  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.
