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Normality of Schubert varieties in affine Grassmannians II: The tamely ramified case

Patrick Bieker

TL;DR

We develop a tamely ramified criterion for the normality of Schubert varieties in twisted affine Grassmannians, tying normality to the order of the algebraic fundamental group of a Levi subgroup via $char(k)$ and the invariant $pi1(M_{barmu}^{der})$. The core method combines a twisted Levi lemma, transversal slices, and a tangent-space criterion to reduce to Levi subgroups and quasi-minuscule cases, enabling a broad classification across absolutely almost simple and product groups, plus corollaries for local models in mixed and equal characteristic. The results yield a comprehensive classification of normal Schubert varieties at absolutely special level for tamely ramified and non-semisimple groups, and extend to a normality criterion for local models, including rank-one semisimple groups. Collectively, the work clarifies when pathological (non-normal) behavior occurs in positive characteristic and provides concrete, computable criteria for normality and local-model normality in arithmetic geometry applications.

Abstract

We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an application, we obtain a similar normality criterion for local models in both equal and mixed characteristic. In particular, we give a classification of normal Pappas-Zhu local models at absolutely special level as well as for adjoint groups of rank 1.

Normality of Schubert varieties in affine Grassmannians II: The tamely ramified case

TL;DR

We develop a tamely ramified criterion for the normality of Schubert varieties in twisted affine Grassmannians, tying normality to the order of the algebraic fundamental group of a Levi subgroup via and the invariant . The core method combines a twisted Levi lemma, transversal slices, and a tangent-space criterion to reduce to Levi subgroups and quasi-minuscule cases, enabling a broad classification across absolutely almost simple and product groups, plus corollaries for local models in mixed and equal characteristic. The results yield a comprehensive classification of normal Schubert varieties at absolutely special level for tamely ramified and non-semisimple groups, and extend to a normality criterion for local models, including rank-one semisimple groups. Collectively, the work clarifies when pathological (non-normal) behavior occurs in positive characteristic and provides concrete, computable criteria for normality and local-model normality in arithmetic geometry applications.

Abstract

We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an application, we obtain a similar normality criterion for local models in both equal and mixed characteristic. In particular, we give a classification of normal Pappas-Zhu local models at absolutely special level as well as for adjoint groups of rank 1.

Paper Structure

This paper contains 40 sections, 43 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Results
  3. A normality criterion
  4. Classification of normal Schubert varieties
  5. Proof strategy
  6. Root systems and Weyl groups
  7. Relative and absolute root datum
  8. Iwahori-Weyl group
  9. Product decompositions
  10. Echellonage root system
  11. Affine Grassmannians and Schubert varieties
  12. Definitions
  13. Basic geometric properties of Schubert varieties
  14. Normality results
  15. Connected components of the twisted partial affine flag variety
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $G$ be a reductive group that satisfies eqn:cond-G and let $x \in \mathscr{B}(G,F)$ be a special vertex. Let $\bar{\mu} \in X_*(T)^+_I$ be a dominant cocharacter. If then the Schubert variety $\mathop{\rm Gr}\nolimits_{G,x, \leq \bar{\mu}}$ is normal. Moreover, if $x$ is an absolutely special vertex and $G$ is split or adjoint, then eq:char-pi1 is also necessary for the normality of $\mathop{

Figures (5)

  • Figure 1: The Bruhat partial order on dominant coweights in type $B_n$ for $n \geq 3$.
  • Figure 2: The Bruhat partial order on dominant coweights in type $C_n$ for $n \geq 2$. The second connected component is obtained by shifting by $\omega_{n}^\vee$.
  • Figure 3: The Bruhat partial order on dominant coweights of type $D_4$.
  • Figure 4: The Bruhat partial order on dominant coweights of type $D_n$, $n\geq 6$ even.
  • Figure 5: The Bruhat partial order on dominant coweights of type $D_n$, $n \geq 5$ odd.

Theorems & Definitions (84)

  • Theorem 1.1: cf. \ref{['thm:main']}
  • Theorem 1.2: cf. \ref{['theo:normality-loc-mod']}
  • Theorem 1.3: Section \ref{['sect--classification']}, cf. also BiekerRicharz:normality
  • Theorem 1.4: \ref{['theo:normality-loc-mod']}
  • Theorem 1.5: \ref{['thm:class-locmod-ss1']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 74 more