Deligne–lusztig Constructions for Division Algebras and the Local Langlands Correspondence, Ii

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×-representations H•(X) by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of D×. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of D× given by θ 7→ H•(X)[θ]. For a broad class of characters θ, we show that the representation H•(X)[θ] is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from θ using χ-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.