A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications
The purpose of this paper is the complete classification of the complex prehomogeneous vector spaces such that the group is reductive with scalar multiplications. - Let G be a connected complex linear algebraic group and let \(\rho\) be a finite dimensional rational representation on the complex vector space V. We say that the triplet \((G,\rho\),V) is a prehomogeneous vector space if V contains a Zariski-dense G-orbit. Especially, we suppose that \(G=GL(1)^{\ell}\times G_ 1\times...\times G_ k\) with \(G_ i\) simple linear algebraic groups, and \(V=V_ 1\oplus...\oplus V_{\ell}\) where the restriction of \(\rho\) on \(V_ j\) is an irreducible representation of \(G_ 1\times...\times G_ k\). The author has already classified all the prehomogeneous vector spaces when \(\ell =1\), i.e., irreducible representations, in the joint work with \textit{M. Sato} [Nagoya Math. J. 65, 1-155 (1977; Zbl 0321.14030)]. In this paper, the author solves the problem of classification when \(\ell \geq 2\). In the last section of this paper, a complete table of prehomogeneous vector spaces and their relative invariants is given.
classification of complex prehomogeneous vector spaces
scientific article; zbMATH DE number 3845752