On unitary submodules in the polynomial representations of rational Cherednik algebras

Details On unitary submodules in the polynomial representations of rational Cherednik algebras On unitary submodules in the polynomial representations of rational Cherednik algebras

2010-10-20

arxiv.org/abs/1010.4245v2

Mathematics

Representation Theory

37 pages, minor changes; IMRN 2011

Scientific paper

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.

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