Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight lambda of G, the T-fixed point subscheme ((Gr) over bar (lambda)(G))(T) of the Schubert variety (Gr) over bar (lambda)(G) in the affine Grassmannian Gr(G) is a finite scheme. We prove that for all such lambda if G is of type A or D and for many of them if G is of type E, there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to lambda and the ring of functions (twisted by certain line bundle on Gr(G)) of ((Gr) over bar (lambda)(G))(T). We use this fact to give a geometrical proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of A, D, E type and lattice vertex algebras. (C) 2009 Elsevier Inc. All rights reserved.