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

Abstract

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. 

 

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