Summary: 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 \(\mathcal{H}_K (\mathbb{B}_d, \mathbb{C}^n) \subseteq \mathrm{Hol}(\mathbb{B}_d, \mathbb{C}^n)\), \(n= \dim \cap_{j=1}^d \ker T^*_j\). We describe this class of \(\mathcal{U}(d)\)-homogeneous operators, equivalently, nonnegative 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 \(2, \dots, d-1\), \(d \geqslant 3\). We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.
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