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

TL;DR

This paper proves that for a connected linear algebraic group over a p-adic field, the action of the R loop group on Schubert cells in the affine Grassmannian is transitive in the tale topology, extending previous results to more general groups.

Contribution

It generalizes the transitivity of the R loop group action on Schubert cells from reductive groups to all connected linear algebraic groups over p-adic fields.

Findings

The R loop group acts transitively on Schubert cells in the R-affine Grassmannian.

The result holds in the tale topology on affinoid perfectoids.

Extension of Fargues-Scholze's result to non-reductive groups.

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 each Schubert cell in the $\mathbb{B}_{\mathrm{dR}}^+$-affine Grassmannian is transitive in the \'{e}tale topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.