Programme on Lie Group2001


Introduction to the Cohomology of

Arithmetic Groups


Professor Armand Borel

Institute for Advanced Study, Princeton, USA

Distinguished Professor, HKU


These lectures will be mainly concerned with the real or complex cohomology space H*(G ) of an arithmetic subgroup G of a semisimple Q-group G. It can be identified to the cohomology of the quotient G \X by G of the symmetric space X of maximal compact subgroups of G, which has allowed one to study H*(G ) by a variety of methods : geometric constructions, de Rham and Lie algebra cohomology, harmonic and automorphic forms, harmonic analysis. I shall try to give an introduction to those and to some of the results they lead to, first when G \X is compact, where they appear in their simplest form, and then in the (more important) case where G \X is not compact, to which the main part of these lectures will be devoted. An important role is played by a compactification of G \X which has the same homotopy type as G \X, and by its boundary.

See A Borel and N. Wallach, Math. Survey 67, AMS 1999, for some of the basic techniques, the cocompact case and a brief survey of the non-cocompact case. Some of the material lectured upon during the first two parts of the program will be assumed. Surveys of the structure theory of semisimple algebraic groups and of reduction theory will be distributed.


Dates / Time:

A series of 8 lectures on Wednesdays and Fridays to be held 10:30am -12:00noon, 

April 25 - May 18, 2001. 


Room 517, Meng Wah Complex