  This expository article is an expanded version of talks given at the "Current
Developments in Mathematics, 2002" conference. It gives an introduction to the
(generalized) conjecture of Rapoport and Goresky-MacPherson which identifies
the intersection cohomology of a real equal-rank Satake compactification of a
locally symmetric space with that of the reductive Borel-Serre
compactification. We motivate the conjecture with examples and then give an
introduction to the various topics that are involved: intersection cohomology,
the derived category, and compactifications of a locally symmetric space,
particularly those above. We then give an overview of the theory of L-modules
and micro-support (see math.RT/0112251) which was developed to solve the
conjecture but has other important applications as well. We end with sketches
of the proofs of three main theorems on L-modules that lead to the resolution
of the conjecture. The text is enriched with many examples, illustrations, and
references to the literature.
