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

$$\displaystyle{\mu (X) =\inf \left \{\mu (U)\,\vert \,U \supseteq X,\text{U open}\right \} =\sup \left \{\mu (K)\,\vert \,K \subseteq X,\text{K compact}\right \}.}$$

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 [61], Hewitt and Ross [69], Chap. IV, or Loomis [121] for a proof of this. Left-invariance of the measure amounts to left-invariance of the corresponding integral,

$$\displaystyle{ \int _{G}f(\gamma g)\,\mathrm{d}\mu _{L}(g) =\int _{G}f(g)\,\mathrm{d}\mu _{L}(g), }$$
(1.1)

for any Haar integrable function f on G.

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

$$\displaystyle{G = \left \{\left (\begin{array}{cc} \,y\,&x\\ 0 & \,1\, \end{array} \right )\Big\vert \,x,y \in \mathbb{R},y > 0\right \},}$$

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

$$\displaystyle{\mathrm{d}\mu _{L} = {y}^{-2}\,\mathrm{d}x\,\mathrm{d}y,\qquad \mathrm{d}\mu _{ R} = {y}^{-1}\,\mathrm{d}x\,\mathrm{d}y.}$$

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 δ(g) > 0 such that

$$\displaystyle{\int _{G}f({g}^{-1}hg)\,\mathrm{d}\mu _{ L}(h) = \delta (g)\int _{G}f(h)\,\mathrm{d}\mu _{L}(h).}$$

If G is a topological group, a quasicharacter is a continuous homomorphism \(\chi: G\longrightarrow {\mathbb{C}}^{\times }\). If |χ(g)| = 1 for all g ∈ G, then χ is a (linear) character or unitary quasicharacter.

FormalPara Proposition 1.1.

The function \(\delta: G\longrightarrow \mathbb{R}_{+}^{\times }\) is a quasicharacter. The measure δ(h)μ L (h) is right-invariant.

The measure \(\delta (h)\mu _{L}(h)\) is a right Haar measure, and we may write \(\mu _{R}(h) = \delta (h)\mu _{L}(h)\). The quasicharacter δ is called the modular quasicharacter.

FormalPara Proof.

Conjugation by first g 1 and then g 2 is the same as conjugation by g 1 g 2 in one step. Thus \(\delta (g_{1}g_{2}) = \delta (g_{1})\,\delta (g_{2})\), so δ is a quasicharacter. Using (1.1),

$$\displaystyle{\delta (g)\int _{G}f(h)\,\mathrm{d}\mu _{L}(h) =\int _{G}f(g \cdot {g}^{-1}hg)\,\mathrm{d}\mu _{ L}(h) =\int _{G}f(hg)\,\mathrm{d}\mu _{L}(h).}$$

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

$$\displaystyle{\int _{G}f(h)\,\delta (h)\,\mathrm{d}\mu _{L}(h) =\int _{G}f(hg)\,\delta (h)\,\mathrm{d}\mu _{L}(h).}$$

Thus, the measure \(\delta (h)\,\mathrm{d}\mu _{L}(h)\) is right-invariant. □

FormalPara Proposition 1.2.

If G is compact, then G is unimodular and \(\mu _{L}(G) < \infty.\)

FormalPara Proof.

Since δ is a homomorphism, the image of δ is a subgroup of \(\mathbb{R}_{+}^{\times }\). Since G is compact, δ(G) is also compact, and the only compact subgroup of \(\mathbb{R}_{+}^{\times }\) is just \(\left \{1\right \}\). Thus δ 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 \(\int _{G}f(g)\,\mathrm{d}\mu _{L}(g)\) by \(\int _{G}f(g)\,\mathrm{d}g\).

FormalPara Proposition 1.3.

If G is unimodular, then the map \(g\longrightarrow {g}^{-1}\) is an isometry.

FormalPara Proof.

It is easy to see that \(g\longrightarrow {g}^{-1}\) turns a left Haar measure into a right Haar measure. If left and right Haar measures agree, then \(g\longrightarrow {g}^{-1}\) multiplies the left Haar measure by a positive constant, which must be 1 since the map has order 2. □

Exercises

FormalPara Exercise 1.1.

Let \(d_{\mathbf{a}}X\) denote the Lebesgue measure on \(\mathrm{Mat}_{n}(\mathbb{R})\). It is of course a Haar measure for the additive group \(\mathrm{Mat}_{n}(\mathbb{R})\). Show that \(\vert \det (X){\vert }^{-n}d_{\mathbf{a}}X\) is both a left and a right Haar measure on \(\mathrm{GL}(n, \mathbb{R})\).

FormalPara Exercise 1.2.

Let P be the subgroup of \(\mathrm{GL}(r + s, \mathbb{R})\) consisting of matrices of the form

$$\displaystyle{p = \left (\begin{array}{cc} g_{1} & X\\ &g_{2} \end{array} \right ),g_{1} \in \mathrm{GL}(r, \mathbb{R}),\;g_{2} \in \mathrm{GL}(s, \mathbb{R}),\quad X \in \mathrm{Mat}_{r\times s}(\mathbb{R}).}$$

Let dg 1 and dg 2 denote Haar measures on \(\mathrm{GL}(r, \mathbb{R})\) and \(\mathrm{GL}(s, \mathbb{R})\), and let \(d_{\mathbf{a}}X\) denote an additive Haar measure on \(\mathrm{Mat}_{r\times s}(\mathbb{R})\). Show that

$$\displaystyle{d_{L}p = \vert \det (g_{1}){\vert }^{-s}\,\mathrm{d}g_{ 1}\,\mathrm{d}g_{2}\,d_{\mathbf{a}}X,\qquad d_{R}p = \vert \det (g_{2}){\vert }^{-r}\,\mathrm{d}g_{ 1}\,\mathrm{d}g_{2}\,d_{\mathbf{a}}X,}$$

are (respectively) left and right Haar measures on P, and conclude that the modular quasicharacter of P is

$$\displaystyle{\delta (p) = \vert \det (g_{1}){\vert }^{s}\vert \det (g_{ 2}){\vert }^{-r}.}$$