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September 10, 2025Open Access

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

Key Points

A duality between twisted affine Demazure modules and certain function rings is established, enhancing understanding of affine Schubert varieties.
The study proves the duality theorem for the group scheme E_6, marking a significant development in the area of algebraic geometry.
Results confirm the smooth locus determination of affine Schubert varieties, supporting the conjecture made by Haines and Richarz.
This research employs the methods of Zhu to explore the properties of twisted affine varieties and their modules.

Abstract

Abstract Let {G} be a special parahoric group scheme of twisted type over the ring of formal power series over C, excluding the absolutely special case of A^ (2) ₂. Using the methods and results of Zhu, we prove a duality theorem for general {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 {G}. Along the way, we also establish the duality theorem for E₆. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of {G}. In particular, this confirms a conjecture of Haines and Richarz.

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