Basic triad in Macdonald theory

Macdonald polynomials are important special functions in modern mathematical physics. Despite this, the widely known ways to define them appear somewhat artificial (ad hoc): for instance, the Schur polynomials have much more direct connection to representation theory. In 2012 Noumi and Shiraishi proposed a unified view on Macdonald polynomials, where they appear at particular spectral parameter values of the generating function, which in turn solves bispectral problem for Ruisenaars-Schneider system. Our observation is that another specialization of the Noumi-Shiraishi function equals to yet another object: (multivariate) Baker-Akhiezer function (BAF), introduced by O.Chalykh in 2013; this function is much more directly constructed based on a root system. Quite surprisingly, with just a small amount of auxiliary symmetry arguments one can go from one object to another – therefore, it makes sense to consider them as forming a triality, or "triad". In my talk I will explain how these three objects are interrelated, as well as highlight the problems that seem natural in view of this connection and which have good chances to be resolved in the near future.