Cambridge Journal of Mathematics

Volume 11 (2023)

Number 2

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

Pages: 441 – 505

DOI: https://dx.doi.org/10.4310/CJM.2023.v11.n2.a2

Authors

Charlotte Chan (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Alexander B. Ivanov (Mathematisches Institut, Universität Bonn, Germany)

Abstract

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.

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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