On a certain class of prehomogeneous vector spaces
Let k be a field of characteristic 0. Let \(X=Aff^ n\) be an n- dimensional k-affine space and let \(S:=\{x\in X\); \(f(x)=0\}\) be an irreducible hypersurface in X defined over k. We denote \(Y:=X-S\). The author presents the following problem: classify all the pairs (G,Y) where G is a connected irreducible k-subgroup of GL(X) which acts on Y transitively. Such pairs become special cases of prehomogeneous vector spaces defined over k. He solves this problem in the case when G splits over k and a quaternion division k-algebra exists. In his proof, he utilizes effectively the property that the roots of b-function of \(f^ s\) are all integers. This property is held under ``castling transform'', hence it is sufficient to classify all the ``reduced'' pairs in the sense of Sato-Kimura.
scientific article; zbMATH DE number 4029751