On a decomposition of $p$-adic Coxeter orbits

TL;DR

This paper studies the structure of certain $p$-adic Deligne--Lusztig spaces associated with classical groups, revealing a decomposition into disjoint unions and extending previous results on conjugacy classes.

Contribution

It proves a decomposition of $X_w(b)$ for classical groups when $b$ is basic and $w$ is Coxeter, and extends results on unramified tori and Frobenius-twisted sections.

Findings

Decomposition of $X_w(b)$ into disjoint unions for classical groups.

Extension of rational conjugacy class results to extended pure inner forms.

A loop version of Frobenius-twisted Steinberg's cross section.

Abstract

We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$ introduced in [Iva21] attached to an unramified reductive group ${\bf G}$ over a non-archimedean local field. We prove that when ${\bf G}$ is classical, $b$ basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms, and prove a loop version of Frobenius-twisted Steinberg's cross section.