Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t=\infty$

Abstract

Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{\lambda }$, where $W_{\lambda }$ is the stabilizer of $\lambda$ in a finite Weyl group $W$. In this paper, we give a module ${\mathbb{K}}_w(\lambda )$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{w\lambda }(q,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1-q^{-r}{)}^{-1}$ for a positive integer $r$. This module ${\mathbb{K}}_w(\lambda )$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_w^{-}(\lambda )$ by the sum of the submodules $V_z^{-}(\lambda )$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{\lambda }$ such that $z>w$ in the Bruhat order $<$ on $W$.

References in corpus (3)

• A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X
• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
• Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

Cited by in corpus (1)

• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds