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 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.
Highlights
The main goal of this paper is to obtain commutative algebras of difference operators containing generalizations of Macdonald–Ruijsenaars operators using special representations of Double Affine Hecke Algebras (DAHAs)
Commutative algebras of difference operators containing the operators constructed by Macdonald were realized by Cherednik with the help of his DAHA and its polynomial representation [5,8]
In the present work we develop a uniform approach to the generalized Macdonald–Ruijsenaars systems based on the special representations of DAHAs
Summary
The main goal of this paper is to obtain commutative algebras of difference operators containing generalizations of Macdonald–Ruijsenaars operators using special representations of Double Affine Hecke Algebras (DAHAs). Upon restriction to Weyl group invariants we obtain commutative algebras of difference operators In this way we recover operator (1.2) (under assumption that h /η ∈ Z) for the case of DAHA of type A and derive further known and new generalized Macdonald–Ruijsenaars operators starting from other root systems. The generalized Macdonald–Ruijsenaars operators which we obtain carry these restrictions This is different from the type A construction of [29] where the authors work in the space of symmetric functions in infinitely many variables and consider some invariant ideals in that algebra.
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