On M. Sato's classification of some reductive prehomogeneous vector spaces
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).
scientific article; zbMATH DE number 5931047