Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type Dd

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D-d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when Y similar or equal to P-1 and successively we extend the result to curves of genus g >= 1.