Deligne-lusztig Constructions for Division Algebras and the Local Langlands Correspondence

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K with invariant 1/n. 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 prove that when n = 2, the p-adic Deligne-Lusztig (ind-)scheme X induces a correspondence θ 7→ H•(X)[θ] between smooth one-dimensional representations of L× and representations of D× that matches the correspondence given by the LLC and JLC.