Macdonald operators at infinity

Abstract

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1 , x 2 , . . . and of two parameters q, t are their eigenfunctions. These operators are defined as limits at N → ∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1 , . . . , x N . They are differential operators in terms of the power sum variables p n = x n 1 + x n 2 + · · · and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall-Littlewood symmetric functions of the variables x 1 , x 2 , . . . . Our result also yields elementary step operators for the Macdonald symmetric functions.

Integrable Hierarchy of the Quantum Benjamin-Ono Equation

“…In our recent publication [9] we proved the following theorem. Another proof can be obtained by using the results of [15], see also [10].…”

Lax Operator for Macdonald Symmetric Functions

“…From the geometric point of view they were studied in [15], see also the works [6,14,19]. Explicit expressions for the limits were given in [1,4,16] and independently in [11]. All these expressions involved Hall-Littlewood symmetric functions [9] in the variables x 1 , x 2 , .…”

“…In the present article we construct a different family of commuting operators on Λ such that the Macdonald symmetric functions are their eigenvectors. Unlike in [1,4,11] our operators are expressed in terms of the Hall-Littlewood symmetric functions corresponding to the partitions with one part only. Our construction uses the Lax operator formalism, see Subsection 2.1 for details.…”

Interpolation Macdonald polynomials and Cauchy-type identities

“…These operators are large-N limits of (slightly renormalized) qdifference operators considered below in Section 7. They are analogues of Nazarov-Sklyanin's "Macdonald operators at infinity" [16], which are diagonalized in the homogeneous basis {P µ ( · ; q, t)}. For the latter operators, Nazarov and Sklyanin derived a nice expression in terms of the Hall-Littlewood symmetric functions.…”