Tensor product structure of affine Demazure modules and limit constructions
Dec, 2004Citations per year
Abstract:
Let \Lg be a simple complex Lie algebra, we denote by \Lhg the corresponding affine Kac--Moody algebra. Let 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 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.Note:
- 21 pages, in the current version there are some faults, on account by oversights, revised Subj-class: Representation Theory: Quantum Algebra
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