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

We prove the irreducibility of the Hurwitz spaces which parametrize equivalence classes of Galois coverings of , 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.