Generalized Calogero-Moser systems from rational Cherednik algebras

Abstract

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.

36 pages; the main change is an improvement of section 7 so that it now deals with an arbitrary complex reflection group; Selecta Math, 2011

References in corpus (10)

• A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
• Unitary representations of rational Cherednik algebras
• Subrepresentations in the Polynomial Representation of the Double Affine Hecke Algebra of type $G{L}_n$ at $t^{k+1}{q}^{r-1}=1$
• On the symmetry of commuting differential operators with singularities along hyperplanes
• Non-semisimple Macdonald polynomials
• Composition Factors of Polynomial Representation of DAHA and Crystallized Decomposition Numbers
• A basis for the polynomial eigenfunctions of deformed Calogero-Moser-Sutherland operators
• The polynomial representation of the double affine Hecke algebra of type $(C_n^{\vee },C_n)$ for specialized parameters
• Reducibility of the polynomial representation of the degenerate double affine Hecke algebra
• Deformation of two body quantum Calogero-Moser-Sutherland models

Cited by in corpus (1)

• Generalized Macdonald-Ruijsenaars systems