We introduce a family of Hardy spaces {H-(sic)} ((sic)EG)1 on the distinguished boundary of the quotient domain Dn/G, where G is a finite pseudoreflection group acting on Dn and Gs1 is the set of equivalence classes of one-dimensional representations of G. We establish a uniform platform to study L-p regularity properties of the generalized Szego projections associated to Hardy spaces .7-C% for every (sic) ? G1.