Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths

Abstract

Let $\lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(\lambda )$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t=0$ and $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{w\lambda }(q,t)$ are explicitly described in terms of QLS paths of shape $\lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $\lambda$, $\mu$, we have an isomorphism $\Theta :\mathrm{QLS}(\lambda +\mu )\to \mathrm{QLS}(\lambda )\otimes \mathrm{QLS}(\mu )$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $\Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.

arXiv admin note: text overlap with arXiv:1802.06339