Study on the path models for extremal weight modules over a quantum affine algebra

M.Kashiwara通过推广可积最高权模的概念，引入了极权模$V(Lambda)$的概念，其中整权$lambda$是与(一般)Kac-Moody代数$g$相关的量子群$U_q(G)$上的极权。当$g$是仿射李代数，且整权$lambda$是零级时，我们得到了量子仿射代数的某些有限维表示，作为极权模$V(Lambda)$的商模。例如，我们用代数几何方法得到了H.Nakajima引入的标准模$M(Lambda)$，它们与V.Chari和A.Pressley引入的量子Weyl模$M(Lambda)$重合。在我们从2002年到2004年的一系列工作中，对于一个零级整权$lambda$，它是仿射李代数$g$的零级基本权的和，我们研究了一种晶体$B(Lambda){CL}$，它是形状为$lambda$的所有Lakshmibai-Seshadri路的晶体(模为$g$的零根)。证明了该晶体与量子仿射代数零能级基本表示的路径模型的张量积同构。从这个结果可以看出，晶体$B(Lambda)_{CL}$可以被认为是标准模数$M(Lambda)$的路径模型。此外，利用我们的上述结果，我们可以通过使用$G$的Weyl群$W$的元素和$W$上的Bruhat序，以组合的方式描述量子仿射代数上的标准模$M(Lambda)$关于与$g$的(典范)有限维约化李子代数$g_0$有关的量子群$U_q(G_0)$的限制的分支规则。

M.Kashiwara introduced the notion of an extremal weight module $V(lambda)$ with an integral weight $lambda$ as an extremal weight over the quantum group $U_q(g)$ associated to a (general) Kac-Moody algebra $g$, by generalizing the notion of an integrable highest weight module. When $g$ is an affine Lie algebra and an integral weight $lambda$ is of level zero, we obtain certain finite-dimensional representations of a quantum affine algebra as quotient modules of the extremal weight module $V(lambda)$. For example, we obtain standard modules $M(lambda)$ introduced by H.Nakajima by an algebro-geometric method, which have turned out to coincide with quantum Weyl modules $M(lambda)$ introduced by V.Chari and A.Pressley. In our series of works from 2002 to 2004, for an integral weight $lambda$ of level zero that is a sum of level-zero fundamental weights for the affine Lie algebra $g$, we studied a certain crystal $B(lambda)_{cl}$, which is (modulo the null root of $g$) the crystal of all Lakshmibai-Seshadri paths of shape $lambda$. As a result, we proved that this crystal $B(lambda)_{cl}$ is isomorphic as a crystal to a tensor product of the path models for level-zero fundamental representations of the quantum affine algebra. From this result, we see that the crystal $B(lambda)_{cl}$ can be thought of as a path model for the standard module $M(lambda)$. Moreover, using our results above, we can describe the branching rule of the standard module $M(lambda)$ over the quantum affine algebra with respect to the restriction to the quantum group $U_q(g_0)$ associated to a (canonical) finite-dimensional reductive Lie subalgebra $g_0$ of $g$ in a combinatorial way, by using elements of the Weyl group $W$ of $g$ and the Bruhat order on $W$.