  We construct a large family of positive-definite kernels $K:
\mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in
the first variable and anti-holomorphic in the second, that are quasi-invariant
with respect to the subgroup $\mbox{M\"ob} \times\cdots\times \mbox{M\"ob}$
($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. The
adjoint of the $n$ - tuples of multiplication operators by the co-ordinate
functions on the Hilbert spaces $\mathcal H_K$ 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 $\mathrm B_r(\mathbb D^n)$ and that
they are mutually pairwise unitarily inequivalent.

Functional Analysis