaking, the proof is divided into the exceptional case (Theorem 0.2) and the classical case (Theorem 0.3).
reviewed by: Wen-Wei Li
Mathematics Subject Classification ID: 11S90
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 5931047
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: classification
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5931047