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Transitivity of the $\mathbb{B}^+_\mathrm{dR}$-loop group action on Schubert cells

Sean Howe

TL;DR

The paper proves that for any connected linear algebraic group $G$ over a $p$-adic field $L$, the action of $G(\mathbb{B}^+_\mathrm{dR})$ on the affine Schubert cell $\mathrm{Gr}_{[\mu]}$ inside the $\mathbb{B}^+_\mathrm{dR}$-affine Grassmannian is transitive in the étale topology, not only in the reductive case but for general $G$. The authors reduce the non-reductive case to the reductive one via Levi decomposition and analyze the corresponding quotients to show that the étale quotient $G(\mathbb{B}^+_\mathrm{dR})/\mathrm{Stab}_{G(\mathbb{B}^+_\mathrm{dR})}(\ast_\mu)$ reproduces $\mathrm{Gr}_{[\mu]}$. This establishes that $\mathrm{Gr}_{[\mu]}$ is étale-locally a homogeneous space for $G(\mathbb{B}^+_\mathrm{dR})$, aligning with the broader geometric framework for the local Langlands program. The results connect with inscribed $\mathbb{B}^+_\mathrm{dR}$-Grassmannians, tangent-bundle constructions, and $p$-adic Hodge-theoretic period domains, highlighting the structural role of $\mathrm{Gr}_{[\mu]}$ in non-reductive settings.

Abstract

For $G$ a connected linear algebraic group over a $p$-adic field, we show that the action of $G(\mathbb{B}^+_\mathrm{dR})$ on Schubert cells in the $\mathbb{B}_\mathrm{dR}^+$-affine Grassmannian is transitive in the étale topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.

Transitivity of the $\mathbb{B}^+_\mathrm{dR}$-loop group action on Schubert cells

TL;DR

The paper proves that for any connected linear algebraic group over a -adic field , the action of on the affine Schubert cell inside the -affine Grassmannian is transitive in the étale topology, not only in the reductive case but for general . The authors reduce the non-reductive case to the reductive one via Levi decomposition and analyze the corresponding quotients to show that the étale quotient reproduces . This establishes that is étale-locally a homogeneous space for , aligning with the broader geometric framework for the local Langlands program. The results connect with inscribed -Grassmannians, tangent-bundle constructions, and -adic Hodge-theoretic period domains, highlighting the structural role of in non-reductive settings.

Abstract

For a connected linear algebraic group over a -adic field, we show that the action of on Schubert cells in the -affine Grassmannian is transitive in the étale topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.
Paper Structure (3 sections, 1 theorem, 16 equations)

This paper contains 3 sections, 1 theorem, 16 equations.

Table of Contents

  1. Statement of the result
  2. Acknowledgements
  3. Proof of main result

Key Result

Theorem 1.0.1

The action of $G(\mathbb{B}^+_\mathrm{dR})$ on $\mathrm{Gr}_{[\mu]}$ is transitive in the étale topology: that is, if $(\mathcal{E}_i, \varphi_i) \in \mathrm{Gr}_{[\mu]}(\mathrm{Spa}(R,R^+))$ for $i=1,2$, then there is an étale cover $\mathrm{Spa}(S,S^+) \rightarrow \mathrm{Spa}(R,R^+)$ and $g \in G

Theorems & Definitions (7)

  • Theorem 1.0.1
  • Remark 1.0.2
  • Remark 1.0.3
  • Example 1.0.4
  • Remark 1.0.5
  • proof : Proof of \ref{['theorem.action-transitive']} in the reductive case
  • proof : Proof of \ref{['theorem.action-transitive']} in the general case