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- W2315873232 abstract "Introduction. Let p': G' -* GL(V) be a finite-dimensional rational representation of a semi-simple algebraic group G', all defined over C. Then, we have p' = Pi ?) ... (Epe and V= V1I ?D * ** Ve where pi is an irreducible representation of G' on Vi for i = 1, . . ., f . Let p be the action of G = GL(1) I X G' on V which is the composition of p' and scalar multiplications GL(1) Ion each irreducible component Vi (i = 1, . . ., f). In this paper, we shall classify all the triplet (G, p, V) such that V decomposes into a finite union of G-orbits. Prof. V. G. Kac ([3], [4]) and the first author ([7], [14], but only the necessary conditions are written in [14]) have already completed independently the case when p is irreducible, i.e., f = 1. Prof. V. G. Kac has also completed the case when each irreducible components are (GL(n) X GL(m), Al ? Al, V(n) ? V(m)) (see [5]) which is the development of the results of Prof. P. Gabriel ([1]). Note that if V decomposes into the finite union of G-orbits, then one of the orbits must be Zariski-dense, and hence (G, p, V) must be a prehomogeneous vector space (abbrev. P. V.). We call such a P. V. a finite prehomogeneous vector space (abbrev. F.P.). This paper consists of the following sections." @default.
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- W2315873232 date "1986-06-01" @default.
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- W2315873232 title "A Classification of the Representations of Reductive Algebraic Groups Which Admit Only a Finite Number of Orbits" @default.
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- W2315873232 doi "https://doi.org/10.2307/2374658" @default.
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