  We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite
Deligne--Lusztig varieties attached to division algebras over local fields. We
also prove the two conjectures of Boyarchenko on these varieties. It is known
that in this setting, the semi-infinite Deligne--Lusztig varieties are
ind-schemes comprised of limits of certain finite-type schemes $X_h$.
Boyarchenko's two conjectures are on the maximality of $X_h$ and on the
behavior of the torus-eigenspaces of their cohomology. Both of these
conjectures were known in full generality only for division algebras with Hasse
invariant $1/n$ in the case $h = 2$ (the "lowest level") by the work of
Boyarchenko--Weinstein on the cohomology of a special affinoid in the
Lubin--Tate tower. We prove that the number of rational points of $X_h$ attains
its Weil--Deligne bound, so that the cohomology of $X_h$ is pure in a very
strong sense. We prove that the torus-eigenspaces of $H_c^i(X_h)$ are
irreducible representations and are supported in exactly one cohomological
degree. Finally, we give a complete description of the homology groups of the
semi-infinite Deligne--Lusztig varieties attached to any division algebra, thus
giving a geometric realization of the unramified supercuspidal representations
of these groups. We expect that the techniques developed in this paper will be
useful in studying these constructions for reductive groups over local fields
in general.
