A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at $t=0$ and Demazure characters

Abstract

We establish the equality of the specialization $E_{w\lambda }(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{w\lambda }(x;q,t)$ at $t=0$ with the graded character $\mathrm{gch}{U}_w^{+}(\lambda )$ of a certain Demazure-type submodule $U_w^{+}(\lambda )$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $\lambda$ 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 $P_{\lambda }(x;q,0)$ of the symmetric Macdonald polynomial $P_{\lambda }(x;q,t)$ 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 a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.

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