Corollary

Tag 0ABX, permanent. Cite it as [Bourbaki, Tag 0ABX].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

For all $g \in G$ ,

\delta(g)^*\omega = (\det \mathrm{Ad}\, g) \omega.

\delta(g)^*\omega = \delta(g)^*\gamma(g)^*\omega = (\mathrm{Int}\, g)^*\omega \text{ and } L(\mathrm{Int}\, g) = \mathrm{Ad}\, g.

Let G be a locally compact group and φ an endomorphism of G. Suppose that there exist open neighbourhoods V, $V'$ of e such that $\phi(V) = V'$ and $\phi|V$ is a local isomorphism of G into G. Let μ be a left Haar measure on G. By Integration, Chapter VII, § 1, Corollary to Proposition 9, there exists a unique number $a > 0$ such that $\phi(\mu|V) = a^{-1}\mu|V'$ . Clearly a is independent of the choice of V, $V'$ and μ. It is called the modulus of φ and is denoted by $\mathrm{mod}_G \phi$ or simply $\mathrm{mod}\, \phi$ . When φ is an automorphism of G, we recover Definition 4 of Integration, Chapter VII, § 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 6. Integration on compact Lie groups

Cites

Definition 4 of Integration, Chapter VII, § 1 not yet on this site: INT VII
Integration, Chapter VII, § 1, Corollary to Proposition 9 not yet on this site: INT VII