Commuting tuple of multiplication operators homogeneous under the unitary group

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