Generalized Macdonald polynomials at higher level

Macdonald polynomials can be introduced as eigenstates of theMacdonald Hamiltonian. This operator coincides with a certain elementof the quantum toroidal gl(1) algebra evaluated in the Fockrepresentation of levels (1,0). Representations of higher levels (r,0)produce generalizations of this Hamiltonian, and their eigenstatesdefine generalized Macdonald polynomials. In a joint work with L.Cassia and A. Stoyan [2508.19704], we extend some of the well-knownformulas obtained for ordinary Macdonald polynomials to their higherlevel counterpart. In particular, we propose a generalization of theformula relating Macdonald polynomials to Whittaker vectors obtained byGarsia, Haiman and Tesler. The talk will start with a brief review ofthe quantum toroidal gl(1) algebra, and its action on the ring ofsymmetric functions. Then, I will introduce Macdonald polynomials andtheir generalizations, and present our main results. I will concludewith a discussion on possible applications to the (q,t)-deformedintegrable hierarchies recently introduced with A. Garbali.