Generalized Calogero-Moser systems from rational Cherednik algebras

Misha Feign

Tuesday 21st October, 2008 15:00-16:00 Mathematics Building, room 204

Abstract

Integrability of quantum Calogero-Moser system can be easily established with the help of Dunkl operators. In algebraic terms, Calogero-Moser operator and its quantum integrals are certain pairwise commuting elements of the rational Cherednik algebra acting in the polynomial representation. We use Dunkl operators and special submodules of the polynomial representation in order to construct integrable generalizations of the Calogero-Moser systems. These generalizations are determined by a Coxeter group and a special parabolic subgroup. In more detail, the submodules consist of the polynomials vanishing on the Coxeter orbit of the set of poins fixed under the action of the parabolic subgroup. Then Dunkl operators at the special value of the parameter can be restricted to the polynomials defined on this orbit, which then define generalized Calogero-Moser system and its quantum integrals. It is possible to describe explicitly both all parabolic subgroups which determine such submodules in the polynomial representation and the corresponding generalized Calogero-Moser systems.

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