Elliptic Ruijsenaars difference operators on bounded partitions

TL;DR

This paper constructs a finite-dimensional basis of eigenfunctions for elliptic Ruijsenaars difference operators on bounded partitions, connecting to Macdonald polynomials in the trigonometric limit.

Contribution

It introduces a truncation method for elliptic Ruijsenaars operators, leading to a finite basis of eigenfunctions expressed via polynomials, bridging to Macdonald polynomials.

Findings

Finite basis of eigenfunctions constructed

Connection to Macdonald polynomials established

Truncation method applied to elliptic operators

Abstract

By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials.