Chapter VIII. Integration

Abstract

An $m$-dimensional manifold $M$ that is oriented admits a notion of integration $f\mapsto\int_Mf\omega$ for any smooth $m$ form. Here $f$ can be any continuous real-valued function of compact support. This notion of integration behaves in a predictable way under diffeomorphism. When $\omega$ satisfies a positivity condition relative to the orientation, the integration defines a measure on $M$. A smooth map $M\to N$ with $\dim M\lt\dim N$ carries $M$ to a set of measure zero.

For a Lie group $G$, a left Haar measure is a nonzero Borel measure invariant under left translations. Such a measure results from integration of $\omega$ if $M=G$ and if the form $\omega$ is positive and left invariant. A left Haar measure is unique up to a multiplicative constant. Left and right Haar measures are related by the modular function, which is given in terms of the adjoint representation of $G$ on its Lie algebra. A group is unimodular if its Haar measure is two-sided invariant. Unimodular Lie groups include those that are abelian or compact or semisimple or reductive or nilpotent.

When a Lie group $G$ has the property that almost every element is a product of elements of two closed subgroups $S$ and $T$ with compact intersection, then the left Haar measures on $G$, $S$, and $T$ are related. As a consequence, Haar measure on a reductive Lie group has a decomposition that mirrors the Iwasawa decomposition, and also Haar measure satisfies various relationships with the Haar measures of parabolic subgroups. These integration formulas lead to a theorem of Helgason that characterizes and parametrizes irreducible finite-dimensional representations of $G$ with a nonzero $K$ fixed vector.

The Weyl Integration Formula tells how to integrate over a compact connected Lie group by first integrating over conjugacy classes. It is a starting point for an analytic treatment of parts of representation theory for such groups. Harish-Chandra generalized the Weyl Integration Formula to reductive Lie groups that are not necessarily compact. The formula relies on properties of Cartan subgroups proved in Chapter VII.