Haar measures and the modular function

Haar measures and the modular function

Let $G$ be a locally compact Hausdorff topological group. A Haar measure on $G$ is a linear functional $\mu :{C}_{0}\left(G\right)\to ℂ$ such that

1. (continuity) $\mu$ is continuous with respect to the topology on ${C}_{0}\left(G\right)$ given by $∥f∥ ∞ =sup f g | g∈G ,$
2. (positivity) If $f\left(g\right)\in {ℝ}_{\ge 0}$ for all $g\in G$ then $\mu \left(f\right)\in {ℝ}_{\ge 0},$
3. (left invariance) $\mu \left({L}_{g}f\right)=\mu \left(f\right),$ for all $g\in G$ and $f\in {C}_{0}\left(G\right).$

(Existence and uniqueness of Haar measure) If $G$ is a locally compact Hausdorff topological group then $G$ has a Haar measure and any two Haar measures are proportional.

Fix a (left) Haar measure $\mu$ on $G.$ A group is unimodular if $\mu$ is also a right Haar measure on $G.$ The modular function is the function $\Delta :G\to {ℝ}_{\ge 0}$ given by $μ f =Δ g μ R g f ,for allf∈ C 0 G .$ The fact that the image of $\Delta$ is in ${ℝ}_{\ge 0}$ is a consequence of the positivity condition in the definition of Haar measure. There are several equivalent ways of defining the modular function $μ f* =μ Δ -1 f or ∫ G f g dμ gh = ∫G f g Δ h dμ g ,orμ f = μR Δf ,$ for all $f\in {C}_{0}\left(G\right),$ where ${\mu }_{R}$ is a right Haar measure on $G.$ The group $G$ is unimodular exactly when $\Delta =1.$

Finite groups, abelian groups, compact groups, semisimple Lie groups, reductive Lie groups, and nilpotent Lie groups are all unimodular.

1. On a Lie group the Haar measure is given by where $\omega$ is the unique positive left invariant $n$ form on $G.$
2. For a Lie group $G$ the modular function is given by $Δ g = det Ad g ,for allg∈G.$

Examples

1. $ℝ,$ under addition. Haar measure is the usual Lebesgue measure $dx$ on $ℝ.$
2. ${ℝ}_{\ge 0},$ under multiplication. Haar measure is given by $\frac{1}{x}dx.$
3. ${\mathrm{GL}}_{n}\left(ℝ\right)$ has Haar measure $\frac{1}{{\left|\mathrm{det}\left({x}_{ij}\right)\right|}^{n}}\prod _{i,j=1}^{n}d{x}_{ij}.$
4. The group ${B}_{n}$ of upper triangular matrices in ${\mathrm{GL}}_{n}\left(ℝ\right)$ has Haar measure $\frac{1}{{\prod _{i=1}^{n}\left|{x}_{ii}\right|}^{i}}\prod _{1\le i This group is not unimodular unless $n=1.$
5. A finite group has Haar measure $\mu \left(f\right)=\frac{1}{\left|G\right|}\sum _{g\in G}f\left(g\right).$