For classical G, basic b and Coxeter w, the p-adic Deligne-Lusztig space X_w(b) decomposes as a disjoint union of translates of an integral p-adic Deligne-Lusztig space, with extensions to tori conjugacy and a loop Steinberg cross section.
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On a decomposition of $p$-adic Coxeter orbits
For classical G, basic b and Coxeter w, the p-adic Deligne-Lusztig space X_w(b) decomposes as a disjoint union of translates of an integral p-adic Deligne-Lusztig space, with extensions to tori conjugacy and a loop Steinberg cross section.