Commuting tuple of multiplication operators homogeneous under the unitary group

Abstract

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