Lie Groups Beyond an Introduction pp 456-486 | Cite as

# Integration

## Abstract

An *m*-dimensional manifold *M* that is oriented admits a notion of integration *f* ↦ *∫* _{ M } *fω* 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 *ω* satisfies a positivity condition relative to the orientation, the integration defines a measure on *M*. A smooth map *M* → *N* with dim *M* < 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 *ω* if *M* = *G* and if the form *ω* 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.

## Keywords

Conjugacy Class Haar Measure Parabolic Subgroup Modular Function Irreducible Character## Preview

Unable to display preview. Download preview PDF.