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，引入了极值权模V(λ)$的概念，其积分权$ λ $作为与（一般）Kac-Moody代数$g$相关的量子群$U_q(g)$上的极值权。当$g$是仿射李代数，且积分权$ λ $为零阶时，我们得到了一个量子仿射代数作为极值权模$V(λ)$的商模的有限维表示。例如，我们用代数-几何方法得到了H.Nakajima引入的标准模$M(lambda)$，它与V.Chari和A.Pressley引入的量子Weyl模$M(lambda)$一致。在我们2002 - 2004年的一系列工作中，对于一个零级积分权$lambda$是仿射李代数$g$的零级基本权$g$的和，我们研究了一个特定的晶体$B(lambda)_{cl}$，它是（对$g$的零根取模）形状为$lambda$的所有Lakshmibai-Seshadri路径的晶体。结果证明了该晶体$B(λ)_{cl}$是量子仿射代数的零级基本表示的路径模型的张量积的晶体同构。从这个结果中，我们看到晶体$B(lambda)_{cl}$可以被认为是标准模块$M(lambda)$的路径模型。此外，利用上述结果，我们可以用Weyl群$W$的元素和$W$上的Bruhat阶，用组合的方式描述量子仿射代数上标准模$M(λ)$关于量子群$U_q(g_0)$约束的分支规则。

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 $W$.