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 étale topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.

13 pages; accepted version. Exposition expanded and details added in the reductive case