On a Linear Structure of the Quotient Variety by a Finite Reflexion Group

Abstract

The objective of the article is to show that the orbit space of a finite reflection group acting on the complexification of the real vector space carries naturally a complex vector space structure Q together with a nondegenerate bilinear form J on it defined over the real number field. For details of the results, one is referred to Theorems I, II and III in the introduction. This structure on the invariants is called the flat structure. Originaly, the flat structure (published later in [7], Oda [24]) is defined on the deformation space of an isolated hypersurface singularity via Gauss-Manin connection and higher residue theory in general. On the other hand, the deformation of a simple singularity is described in terms of a simple Lie algebra. Namely, the deformation space is given by the quotient space of the Cartan algebra by the action of the Weyl group (Brieskorn [2], Slodowy [33]). This suggests that the flat structure is described in terms only of the Weyl group in that case. Actually, such a description is achieved in the present article so that the above mentioned flat structure on the quotient space of a finite reflection group is obtained. The key fact in the construction of the flat metric is the regularity of the eigenvector for exp (2n*f^\/K) of a Coxeter element with the Coxeter % = h (cf. [6], [15], [34]). Since the present article was written (cf. the Footnote *)), there have been several developments in the study of the flat structure and its generalization, as will be summarized below. Nevertheless the present article