Deligne–Lusztig constructions for division algebras and the local Langlands correspondence

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 D be the quaternion algebra over a local non-Archimedean field K of positive characteristic, and let X be the p  -adic Deligne–Lusztig ind-scheme associated to D×D×. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified quadratic extension of K   and representations of D×D× given by θ↦Hi(X)[θ]θ↦Hi(X)[θ]. We show that this correspondence is a bijection (after a mild restriction of the domain and target), and matches the bijection given by local Langlands and Jacquet–Langlands.