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Part 1
Compact Groups

Daniel Bump Graduate Texts in Mathematics Lie Groups 2nd ed. 2013 10.1007/978-1-4614-8024-2_1

Springer Science+Business Media New York 2013

1. Haar Measure

Daniel Bump 1

(1)

Department of Mathematics, Stanford University, Stanford, CA, USA

Abstract

If G is a locally compact group, there is, up to a constant multiple, a unique regular Borel measure L that is invariant under left translation. Here left translation invariance means that ( X ) = ( gX ) for all measurable sets X .

If G is a locally compact group, there is, up to a constant multiple, a unique regular Borel measure L that is invariant under left translation. Here left translation invariance means that ( X ) = ( gX ) for all measurable sets X . Regularity means that

Such a measure is called a left Haar measure . It has the properties that any compact set has finite measure and any nonempty open set has measure > 0.

We will not prove the existence and uniqueness of the Haar measure. See for example Halmos [] for a proof of this. Left-invariance of the measure amounts to left-invariance of the corresponding integral,

(1.1)

for any Haar integrable function f on G .

There is also a right-invariant measure, R , unique up to constant multiple, called a right Haar measure . Left and right Haar measures may or may not coincide. For example, if

then it is easy to see that the left- and right-invariant measures are, respectively,

They are not the same. However, there are many cases where they do coincide, and if the left Haar measure is also right-invariant, we call G unimodular .

Conjugation is an automorphism of G , and so it takes a left Haar measure to another left Haar measure, which must be a constant multiple of the first. Thus, if g G , there exists a constant ( g ) > 0 such that

If G is a topological group, a quasicharacter is a continuous homomorphism . If |( g )| = 1 for all g G , then is a (linear) character or unitary quasicharacter .

Proposition 1.1.

The function is a quasicharacter. The measure (h) L (h) is right-invariant.

The measure is a right Haar measure, and we may write . The quasicharacter is called the modular quasicharacter.

Proof.

Conjugation by first g 1 and then g 2 is the same as conjugation by g 1 g 2 in one step. Thus , so is a quasicharacter. Using (),

Replace f by f in this identity and then divide both sides by ( g ) to find that

Thus, the measure is right-invariant.

Proposition 1.2.

If G is compact, then G is unimodular and

Proof.

Since is a homomorphism, the image of is a subgroup of . Since G is compact, ( G ) is also compact, and the only compact subgroup of is just . Thus is trivial, so a left Haar measure is right-invariant. We have mentioned as an assumed fact that the Haar volume of any compact subset of a locally compact group is finite, so if G is finite, its Haar volume is finite.

If G is compact, then it is natural to normalize the Haar measure so that G has volume 1.

To simplify our notation, we will denote by .

Proposition 1.3.

If G is unimodular, then the map is an isometry.

Proof.

It is easy to see that turns a left Haar measure into a right Haar measure. If left and right Haar measures agree, then multiplies the left Haar measure by a positive constant, which must be 1 since the map has order 2.

Exercises

Exercise 1.1.

Let denote the Lebesgue measure on . It is of course a Haar measure for the additive group . Show that is both a left and a right Haar measure on .

Exercise 1.2.

Let P be the subgroup of consisting of matrices of the form

Let d g 1 and d g 2 denote Haar measures on and , and let denote an additive Haar measure on

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