On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians

Abstract

Quite some years ago, Oleg Chalykh has built a nice theory from the observation that the Macdonald polynomial reduces at $t=q^{-m}$ to a sum over permutations of simpler polynomials called Baker-Akhiezer functions, which can be unambiguously constructed from a system of linear difference equations. Moreover, he also proposed a generalization of these polynomials to the twisted Baker-Akhiezer functions. Recently, in a private communication Oleg suggested that these twisted Baker-Akhiezer functions could provide eigenfunctions of the commuting Hamiltonians associated with the $(-1,a)$ rays of the Ding-Iohara-Miki algebra. In the paper, we discuss this suggestion and some evidence in its support.

14 pages, LaTeX

References in corpus (4)

• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems

Cited by in corpus (5)

• Generating twisted Cherednik eigenfunctions
• Twisted Cherednik spectrum as a $q,t$-deformation
• Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems
• Diamond of triads
• Twisted Baker-Akhiezer function from determinants