Macdonald operators at infinity
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 \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the HallLittlewood symmetric functions of the variables x 1,x 2,. Our result also yields elementary step operators for the Macdonald symmetric functions.
macdonald symmetric functions