Polytopes Associated to Demazure Modules of Symmetrizable Kac–Moody Algebras of Rank Two

AbstractLet ω1,ω2 be the two fundamental weights of a symmetrizable Kac–Moody algebra g of rank two (hence necessarily affine or finite), and τ an element of the Weyl group. In this paper we construct polytopes Pτ(ω1),Pτ(ω2)⊂Rl(τ) and a linear map ξ: Rl(τ)→h* such that for any dominant weight λ=k1ω1+k2ω2, we have CharEτ(λ)=eλ∑eξ(x), where the sum is over all the integral points x, of the polytope k1Pτ(ω1)+k2Pτ(ω2). Furthermore, we show that there exists a flat deformation of the Schubert variety Sτ into the toric variety defined by Pτ(ω1),Pτ(ω2)