2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67909This 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.71 pages, 30 figures, AMS-LaTeX, uses XyPic 3.7 package; to appear in the proceedings of the Harvard/M.I.T. conference "Current Developments in Mathematics, 2002", held in November 2002; minor errors fixed, added references, re-formatted to CDM styleRepresentation TheoryDifferential Geometry11F75, 22E40, 32S60, 55N33 (Primary) 14G35, 22E45 (Secondary)text