  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 $p$-adic Deligne-Lusztig
ind-scheme associated to a division algebra $D$ of invariant k/n over a
non-Archimedean local field $K$. We study its homology groups by establishing a
Deligne-Lusztig theory for families of finite unipotent groups that arise as
subquotients of $D^\times$. The homology of $X$ induces a natural
correspondence between quasi-characters of the (multiplicative group of the)
unramified degree-$n$ extension of $K$ and representations of $D^{\times}$. For
a broad class of characters $\theta,$ we show that the representation
$H_\bullet(X)[\theta]$ is irreducible and concentrated in a single degree.
Moreover, we show that this correspondence matches the bijection given by local
Langlands and Jacquet-Langlands. As a corollary, we obtain a geometric
realization of Jacquet-Langlands transfers between representations of division
algebras.
