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 Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the $D^\times$-representations $H_\bullet (X)$ by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^\times$. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of $D^{\times }$ given by $\theta \mapsto H_\bullet (X)[\theta ]$. For a broad class of characters $\theta ,$ we show that the representation $H_\bullet (X)[\theta ]$ is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from $\theta$ using $\chi$-data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras.