VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Quasi-polynomial generalizations of Macdonald polynomials
Jasper Stokman (Universiteit van Amsterdam)
Wednesday 9th June, 2021 13:30-14:30 Zoom seminar hosted by ICMS
Double affine Hecke algebras (DAHAs) have applications in various areas of mathematics and mathematical physics, including integrable systems. A central role in the theory is played by Cherednik's polynomial DAHA-representation. It provides a commuting family of q-difference-reflection operators called Cherednik operators, acting on a space of Laurent polynomials in several variables. Their simultaneous eigenfunctions are the non-symmetric Macdonald polynomials. The polynomial representation is an example of a so-called standard Y-cyclic DAHA-representation. This is reflected by the fact that the polynomial representation has a generator which is a simultaneous eigenfunction of the Cherednik operators, namely the constant polynomial 1 (it is the non-symmetric Macdonald polynomial of degree zero). Quasi-monomials are monomials for which non-integral exponents are allowed, and quasi-polynomials are linear combinations of quasi-monomials. In this talk I will give realizations of all standard Y-cyclic DAHA-representations on spaces of quasi-polynomials. I will also introduce the corresponding quasi-polynomial generalizations of the Macdonald polynomials. In the last part of talk I indicate some applications, with a focus on those related to integrability. This is joint work with Siddhartha Sahi and Vidya Venkateswaran.
For more information, and how to access the seminar through Zoom, see the webpage of the events here.