  Let X be a scheme of finite type over a Noetherian base scheme S admitting a
dualizing complex, and let U be an open subset whose complement has codimension
at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent
sheaves by showing that a coherent middle extension (or intersection
cohomology) functor from perverse sheaves on U to perverse sheaves on X may be
defined for a much broader class of perversities than has previously been
known. We also introduce a derived category version of the coherent middle
extension functor.
  Under suitable hypotheses, we introduce a construction (called
"S2-extension") in terms of perverse coherent sheaves of algebras on X that
takes a finite morphism to U and extends it in a canonical way to a finite
morphism to X. In particular, this construction gives a canonical
"S2-ification" of appropriate X. The construction also has applications to the
"Macaulayfication" problem, and it is particularly well-behaved when X is
Gorenstein.
  Our main goal, however, is to address a conjecture of Lusztig on the geometry
of special pieces (certain subvarieties of the unipotent variety of a reductive
algebraic group). The conjecture asserts in part that each special piece is the
quotient of some variety (previously unknown in the exceptional groups and in
positive characteristic) by the action of a certain finite group. We use
S2-extension to give a uniform construction of the desired variety.
