Functional models of Γn-contractions and characterization of Γn-isometries

We show that a weighted Bergman space on the polydisc splits up into the orthogonal direct sum of subspaces, each corresponding to an irreducible representation of the symmetric group. The multiplication by the n-tuple of elementary symmetric functions is a Γn-contraction on each of these subspaces. A necessary condition for a commuting n-tuple to be a Γn-contraction is obtained in terms of pencil of operators. We provide a model theory for Γn-isometries. A Beurling-Lax-Halmos type representation for the invariant subspaces of Γn-isometries is given.