## Current Developments in Mathematics

- Current Developments in Mathematics
- Volume 2002 (2002), 219-289.

### On the Cohomology of Locally Symmetric Spaces and of their Compactifications

#### Abstract

This expository article 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 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.

#### Article information

**Source**

Current Developments in Mathematics Volume 2002 (2002), 219-289.

**Dates**

First available in Project Euclid: 29 June 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.cdm/1088530402

**Mathematical Reviews number (MathSciNet)**

MR2062320

**Zentralblatt MATH identifier**

1132.14018

#### Citation

Saper , Leslie. On the Cohomology of Locally Symmetric Spaces and of their Compactifications. Current Developments in Mathematics 2002 (2002), 219--289. http://projecteuclid.org/euclid.cdm/1088530402.