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 $A^{(2)}_{2\ell }$ . 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.