Tensor product structure of affine Demazure modules and limit constructions

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