Generating Systems for Finite Irreducible Complex Reflection Groups

Abstract

For each finite irreducible complex reflection group G in GL(n,C), we construct a system E_G(z) of differential equations on Z $\simeq$ P^n-1 of rank n with the monodromy group G, and with the following generating property: If a system E′(z) on Z of rank n has a finite monodromy group and a projective monodromy group which is a subgroup of P(G), there is an algebraic transformation
E′(z) = θ(z)^1/kE_G(σ(z)),
where k is an integer, θ(z) a rational function on Z, and σ(z) a rational map of Z to Z. For n = 2, 3, we give explicit forms of E_G(z). Several examples of the above algebraic transformation are also given.