Deligne--Lusztig constructions for division algebras and the local Langlands correspondence, II

Abstract

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups, analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let $X$ be the $p$-adic Deligne-Lusztig ind-scheme associated to a division algebra $D$ of invariant k/n over a non-Archimedean local field $K$. We study its homology groups by establishing a Deligne-Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^{\times }$. The homology of $X$ induces a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-$n$ extension of $K$ and representations of $D^{\times }$. For a broad class of characters $\theta ,$ we show that the representation $H_{\bullet }(X)[\theta ]$ is irreducible and concentrated in a single degree. Moreover, we show that this correspondence matches the bijection given by local Langlands and Jacquet-Langlands. As a corollary, we obtain a geometric realization of Jacquet-Langlands transfers between representations of division algebras.

27 pages. Version 2: removed Section 7.2 of v1

References in corpus (2)

• Deligne-Lusztig constructions for unipotent and p-adic groups
• Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences

Cited by in corpus (1)

• The cohomology of semi-infinite Deligne--Lusztig varieties