Lie groups: Modular function

This lecture is part of an online graduate course on Lie groups.

We discuss the modular function of a Lie group, which controls the relation between right and left invariant measures. We show how to use it to tell when a homogeneous space G/H has a G-invariant measure.

I will pause the lectures on Lie groups after this lecture, but will probably continue them sometime later.

[periodic]

Excellent lectures, and it will be great if there’s an introduction on Youtube in this style done for abstract harmonic analysis topics. They are fascinating.

[klmnps]

Dear Professor Borcherds, it seems that this video in misplaced, it is not in the play list of Lie groups and Lie algebras.

[lukasjuhrich]

Dear Mr. Borcherds,
I'm a bit confused about the „transport“ of our n-form at e to an n-form at g (ca. 2:30). Although the left (or right) translation with g induces a pushforward of the tangent spaces TeG -> TgG, don't we need the _pullback_ for the differential forms?
Or do you actually mean the „covariant“ exterior algebra of the tangent bundle when you wrote Λ^n?
If the former, I would have expected that we had to take multiplication with g^{-1}, which maps g to e, and hence induces a _pullback_ ΛeG -> ΛgG.

Or did I get something confused here?

[kilogods]

Why is the nth exterior power of the tangent space 1 dimensional?