Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

Motivated by the quest to unify various superintegrable formulas in matrix models, naturally arises the question of commutative families of operators (Hamiltonians) in DIM algebra, and therefore also of their eigenfunctions. We discuss interrelations between eigenfunctions of the Hamiltonians associated with the commutative (integer ray) subalgebras of the Ding-Iohara-Miki algebra and those of the twisted Cherednik system. In the case of t=q^{−m} with natural m, eigenfunctions of the first system of Hamiltonians are the twisted Baker-Akhiezer functions (BAFs) introduced by O. Chalykh, while eigenfunctions of the twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials.