Macdonald operators at infinity

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}$ and of two parameters $q,t$ are their eigenfunctions. These operators are defined as limits at $N\to \mathrm{\infty }$ of renormalised Macdonald operators acting on symmetric polynomials in the variables $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_N$. They are differential operators in terms of the power sum variables $p_n=x_1^n+x_2^n+\mathrm{.}\mathrm{.}\mathrm{.}$ 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 $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}$. Our result also yields elementary step operators for the Macdonald symmetric functions.