On unitary submodules in the polynomial representations of rational Cherednik algebras

TL;DR

This paper studies the unitarity of minimal submodules in polynomial representations of rational Cherednik algebras, establishing convergence of Gaussian inner products for types B, D, and A, with partial results for exceptional groups.

Contribution

It provides new proofs of unitarity for minimal submodules in types B, D, and A, and extends partial results to exceptional Coxeter groups and type B with unequal parameters.

Findings

Gaussian integrals converge for minimal submodules in types B and D

Minimal submodules in type A are unitary, with a new proof provided

Partial unitarity results obtained for exceptional Coxeter groups

Abstract

We consider representations of rational Cherednik algebras which are particular ideals in the ring of polynomials. We investigate convergence of the integrals which express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at most one exception, hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type A was obtained by Etingof and Stoica, we give a different proof of convergence of the Gaussian product in this case. We also obtain partial results on unitarity of the minimal submodule in the case of exceptional Coxeter groups and group B with unequal parameters.