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.
KeywordsConjugacy Class Haar Measure Parabolic Subgroup Modular Function Irreducible Character
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