Nicolle Gonzalez: Crystals for nonsymmetric Macdonald polynomials.

Abstract

Nonsymmetric Macdonald polynomials were introduced independently by Opdam and Macdonald and then generalized by Cherednik. These polynomials not only generalize the two-parameter symmetric Macdonald polynomials, when specialized to $t=0$ they were shown by Sanderson and Ion to arise as characters of affine Demazure modules. In this talk, I will present a new crystal-theoretic proof of this fact that is based on joint work with Sami Assaf. I will describe a purely combinatorial construction for affine and finite Demazure crystals as well as certain embedding operators which parallel the nested filtrations of Demazure modules. At the level of characters, these operators provide crystal-theoretic lifts of the Macdonald recursion operators of Knop and Sahi. This yields a new proof for the results of Sanderson as well as explicit formulas for the Demazure character expansion of specialized nonsymmetric Macdonald polynomials in terms of certain lowest weight elements. No prior knowledge of Demazure modules or crystals will be assumed.