Polytopes associated to demazure modules of symmetrizable Kac-Moody algebras of rank two

Let omega(1), omega(2), be the two fundamental weights of a symmetrizable Kac-Moody algebra g of rank two (hence necessarily affine or finite), and tau an element of the Weyl group. In this paper we construct polytopes P-tau(omega(1)), P-tau(omega(2)) subset of R-l(tau) and a linear map xi: R-l(tau) --> h* such that for any dominant weight lambda = k(1)omega(1) + k(2) w(2), we have Char E-tau(lambda) = e(lambda)Sigma e(xi(x)) where the sum is over all the integral paints x, of the polytope k(1)P(tau)(omega(1)) + k(2)P(tau)(omega(2)). Furthermore, we show that there exists a flat deformation of the Schubert variety S-tau into the toric variety defined by (C) 2000 Academic Press.