Interpolation Macdonald operators at infinity

TL;DR

This paper provides explicit formulas for a sequence of commuting operators diagonalized by interpolation Macdonald functions, connecting Hall-Littlewood functions and their inhomogeneous variants, extending recent operator results.

Contribution

It introduces explicit formulas for the operators $A^k$, involving new inhomogeneous Hall-Littlewood functions, extending Nazarov-Sklyanin and Sekiguchi-Debiard results.

Findings

Explicit formula for operators $A^k$ involving Hall-Littlewood functions

Construction and identification of inhomogeneous Hall-Littlewood functions as degenerations

Extension of recent Macdonald and Sekiguchi-Debiard operator results

Abstract

We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence $A^1, A^2, \ldots$ of commuting operators that are diagonalized by them. Such a sequence of operators arises in the projective limit of finite families of commuting q-difference operators studied by Okounkov, Knop and Sahi. The main theorem is an explicit formula for the operators $A^k$. Our formula involves the family of Hall-Littlewood functions and a new family of inhomogeneous Hall-Littlewood functions, for which we give an explicit construction and identify as a degeneration of the interpolation Macdonald functions in the regime $q \rightarrow 0$. This article is inspired by the recent papers of Nazarov-Sklyanin on Macdonald and Sekiguchi-Debiard operators, and our main theorem is an extension of their results.