Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups

We prove the irreducibility of the Hurwitz spaces which parametrize equivalence classes of Galois coverings of P1, whose Galois group is an arbitrary Weyl group, and the local monodromies are reflections. This generalizes a classical theorem due to Lüroth, Clebsch and Hurwitz.