## A family of homogeneous operators in the Cowen–Douglas class over the poly-disc

### Volume 271 / 2023

#### Abstract

We construct a large family of positive definite kernels $K: \mathbb D^n\times \mathbb D^n \to \mathrm M (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb$\,\times \cdots \times\,$Möb ($n$ times) of the bi-holomorphic automorphism group of $\mathbb D^n$. The adjoint of the $n$-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space $\mathcal H_K$ determined by $K$. We show that these $n$-tuples are irreducible, are in the Cowen–Douglas class $\mathrm B_r(\mathbb D^n)$ and are mutually pairwise unitarily inequivalent.