Chevalley's theorem for the complex crystallographic groups

Abstract

We prove that, for the irreducible complex crystallographic Coxeter group W, the following conditions are equivalent: a) W is generated by reflections; b) the analytic variety X/W is isomorphic to a weighted projective space. The result is of interest, for example, in application to topological conformal field theory. We also discuss the status of the above statement for other types of complex crystallographic group W and certain generalizations of the statement. It is impossible to read this paper without first reading our paper [5] which contains all the notations and the data on affine root systems and complex crystallographic Coxeter groups. All the data needed on the modular functions theory is collected in §4.