Toeplitz operators on the proper images of bounded symmetric domains

Let be a bounded symmetric domain in Cⁿ and f: ^ be a proper holomorphic mapping factored by (automorphisms) a finite complex reflection group G. We define an appropriate notion of the Hardy space H² (^) on ^ which can be realized as a closed subspace of an L²-space on the Silov boundary of ^. We study various algebraic properties of Toeplitz operators (such as the finite zero product property, commutative and semi-commutative property etc. ) on H² (^). We prove a Brown-Halmos type characterization for Toeplitz operators on H² (^), where ^ is an image of the open unit polydisc in Cⁿ under a proper holomorphic mapping factored by an irreducible finite complex reflection group.