Tensor product structure of affine Demazure modules and limit constructions

Details Tensor product structure of affine Demazure modules and limit constructions Tensor product structure of affine Demazure modules and limit constructions

2004-12-21

arxiv.org/abs/math/0412432v3

Mathematics

Representation Theory

24 pages, in the current version we added the case of twisted affine Kac--Moody algebras

Scientific paper

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.

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