Generalized Macdonald-Ruijsenaars systems

TL;DR

This paper generalizes Macdonald-Ruijsenaars systems using polynomial representations of Double Affine Hecke Algebras, constructing new integrable difference operators and quantum integrals for types A_n and (C_n^\/vee,C_n).

Contribution

It introduces a unified framework for constructing generalized Macdonald-Ruijsenaars systems via DAHA submodules, extending known results to new algebraic types.

Findings

Constructed submodules as ideals vanishing on affine planes.

Derived new and known integrable difference operators.

Provided explicit quantum integrals for the 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.