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

DOI10.1515/CRELLE-2022-0060arXiv2003.04399MaRDI QIDQ6336427

Alexander B. Ivanov

Publication date: 9 March 2020

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 $LGightarrowL(G/B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in 'etale topology) of Bouthier--v{C}esnaviv{c}ius. 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 $dot{X}_{dotw}(b)$ are pro-'etale torsors over clopen subsets of $X_w(b)$, and analyze some examples.

Mathematics Subject Classification ID

Representation theory for linear algebraic groups (20G05) Reflection and Coxeter groups (group-theoretic aspects) (20F55) Classical groups (algebro-geometric aspects) (14L35) Rigid analytic geometry (14G22)

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