On reducing submodules of Hilbert modules with [formula omitted]-invariant kernels

Abstract

Fix a bounded domain Ω⊆Cn and a positive definite kernel K on Ω, both invariant under Sn, the permutation group on n symbols. Let H⊆Hol(Ω) be the Hilbert module determined by K. We show that H splits into orthogonal direct sum of subspaces PpH indexed by the partitions p of n. We prove that each submodule PpH is a locally free Hilbert module of rank equal to square of the dimension χp(1) of the irreducible representation corresponding to p. Given two partitions p and q, we show that if χp(1)≠χq(1), then the sub-modules PpH and PqH are not unitarily equivalent. We prove that if H is a contractive analytic Hilbert module on Ω, then the Taylor joint spectrum of the n-tuple of multiplication operators by elementary symmetric polynomials on PpH is clos(s(Ω)), where s:Cn→Cn is the symmetrization map. It is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials C[z]Sn in n variables, equipped with the sup norm on clos(s(Ω)).