Macdonald Polynomials and Algebraic Integrability

Abstract

We construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t=q k , k ¥ Z. This leads to a new, more elementary proof of several Macdonald conjectures, proved first by Cherednik. We also establish the algebraic integrability of Macdonald operators at t=q k (k ¥ Z), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including the BC n case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of the A n root system where the previously known methods do not work.

Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

“…We should mention that there is a small discrepancy between his and our formulas because of the misprint in the formula (7.3) in [37]. It is interesting that the form of the operator (33) is invariant under the simultaneous interchange of q ↔ t and x ↔ y.…”

Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

“…Moreover, this approach suggested using techniques from integrable systems (such as the Darboux transformation) to construct extensions of the Askey-Wilson polynomials which satisfy higher-order q-difference equations [12]. In the multivariable case, algebro-geometric methods were used by Chalykh [2], within the context of symmetric functions, to give more elementary proofs of several of Macdonald's conjectures. It is a challenging problem to construct a Baker-Akhiezer type function for the operators discussed in this paper and prove the duality using algebro-geometric tools.…”

Bethe Ansatz for the Ruijsenaars Model of BC1-Type

“…[2,3]. We begin with two elementary results about a three-term difference operator with meromorphic coefficients:…”