Fusion Product Structure of Demazure Modules

Let oe"currency sign be a finite-dimensional complex simple Lie algebra. Given a non-negative integer a"", we define to be the set of dominant weights lambda of oe"currency sign such that a""I >(0)+lambda is a dominant weight for the corresponding untwisted affine Kac-Moody algebra . For the current algebra oe"currency sign[t] associated to oe"currency sign, we show that the fusion product of an irreducible oe"currency sign-module V(lambda) such that and a finite number of special family of oe"currency sign-stable Demazure modules of level a"" (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171-198 (2006), Adv. Math. 211(2), 566-593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the oe"currency sign[t]-module structure of the irreducible -module V(a"" I >(0) + lambda) as a semi-infinite fusion product of finite dimensional oe"currency sign[t]-modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566-593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).