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

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 $Theta : mathrm{QLS}(lambda + mu) rightarrow mathrm{QLS}(lambda) otimes mathrm{QLS}(mu)$ 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.