Generalized Macdonald–Ruijsenaars systems

AbstractWe 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 An, (Cn∨,Cn). 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 An and (Cn∨,Cn) 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