Tensor product structure of affine Demazure modules and limit constructions

Abstract

Let $\Lg$ be a simple complex Lie algebra, we denote by $\Lhg$ the corresponding affine Kac--Moody algebra. Let $\Lambda_0$ be the additional fundamental weight of $\Lhg$. For a dominant integral $\Lg$--coweight $\lam^\vee$, the Demazure submodule $V_{-\lam^\vee}(m\Lam_0)$ is a $\Lg$--module. For any partition of $\lam^\vee=\sum_j \lam_j^\vee$ as a sum of dominant integral $\Lg$--coweights, the Demazure module is (as $\Lg$--module) isomorphic to $\bigotimes_j V_{-\lam^\vee_j}(m\Lam_0)$. For the ``smallest'' case, $\lam^\vee=\om^\vee$ a fundamental coweight, we provide for $\Lg$ of classical type a decomposition of $V_{-\om^\vee}(m\Lam_0)$ into irreducible $\Lg$--modules, so this can be viewed as a natural generalization of the decomposition formulas in \cite{KMOTU} and \cite{Magyar}. A comparison with the $U_q(\Lg)$--characters of certain finite dimensional $U_q'(\Lhg)$--modules (Kirillov--Reshetikhin--modules) suggests furthermore that all quantized Demazure modules $V_{-\lam^\vee,q}(m\Lam_0)$ can be naturally endowed with the structure of a $U_q'(\Lhg)$--module. Such a structure suggests also a combinatorially interesting connection between the LS--path model for the Demazure module and the LS--path model for certain $U_q'(\Lhg)$--modules in \cite{NaitoSagaki}. For an integral dominant $\Lhg$--weight $\Lambda$ let $V(\Lam)$ be the corresponding irreducible $\Lhg$--representation. Using the tensor product decomposition for Demazure modules, we give a description of the $\Lg$--module structure of $V(\Lam)$ as a semi-infinite tensor product of finite dimensional $\Lg$--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.