Macdonald Duality and the proof of the Quantum Q-system conjecture

TL;DR

This paper proves that time-evolved Macdonald operators form a representation of quantum Q-systems in the $q$-Whittaker limit for all affine types, using duality properties and a Fourier transform approach.

Contribution

It establishes the connection between Macdonald operators and quantum Q-systems across all affine types, extending previous results and providing a unified proof framework.

Findings

Macdonald operators form quantum Q-systems in the $q$-Whittaker limit

Duality of Macdonald and Koornwinder polynomials is crucial

Fourier transformed picture simplifies the proof

Abstract

The $SL(2,\mathbb Z)$-symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type $A$. We prove for all twisted and untwisted affine algebras of type $ABCD$ that the time-evolved $q$-difference Macdonald operators, in the $t\to\infty$ $q$-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the $q$-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic $q$-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the "Fourier transformed" picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further.