The Harish-Chandra integral

Colin McSwiggen

Published 2018-06-28Version 1

This paper introduces Harish-Chandra's integral formula for compact, connected, semisimple Lie groups. It is intended for mathematicians and physicists who are familiar with the basics of Lie groups and Lie algebras but who may not be specialists in representation theory. I present Harish-Chandra's proof of the formula in contemporary language, work out the integrals for all compact classical groups in detail, and show how to generalize the formula to compute similar integrals over arbitrary compact Lie groups.