  Consider the middle perversity intersection cohomology groups of various
compactifications of a Hermitian locally symmetric space. Rapoport and
independently Goresky and MacPherson have conjectured that these groups
coincide for the reductive Borel-Serre compactification and the
Baily-Borel-Satake compactification. This paper describes the theory of
L-modules and how it is used to solve the conjecture. More generally we
consider a Satake compactification for which all real boundary components are
equal-rank. Details will be given elsewhere (math.RT/0112251). As another
application of L-modules, we prove a vanishing theorem for the ordinary
cohomology of a locally symmetric space. This answers a question raised by
Tilouine.
