A classification of 2-simple prehomogeneous vector spaces of type I
The purpose of this paper is the classification of 2-simple prehomogeneous vector spaces of type I. A prehomogeneous vector space (G,\(\rho\),V) is called a 2-simple prehomogeneous vector space when (1) \(G=GL(1)^{\ell}\times G_ 1\times G_ 2\) with simple algebraic groups \(G_ 1\) and \(G_ 2\), (2) \(\rho\) is the composition of a rational representation \(\rho '\) of \(G_ 1\times G_ 2\) of the form \[ \rho '=\rho_ 1\otimes \rho_ 1'+...+\rho_ k\otimes \rho_ k'+(\sigma_ 1+...+\sigma_ s)\otimes 1+1\otimes (\tau_ 1+...+\tau_{\ell}) \] with \(k+s+t=\ell\), where \(\rho_ i\), \(\sigma_ i\) (resp. \(\rho_ j'\), \(\tau_ j)\) are nontrivial irreducible representations of \(G_ 1\) (resp. \(G_ 2)\), and the scalar multiplications \(GL(1)^{\ell}\) on each irreducible component \(V_ i\) for \(i=1,...,\ell\) where \(V=V_ 1\oplus...\oplus V_{\ell}\). We say that a 2-simple prehomogeneous vector space (G,\(\rho\),V) is of type I if \(k\geq 1\) and at least one of \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) is a non-trivial prehomogeneous vector space. If \(k\geq 1\) and all \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) are trivial prehomogeneous vector spaces, it is called of type II. In a previous paper [\textit{T. Kimura}, \textit{S.-i. Kasai}, \textit{M. Taguchi} and \textit{M. Inuzuka}, Trans. Am. Math. Soc. 308, No.2, 433-494 (1988)], all 2-simple prehomogeneous vector spaces of type II have been classified. Together with the previous paper, this paper completes a classification of all 2-simple prehomogeneous vector spaces.
2-simple prehomogeneous vector spaces of type I
scientific article; zbMATH DE number 4075392
scientific article