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

Abstract

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 + (<Jy+ ■■■ +os) ® 1 + 1 ® (tx + • • • +t,) of the group G, X G2 where p,, Oj (resp. p,', Ty) are nontrivial irreducible representations of Gy (resp. G2). We write (G, p', V) instead of (G, p, V) for simplicity. To avoid confusion, we write + instead of ©. Clearly each (GL(1) X Gy X G2, p, ® p,') (i = l,...,k)is an irreducible 2-simple P.V. If at least one of them is a nontrivial P.V. (see Definition 1.1), we call it a 2-simple P.V. of type I. On the other hand, if all of them are trivial P.V.'s, we call it a 2-simple P.V. of type II. In this case, we have G2 = SL{n),