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 Int. Math. Res. Not. IMRN, 2012, Volume 2012, Issue 15, Pages 3375–3414 (Mi imrn8)

On unitary submodules in the polynomial representations of rational Cherednik algebras

M. Feigina, C. Shramovb

a School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK
b Steklov Mathematical Institute, Gubkina str., 8, Moscow 119991, Russia

Abstract: We consider representations of rational Cherednik algebras that are particular ideals in the ring of polynomials. We investigate convergence of the integrals that 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.

 Funding Agency Grant Number Engineering and Physical Sciences Research Council EP/F032889/1 British Council PMI2 Russian Foundation for Basic Research 08-01-00395-a11-01-00185-a11-01-00336-a Ministry of Education and Science of the Russian Federation 4713.2010.111.G34.31.0023 The work of M. F. was partially supported by the Engineering and Physical Sciences Research Council (grant number EP/F032889/1), M. F. also acknowledges support of the British Council (PMI2 Research Cooperation award). The work of C. S. was partially supported by the Russian Foundation for Basic Research (grant numbers 08-01-00395-a, 11-01-00185-a, 11-01-00336-a, and Scientific School grant number 4713.2010.1) and by AG Laboratory GU-HSE, RF government grant 11 11.G34.31.0023.

DOI: https://doi.org/10.1093/imrn/rnr140

Bibliographic databases:

Document Type: Article
Revised: 18.04.2011
Accepted:21.06.2011
Language: English