Commuting tuple of multiplication operators homogeneous under the unitary group

Let (Formula presented.) be the group of (Formula presented.) unitary matrices. We find conditions to ensure that a (Formula presented.) -homogeneous (Formula presented.) -tuple (Formula presented.) is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space (Formula presented.), (Formula presented.). We describe this class of (Formula presented.) -homogeneous operators, equivalently, nonnegative kernels (Formula presented.) quasi-invariant under the action of (Formula presented.). We classify quasi-invariant kernels (Formula presented.) transforming under (Formula presented.) with two specific choice of multipliers. A crucial ingredient of the proof is that the group (Formula presented.) has exactly two inequivalent irreducible unitary representations of dimension (Formula presented.) and none in dimensions (Formula presented.), (Formula presented.). We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.