We introduce a family of Hardy spaces on the distinguished boundary of the quotient domain Dⁿ/G, where G is a finite pseudoreflection group acting on the polydisc Dⁿ. We establish a uniform platform to study L^p regularity properties of the generalized Szego projections associated to these Hardy spaces.

The first named author is supported by Centre for Applicable Mathematics, Tata Institute of Fundamental Research. The second named author acknowledges the financial support from CV Raman postdoctoral fellowship from Indian Institute of Science and postdocotoral fellowship from Silesian University in Opava under GA CR grant no. 21-27941S.