The functional equation of zeta distributions associated with prehomogeneous vector spaces

Abstract.A classification of 2-simple prehomogeneous vector spaces is completed by using some P.V.-equivalences together with [3]. Some part is very different from the previous classification of the irreducible or simple cases [1,2], and some new method is necessary. This result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces.Introduction. In M. Sato and T. Kimura [1], all irreducible prehomogeneous vector spaces (abbreviated irreducible P.V.'s) are classified up to castling-equivalence, and it is proved that any irreducible P.V. is castling-equivalent to a 2-simple P.V. (or to (SL(tn) X SL{m) X GL(2), Aj ® A: ® Ax) with m = 2,3). Therefore, as a step to a classification of the general case, it is natural to classify all nonirreducible 2-simple P.V.'s. In this paper, we shall classify all 2-simple P.V.'s of type II and give the complete list of them up to strong equivalence in the sense of Definition 4, p. 36 in [1]. However we need various P.V.-equivalences (cf. Propositions 1.7; 1.12; 1.34; 1.36; 1.37, Theorem 1.16 and its Corollary; Corollary 1.26, Definition 1.31, Remark 1.32) to carry out the classification, and hence, first we shall prove them. We also investigate the regularity in some cases. Since all 2-simple P.V.'s of type I are already classified in [3], this completes the classification of all 2-simple P.V.'s. Here we say that (G, p, V) (or simply (G, p)) is a 2-simple P.V. if it is a prehomogeneous vector space of the following form: G = GL(l)k+s+t xGy x G2 where Gy and G2 are simple algebraic groups, p = the composition of the scalar multiplications GL(l)k+s + ' on each irreducible component and the representation p' = px ® p[ + ■■■ +pk® p'k + (

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The functional equation of zeta distributions associated with prehomogeneous vector spaces

Abstract: Let (G, ρ, V) be a triple of a linear algebraic group G and a rational representation ρ on a finite dimensional vector space V, all defined over the complex number field C.

Cited by 4 publications

References 7 publications

A classification of 2-simple prehomogeneous vector spaces of type I

A classification of 2-simple prehomogeneous vector spaces of type I

A Classification Theory of Prehomogeneous Vector Spaces

Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type 𝐼𝐼

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