Tensor product structure of affine Demazure modules and limit constructions

Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra, we denote by $\widehat{\mathfrak{g}}$ the affine Kac-Moody algebra associated to the extended Dynkin diagram of $\mathfrak{g}$. Let $\Lambda_{0}$ be the fundamental weight of $\widehat{\mathfrak{g}}$ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral $\mathfrak{g}$-coweight $\lambda^{\vee}$, the Demazure submodule $V_{-\lambda^{\vee}}(m\Lambda_{0})$ is a $\mathfrak{g}$-module. We provide a description of the $\mathfrak{g}$-module structure as a tensor product of "smaller" Demazure modules. More precisely, for any partition of $\lambda^{\vee} = \sum_{j} \lambda_{j}^{\vee}$ as a sum of dominant integral $\mathfrak{g}$-coweights, the Demazure module is (as $\mathfrak{g}$-module) isomorphic to $\bigotimes_{j} V_{-\lambda^{\vee}_{j}}(m\Lambda_{0})$. For the "smallest" case, $\lambda^{\vee} = \omega^{\vee}$ a fundamental coweight, we provide for $\mathfrak{g}$ of classical type a decomposition of $V_{-\omega^{\vee}}(m\Lambda_{0})$ into irreducible $\mathfrak{g}$-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the $U_{q}(\mathfrak{g})$-characters of certain finite dimensional $U_{q}'(\widehat{\mathfrak{g}})$-modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules $V_{-\lambda^{\vee}, q}(m\Lambda_{0})$ can be naturally endowed with the structure of a $U_{q}'(\widehat{\mathfrak{g}})$-module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the "smallest" Demazure modules are, when viewed as $\mathfrak{g}$-modules, isomorphic to some KR-modules. For an integral dominant $\widehat{\mathfrak{g}}$-weight $\Lambda$ let $V(\Lambda)$ be the corresponding irreducible $\widehat{\mathfrak{g}}$-representation. Using the tensor product decomposition for Demazure modules, we give a description of the $\mathfrak{g}$-module structure of $V(\Lambda)$ as a semi-infinite tensor product of finite dimensional $\mathfrak{g}$-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.