On a decomposition of $p$-adic Coxeter orbits

Abstract

We analyze the geometry of some p-adic Deligne-Lusztig spaces Xw(b) introduced in [Iva21] attached to an unramified reductive group G over a non-archimedean local field. We prove that when G is classical, b basic and w Coxeter, Xw(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. Contents 1. Introduction 1 2. Setup and preliminaries 4 3. Rational and stable conjugacy classes of unramified tori 9 4. The maps Ẋ w(b) → X w (b) 15 5. Variant of Steinberg's cross section 16 6. Decomposition into integral p-adic spaces 21 7. Estimates via Newton polygons 27 References 40