Interpolation Macdonald operators at infinity

Abstract

We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence A1,A2,… of commuting operators whose common eigenfunctions are the interpolation Macdonald functions. 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 Ak. 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 when q→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.