Discrete harmonic analysis on a Weyl alcove

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.

52 pages, updated references and modified according to the comments of the referee

References in corpus (7)

• Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
• The Dynamics of the One-Dimensional Delta-Function Bose Gas
• Trigonometric Cherednik algebra at critical level and quantum many-body problems
• Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle
• The propagator of the attractive delta-Bose gas in one dimension
• Unitary representations of affine Hecke algebras related to Macdonald spherical functions
• A discrete Fourier transform associated with the affine Hecke algebra

Cited by in corpus (5)

• Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove
• Orthogonality of Macdonald polynomials with unitary parameters
• Discrete Fourier transform associated with generalized Schur polynomials
• Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
• On the basic representation of the double affine Hecke algebra at critical level