Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules

Let $\mathscr{G}$ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb{C}$, excluding the absolutely special case of $A_{2\ell}^{(2)}$. Using the methods and results of Zhu, we prove a duality theorem for general $\mathscr{G}$ : there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for $\mathscr{G}$. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of $\mathscr{G}$. In particular, this confirms a conjecture of Haines and Richarz.