Demazure-Lusztig Operators and Metaplectic Whittaker Functions on Covers of the General Linear Group

There are two different approaches to constructing Whittaker functions of metaplectic groups over non-archimedean local fields. One approach, due to Chinta and Offen for the general linear group and to McNamara in general, represents the spherical Whittaker function in terms of a sum over a Weyl group. The second approach, by Brubaker, Bump and Friedberg and separately by McNamara, expresses it as a sum over a highest weight crystal. This work builds a direct, combinatorial connection between the two approaches. This is done by exploring both in terms of Demazure and Demazure-Lusztig operators associated to the Weyl group of an irreducible root system. The relevance of Demazure and Demazure-Lusztig operators is indicated by results in the non-metaplectic setting: the Demazure character formula, Tokuyama's theorem and the work of Brubaker, Bump and Licata in describing Iwahori-Whittaker functions. The first set of results is joint work with Gautam Chinta and Paul E. Gunnells. We define metaplectic Demazure and Demazure-Lusztig operators for a root system of any type. We prove that they satisfy the same Braid relations and quadratic relations as their nonmetaplectic analogues. Then we prove two formulas for the long word in the Weyl group. One is a metaplectic generalization of Demazure's character formula, and the other connects the same expression to Demazure-Lusztig operators. Comparing the two results to McNamara's construction of metaplectic Whittaker functions results in a formula for the Whittaker functions in the spirit of the Demazure character formula. The second set of results relates to Tokuyama's theorem about the crystal description of type A characters. We prove a metaplectic generalization of this theorem. This establishes a combinatorial link between the two approaches to constructing Whittaker functions for metaplectic covers of any degree. The metaplectic version of Tokuyama's theorem is proved as a special case of a stronger result: a crystal description of polynomials produced by sums of Demazure-Lusztig operators acting on a monomial. These results make use of the Demazure and Demazure-Lusztig formulas above, and the branching structure of highest weight crystals of type A. The polynomials produced by sums of Demazure-Lusztig operators acting on a monomial are related to Iwahori fixed Whittaker functions in the nonmetaplectic setting.