On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $\mathrm{GL}_n$

For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from classical Deligne–Lusztig theory by using the loop space functor. We study this construction in the special case that $G$ is an inner form of $\mathrm{GL}_n$ and the loop Deligne–Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L‑parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and—in a sense—quite explicit way to realize special cases of the local Langlands and Jacquet–Langlands correspondences.

2010 Mathematics Subject Classification

Primary 11G25. Secondary 14F20, 20G25.

Full Text (PDF format)

Charlotte Chan is partially supported by the DFG via the Leibniz Prize of Peter Scholze, by NSF grant DMS-1641185 (US Junior Oberwolfach Fellow), and by an NSF Postdoctoral Research Fellowship, DMS-1802905.

Alexander B. Ivanov is supported by the DFG via the Leibniz Preis of Peter Scholze.

Received 27 January 2021

Published 6 June 2023