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

Let ω1, ω2 be the two fundamental weights of a symmetrizable Kac–Moody algebra 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) l(τ) and a linear map ξ: l(τ) → * such that for any dominant weight λ = k1ω1 + k2ω2, we have Char Eτ(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).