Geodesic
Macdonald operators at infinity
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 23-44.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables $x_1,x_2,\dots$ and of two parameters $q$, $t$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalized Macdonald operators acting on symmetric polynomials in the variables $x_1,\dots ,x_N$. 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 Hall-Littlewood symmetric functions of the variables $x_1,x_2,\dots$ Our result also yields elementary step operators for the Macdonald symmetric functions.
Classification : 05E05
Keywords: Macdonald symmetric functions 
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     title = {Macdonald operators at infinity},
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Nazarov, M.L.; Sklyanin, E.K. Macdonald operators at infinity. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 23-44. http://geodesic.mathdoc.fr/item/JAC_2014__40_1_a11/