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

Abstract

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T-fixed point subscheme ( Gr ¯ G λ ) T of the Schubert variety Gr ¯ G λ in the affine Grassmannian Gr G is a finite scheme. We prove that for all such λ 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 λ and the ring of functions (twisted by certain line bundle on Gr G ) of ( Gr ¯ G λ ) 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.