Generalized Macdonald-Ruijsenaars systems

Abstract

We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in types A_n, (C_n^\vee,C_n). We obtain commutative algebras of difference operators given by the action of invariant combinations of Cherednik-Dunkl operators in the corresponding quotient modules of the polynomial representation. This gives known and new generalized Macdonald-Ruijsenaars systems. Thus in the cases of DAHAs of types A_n and (C_n^\vee,C_n) we derive Chalykh-Sergeev-Veselov operators and a generalization of the Koornwinder operator respectively, together with complete sets of quantum integrals in the explicit form.

arXiv comment50 pages; v2: the main change is the addition of theorems 4.5, 5.7 on quantum integrals

References in corpus (9)

• Self-dual Koornwinder-Macdonald polynomials
• Macdonald difference operators and Harish-Chandra series
• Subrepresentations in the Polynomial Representation of the Double Affine Hecke Algebra of type $G{L}_n$ at $t^{k+1}{q}^{r-1}=1$
• Composition Factors of Polynomial Representation of DAHA and Crystallized Decomposition Numbers
• The polynomial representation of the double affine Hecke algebra of type $(C_n^{\vee },C_n)$ for specialized parameters
• Lectures on the affine Hecke algebras and Macdonald conjectures
• Generalized Calogero-Moser systems from rational Cherednik algebras
• Nonsymmetric Koornwinder polynomials and duality
• Macdonald polynomials and algebraic integrability