A combinatorial formula for Sahi, Stokman, and Venkateswaran's generalization of Macdonald polynomials

Abstract

Sahi, Stokman, and Venkateswaran have constructed, for each positive integer n, a family of Laurent polynomials depending on parameters q and k (in addition to ⌊n/2⌋ “metaplectic parameters”), such that the n=1 case recovers the nonsymmetric Macdonald polynomials and the q→0,∞ limits with a choice of metaplectic parameters (as Gauss sums) yield metaplectic Iwahori Whittaker functions. In this paper, we study these new polynomials, which we call SSV polynomials, in the case of GLr. We apply a result of Ram and Yip in order to give a combinatorial formula for the SSV polynomials and their symmetrized variants in terms of alcove walks. The result is used to precisely characterize the support of each SSV polynomial, showing that SSV polynomials have fewer terms than the corresponding Macdonald polynomials. Finally, we calculate the q→0 and q→∞ limits of the SSV polynomials and observe that our combinatorial formula can be written in terms of alcove walks with only positive and negative folds respectively. We conclude by giving the correspondence between SSV polynomials and metaplectic Iwahori Whittaker functions, part of which is due to Sahi, Stokman, and Venkateswaran.