Crystal structure of the set of Lakshmibai-Seshadri paths of a level-zero shape for an affine Lie algebra

Abstract

Let $\lambda = \sum_{i \in I_{0}} m_{i} \varpi_{i}$, with $m_{i} \in \mathbb{Z}_{\ge 0}$ for $i \in I_{0}$, be a level-zero dominant integral weight for an affine Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, where the $\varpi_{i}$, $i \in I_{0}$, are the level-zero fundamental weights, and let $\mathbb{B}(\lambda)$ be the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. First, we give an explicit description of the decomposition of the crystal $\mathbb{B}(\lambda)$ into a disjoint union of connected components, and show that all the connected components are pairwise ``isomorphic'' (up to a shift of weights). Second, we ``realize'' the connected component $\mathbb{B}_{0}(\lambda)$ of $\mathbb{B}(\lambda)$ containing the straight line $\pi_{\lambda}$ as a specified subcrystal of the affinization $\hat{\mathbb{B}(\lambda)_{\mathrm{cl}}}$ (with weight lattice $P$) of the crystal $\mathbb{B}(\lambda)_{\mathrm{cl}} \simeq \bigotimes_{i \in I_{0}} \bigl(\mathbb{B}(\varpi_{i})_{\mathrm{cl}} \bigr)^{\otimes m_{i}}$ (with weight lattice $P_{\mathrm{cl}} = P/(\mathbb{Q}\delta \cap P)$, where $\delta$ is the null root of $\mathfrak{g}$), which was studied in a previous paper.