A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

Abstract

Abstract. We establish the equality of the specialization E wλ (x ; q, 0) of the nonsymmetric Macdonald polynomial E wλ (x ; q, t) at t = 0 with the graded character gch U + w (λ) of a certain Demazuretype submodule U + w (λ) of a tensor product of "single-column" Kirillov-Reshetikhin modules for an untwisted affine Lie algebra, where λ is a dominant integral weight and w is a (finite) Weyl group element; this generalizes our previous result, that is, the equality between the specialization P λ (x ; q, 0) of the symmetric Macdonald polynomial P λ (x ; q, t) at t = 0 and the graded character of a tensor product of single-column Kirillov-Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of nonsymmetric Macdonald polynomials: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.

Demazure character formula for semi-infinite flag varieties

“…The aim of this paper is two-fold: one is to extend Braverman-Finkelberg's cohomology formula of line bundles to include some naturally twisted sheaves, and the other is to generalize their results to all Schubert varieties so that the situation becomes more satisfactory from representation-theoretic viewpoints. It turns out that such an extension provides a natural realization of certain specializations of non-symmetric Macdonald polynomials, together with difference equations characterizing them, generalizing their links to the representation theory of current algebras as discovered by Braverman-Finkelberg [6], Lenart-Naito-Sagaki-Schilling-Shimozono [28,29,30], Cherednik-Orr [12], Naito-Nomoto-Sagaki [31], and Feigin-Makedonskyi [16].…”

A Weyl Module Stratification of Integrable Representations

“…Both local Weyl modules and global Weyl modules constitute a basis in K 0 of the category of graded g[z]-modules. It is shown ( [27,31,4,32,35]) that these bases are orthogonal to each other with respect to Ext • scalar product (see Theorem 2.2) and their characters are related to the specializations of Macdonald polynomials at t = 0. It appears that characters of some natural representations of g[z] can be expressed via Weyl module characters with positive coefficients.…”

Generalized Weyl modules and nonsymmetric q-Whittaker functions

“…This may be viewed as an extension of results of [CI,I1,Sa] expressing the v = 0 specializations of Macdonald polynomials as graded characters. We also mention the related works [NNS, NS] and [LNSSS1]- [LNSSS4] in this direction. The works [I1,LNSSS4,Sa] realize the nonsymmetric Macdonald polynomials E λ (X; q, 0) for arbitrary λ ∈ X as graded characters of Demazure submodules of Weyl modules, while [NNS] considers the v = ∞ specializations for λ ∈ X (which are essentially our E w0 λ (X; q, 0)-see (2.9)).…”

“…We also mention the related works [NNS, NS] and [LNSSS1]- [LNSSS4] in this direction. The works [I1,LNSSS4,Sa] realize the nonsymmetric Macdonald polynomials E λ (X; q, 0) for arbitrary λ ∈ X as graded characters of Demazure submodules of Weyl modules, while [NNS] considers the v = ∞ specializations for λ ∈ X (which are essentially our E w0 λ (X; q, 0)-see (2.9)). In type A, the polynomials E σ λ have been studied recently by Alexandersson [A], using the combinatorics of non-attacking fillings with general basement (determined by σ), which results in a natural extension to E σ λ of the Haglund-Haiman-Loehr formula for E λ [HHL].…”