A Family of Homogeneous Operators In The Cowen-Douglas Class Over The Poly-disc

Abstract

We construct a large family of positive-definite kernels K : D n × D n → M(r, C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb × • • • × Möb (n times) of the bi-holomorphic automorphism group of D n . The adjoint of the n -tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces HK determined by K is then homogeneous with respect to this subgroup. We show that these n -tuples are irreducible, are in the Cowen-Douglas class Br(D n ) and that they are mutually pairwise unitarily inequivalent.