Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

Let $\mathcal U(d)$ be the group of $d\times d$ unitary matrices. We find conditions to ensure that a $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol T$ is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq \mbox{\rm Hol}(\mathbb B_d, \mathbb C^n)$, $n= \dim \cap_{j=1}^d \ker T^*_{j}.$ We describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative kernels $K$ quasi-invariant under the action of $\mathcal U(d)$. We classify quasi-invariant kernels $K$ transforming under $\mathcal U(d)$ with two specific choice of multipliers. A crucial ingredient of the proof is that the group $SU(d)$ has exactly two inequivalent irreducible unitary representations of dimension $d$ and none in dimensions $2, \ldots , d-1$, $d\geq 3$. We obtain explicit criterion for boundedness, reducibility and mutual unitary equivalence among these operators.