Arc-descent for the perfect loop functor and $p$-adic Deligne--Lusztig spaces

Abstract

We prove that the perfect loop functor $LX$ of a quasi-projective scheme $X$ over a local non-archimedean field $k$ satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group $G$, the map $LG\to L(G/B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in étale topology) of Bouthier--Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne--Lusztig spaces $X_w(b)$ attached to unramified $p$-adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces ${\dot{X}}_{\dot{w}}(b)$ are pro-étale torsors over clopen subsets of $X_w(b)$, and analyze some examples.

arXiv comment45pages; v3; title changed; introduction rewritten completely; Remark 3.2 added; Proposition 11.9 strengthened; bibiliography updated; minor format changes

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