An explicit formula for the specialization of nonsymmetric Macdonald polynomials at $t = \infty$

In this paper, we give an explicit description of the specialization $E_{\mu}(q, \infty)$ of the nonsymmetric Macdonald polynomial $E_{\mu}(q, t)$ at $t = \infty$ for an arbitrary untwisted affine root system in terms of the quantum Bruhat graph associated to the underlying finite root system and finite Weyl group, where $\mu$ is an integral weight in the finite Weyl group orbit of a dominant integral weight $\lambda$. To be more precise, we interpret the specialization $E_{\mu}(q, \infty)$ as a graded character of the set of quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$ whose final direction belongs to an explicitly specified subset of the finite Weyl group. Here we note that the set of all QLS paths of shape $\lambda$ provides an explicit realization of the crystal basis of the quantum Weyl module $W_{q}(\lambda)$ over a quantum affine algebra, which is a quantum analog of the local Weyl module $W(\lambda)$ over a current algebra.