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 $\widehat{\mathbb{B}(\lambda {)}_{\mathrm{cl}}}$ (with weight lattice $P$) of the crystal $\mathbb{B}(\lambda {)}_{\mathrm{cl}}\simeq {⨂}_{i\in I_0}(\mathbb{B}({\varpi }_i{)}_{\mathrm{cl}}{)}^{\otimes m_i}$ (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.