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

Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $\lambda$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^\lambda)^T$ of the Schubert variety $\bar{Gr}_G^\lambda$ in the affine Grassmannian $Gr_G$ is a finite scheme. We prove that 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 $(\bar{Gr}_G^\lambda)^T$. We use this fact to give a geometric proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of $A,D,E$ type and lattice vertex algebras.