From quantum toroidal algebras to wreath Macdonald operators

Hybrid. Contact: Christian Kaiser (kaise @ mpim-bonn.mpg.de)

In this talk, I will explain how quantum toroidal algebras lead to a novel family of commuting difference operators whose joint eigenfunctions are the wreath Macdonald polynomials. The latter polynomials arise from the study of symplectic quotient singularities involving the wreath products of symmetric groups with an arbitrary finite cyclic group. When the cyclic group is trivial, one recovers the usual (type A) Macdonald difference operators and polynomials, whose theory has been extensively developed. In contrast, very little is known about wreath Macdonald polynomials in general. Our 'wreath Macdonald operators' provide a new, more direct characterization of wreath Macdonald polynomials for arbitrary finite cyclic groups. This is joint work with Mark Shimozono and Joshua Wen.