  Let $K$ be a local non-Archimedean field of positive characteristic and let
$L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and
Jacquet-Langlands correspondences, to each sufficiently generic multiplicative
character of $L$, one can associate an irreducible representation of the
multiplicative group of the central division algebra $D$ of invariant $1/n$
over $K$.
  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 between
smooth one-dimensional representations of $L^\times$ and representations of
$D^\times$ that matches the correspondence given by the LLC and JLC.
