Extremal weight modules of quantum affine algebras

Abstract

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra, and let $\mathbf{U}_q (\widehat{\mathfrak{g}})$ be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a $\mathbf{U}_q (\widehat{\mathfrak{g}})$-module $V(\lambda)$, having a global crystal base for an integrable weight $\lambda$ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl module introduced by Chari-Pressley [6]. In [12, §13] Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when $\widehat{\mathfrak{g}}$ is an untwisted affine Lie algebra of a simple Lie algebra $\mathfrak{g}$ of type $ADE$, using a result of Beck-Chari-Pressley [5]. As a by-product, we also show that the extremal weight module is isomorphic to a universal standard module, defined via quiver varieties by the author [16, 18]. This result was conjectured by Varagnolo-Vasserot [19] and Chari-Pressley [6] in a less precise form. Furthermore, we give a characterization of global crystal bases by an almost orthogonality propery, as in the case of global crystal base of highest weight modules.