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

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.