Discrete harmonic analysis on a Weyl alcove

TL;DR

This paper develops an explicit integrable discrete Laplacian on the Weyl alcove using double affine Hecke algebra representations, employing Bethe Ansatz to diagonalize it with Macdonald spherical functions.

Contribution

It introduces a novel discrete Laplacian linked to the double affine Hecke algebra at q=1, providing a new integrable discretization of the Laplace operator with explicit diagonalization.

Findings

Constructed an explicit integrable discrete Laplacian on the Weyl alcove.

Diagonalized the Laplacian using Bethe Ansatz and Macdonald spherical functions.

Connected the Laplacian to the double affine Hecke algebra at the critical level.

Abstract

We introduce a representation of the double affine Hecke algebra at the critical level q=1 in terms of difference-reflection operators and use it to construct an explicit integrable discrete Laplacian on the Weyl alcove corresponding to an element in the center. The Laplacian in question is to be viewed as an integrable discretization of the conventional Laplace operator on Euclidian space perturbed by a delta-potential supported on the reflection hyperplanes of the affine Weyl group. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.