[Paper Review] Arc-descent for the perfect loop functor and $p$-adic Deligne--Lusztig spaces

Alexander B. Ivanov|arXiv (Cornell University)|Mar 9, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes arc-descent for the perfect loop functor on quasi-projective schemes over non-archimedean local fields, proving that $LX$ is an arc-sheaf and extending this to $v$-topological surjectivity for $LG \to L(G/B)$ in the unramified reductive group case. It introduces a new, well-behaved notion of $p$-adic Deligne–Lusztig spaces $X_w(b)$, showing they are ind-representable in many cases and that their covering spaces $\dot{X}_{\dot{w}}(b)$ are pro-étale torsors over clopen subsets.

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 $LG ightarrow L(G/B)$ 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.

Motivation & Objective

• To establish arc-descent for the perfect loop functor $LX$ on quasi-projective schemes over $\mathbb{Q}_p$.
• To prove that the map $LG \to L(G/B)$ is a $v$-surjection for unramified reductive groups $G$, extending equal-characteristic results to mixed characteristic.
• To define a new, well-behaved notion of $p$-adic Deligne–Lusztig spaces $X_w(b)$ in the $v$-topology.
• To show that these spaces are ind-representable in various cases, addressing a question of Boyarchenko.
• To analyze the covering spaces $\dot{X}_{\dot{w}}(b)$ as pro-étale torsors over clopen subsets of $X_w(b)$.

Proposed method

• Prove that vector bundles over $W(R)[p^{-1}]$ form an arc-stack in $R$ for perfect $R$, using perfectoid techniques from [SW20].
• Deduce arc-descent for $LX$ on quasi-projective $X$ by analyzing open and closed immersions and the behavior of vector bundles.
• Show that all finite locally free $W(A)[p^{-1}]$-modules are free when $\operatorname{Spec} A$ is a disjoint union of spectra of valuation rings.
• Use the $v$-topological surjectivity of $LG \to L(G/B)$ to define $X_w(b)$ via a Cartesian diagram involving the graph of the geometric Frobenius and left multiplication by $b$.
• Establish ind-representability of $X_w(b)$ by analyzing the structure of the loop spaces and their covering spaces.
• Show that $\dot{X}_{\dot{w}}(b) \to X_w(b)$ is a pro-étale torsor by analyzing the Lang map and the action of $G_0(k)$ on the pullback of $\dot{w}LU$.

Experimental results

Research questions

• RQ1 Does the perfect loop functor $LX$ satisfy arc-descent for quasi-projective schemes over $\mathbb{Q}_p$?
• RQ2 Is the map $LG \to L(G/B)$ a $v$-surjection for unramified reductive groups $G$?
• RQ3 Can a well-defined notion of $p$-adic Deligne–Lusztig spaces $X_w(b)$ be constructed in the $v$-topology?
• RQ4 Are these $X_w(b)$ ind-representable in natural cases, particularly for $b=1$?
• RQ5 Are the covering spaces $\dot{X}_{\dot{w}}(b)$ pro-étale torsors over clopen subsets of $X_w(b)$?

Key findings

• The perfect loop functor $LX$ is an arc-sheaf for any quasi-projective scheme $X$ over $\mathbb{Q}_p$, strengthening Drinfeld's fpqc result.
• All finite locally free $W(A)[p^{-1}]$-modules of constant rank are free when $A$ is a perfect ring whose spectrum is a disjoint union of spectra of valuation rings.
• The map $LG \to L(G/B)$ is a $v$-surjection for unramified reductive groups $G$ over $\mathbb{Q}_p$, providing a mixed-characteristic analog of an equal-characteristic étale result.
• The $p$-adic Deligne–Lusztig space $X_w(b)$ is defined via a Cartesian diagram involving the loop space of the flag variety and the geometric Frobenius, yielding a well-behaved $v$-sheaf.
• For $b=1$, the covering space $\dot{X}_{\dot{w}}(1)$ is isomorphic to a torsor over the classical Deligne–Lusztig variety for $\operatorname{GL}_3$ and the longest Weyl group element.
• The covering spaces $\dot{X}_{\dot{w}}(b)$ are pro-étale torsors over clopen subsets of $X_w(b)$, with the torsor structure arising from the Lang map and $G_0(k)$-action.