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

TL;DR

This paper proves a duality between twisted affine Demazure modules and function rings of fixed point subschemes in affine Schubert varieties, and determines the smooth locus of these varieties, confirming a conjecture of Haines and Richarz.

Contribution

It establishes a duality theorem for twisted affine Demazure modules and affine Schubert varieties, and characterizes the smooth locus, extending previous results and confirming a conjecture.

Findings

Duality between twisted affine Demazure modules and function rings of fixed point subschemes.

Determination of the smooth locus of affine Schubert varieties.

Confirmation of Haines and Richarz's conjecture.

Abstract

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.