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

pi → X ′ f → Y be a covering of smooth, projective complex curves such that pi is a degree 2 étale covering and f is a degree d covering, with monodromy group Sd, branched in n + 1 points one of which is a spe-cial point whose local monodromy has cycle type given by the partition e = (e1,..., er) of d. We study such coverings whose monodromy group is either W(Bd) or wN(W(Bd))(G1)w−1 for some w ∈ W(Bd), where W(Bd) is the Weyl group of type Bd, G1 is the subgroup of W(Bd) generated by reflections with respect to the long roots εi − ε j and N(W(Bd))(G1) is the normalizer of G1. We prove that in both cases the corresponding Hurwitz spaces are not connected and hence are not irreducible. In fact, we show that if n+ |e | ≥ 2d, where |e | =∑ri=1(ei − 1), they have 22g − 1 connected components