A classification of a certain class of reductive prehomogeneous vector spaces. II
[Part I cf. Commun. Algebra 17, No.6, 1425-1447 (1989; Zbl 0671.20037).] The purpose of this paper is to classify all reductive prehomogeneous vector spaces \((\tilde G,{\tilde\rho},\tilde V)\) when each irreducible component \((GL(1)\times G,\Lambda_ 1\otimes \rho_ i,V_ i)\) is castling equivalent to a non-trivial reduced irreducible prehomogeneous vector space. Let G be a connected semi-simple linear algebraic group. Every finite dimensional rational representation \(\rho\) : \(G\to GL(V)\) of G can be written as a direct sum \(\rho =\rho_ 1\oplus...\oplus \rho_ k\) of irreducible rational representations \(\rho_ i: G\to GL(V_ i)\) of G in \(V_ i\) \((i=1,...,k)\). Let \({\tilde \rho}\) be the action of \(\tilde G=GL(1)^ k\times G\) on \(\tilde V=V_ 1\oplus...\oplus V_ k\) which is the composition of \(\rho =\rho_ 1\oplus...\oplus \rho_ k\) and independent scalar multiplications \(GL(1)^ k\) on each irreducible component \(V_ i\). We call such a triplet \((\tilde G,{\tilde\rho},\tilde V)\) a reductive prehomogeneous vector space if it is aprehomogeneous vector space. All the reductive prehomogeneous vector spaces are classified when each irreducible component \((GL(1)\times G,\Lambda \otimes \rho_ i,V_ i)\) \((i=1,...,k)\) is a regular prehomogeneous vector space.
connected semi-simple linear algebraic group
scientific article; zbMATH DE number 4127501
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