Littelmann Paths for the Basic Representation of an Affine Lie Algebra

Let g be a complex simple Lie algebra, and ĝ the corresponding untwisted affine Lie algebra. Let V̂ (Λ0) be the basic irreducible level-one representation of ĝ, and V̂λ∨(Λ0) the Demazure module corresponding to the translation −λ ∨ in the affine Weyl group. Suppose λ∨ is a sum of minuscule coweights of g (which exist if g is of classical type or E6, E7). We give a new model for the crystal graphs of V̂ (Λ0) and V̂λ∨(Λ0) which combines Littelmann’s path model and the Kyoto path model. As a corollary, we prove that Vλ∨(Λ0) is isomorphic as a g-module to a tensor product of fundamental representations of g.