Abstract:
Macdonald polynomials can be defined as common eigenfunctions of a system of commuting difference operators — the Ruijsenaars-Schneider system. They admit a SL(2,Z) group of symmetries and SL(2,Z) is the mapping class group of a torus. One may ask, can Macdonald polynomials be generalized to admit symmetries of mapping class groups of higher genus surfaces? We give positive answer for genus 2, and construct new polynomials — genus 2 Macdonald polynomials. They are common eigenfunctions of an interesting system of difference operators, and we prove that they admit an action of the genus 2 mapping class group. Time permitting, we will talk about generalizations to higher genus, algebraic aspects, and applications to knot theory.
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