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 $\text{C}$ and $T$ be a maximal torus of $G$. For a dominant coweight $\lambda$ of $G$, the $T$-fixed point subscheme $({\overline{Gr}}_G^{\lambda }{)}^T$ of the Schubert variety ${\overline{Gr}}_G^{\lambda }$ in the affine Grassmannian $G{r}_G$ 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 $G{r}_G$) of $({\overline{Gr}}_G^{\lambda }{)}^T$. 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.