Tensor product structure of affine Demazure modules and limit constructions

Fourier, G. and Littelmann, P. (2006) Tensor product structure of affine Demazure modules and limit constructions. Nagoya Mathematical Journal, 182, pp. 171-198.

Abstract

Let g be a simple complex Lie algebra, we denote by g ˆ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ 0 be the fundamental weight of g ˆ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight λ ∨, the Demazure submodule V −λ ∨(m Λ 0) is a g-module. We provide a description of the g-module structure as a tensor product of "smaller" Demazure modules. More precisely, for any partition of λ ∨= ∑ j λ ∨ j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to ⨂ j V −λ ∨ j(m Λ 0). For the "smallest" case, λ ∨= ω ∨ a fundamental coweight, we provide for g of classical type a decomposition of V −ω ∨(m Λ 0) into irreducible 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(g)- characters of certain finite dimensional U ′ q( g ˆ)-modules (Kirillov-Reshetikhin- modules) suggests furthermore that all quantized Demazure modules V −λ ∨,q(m Λ 0) can be naturally endowed with the structure of a U ′ q( g ˆ)-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 g-modules, isomorphic to some KR-modules. For an integral dominant g ˆ-weight Λ let V(Λ) be the corresponding irreducible g ˆ- representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V(Λ) as a semi- infinite tensor product of finite dimensional 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fourier, Dr Ghislain
• Authors: Fourier, G., and Littelmann, P.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Nagoya Mathematical Journal
• ISSN: 0027-7630
• ISSN (Online): 2152-6842