Haar measures and the modular function

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 April 2010

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 $${\parallel f\parallel }_{\infty }=\sup \left\{\left|f\left(g\right)\right||g\in G\right\},$$
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 $$\mu \left(f\right)=\Delta \left(g\right)\mu \left(R_{g}f\right),\text{for all}f\in C_0\left(G\right).$$ 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 $$\mu \left(f*\right)=\mu \left({\Delta }^{-1}f\right)\text{or}{\int }_{G}f\left(g\right)d\mu \left(gh\right)={\int }_{G}f\left(g\right)\Delta \left(h\right)d\mu \left(g\right),\text{or}\mu \left(f\right)={\mu }_R\left(\Delta f\right),$$ 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 $$\mu \left(f\right)={\int }_G{f}_{\omega },\text{for all }f\in C_0\left(G\right),$$ where $\omega$ is the unique positive left invariant $n$ form on $G.$
2. For a Lie group $G$ the modular function is given by $$\Delta \left(g\right)=\left|\det {\mathrm{Ad}}_g\right|,\text{for all}g\in G.$$

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 $\frac{1}{{\left|\det \left(x_{ij}\right)\right|}^n}\prod _{i,j=1}^{n}d{x}_{ij}.$
4. The group $B_n$ of upper triangular matrices in ${\mathrm{GL}}_n\left(ℝ\right)$ has Haar measure $\frac{1}{{\prod _{i=1}^n\left|x_{ii}\right|}^i}\prod _{1\le i<j\le n}d{x}_{ij}.$ This group is not unimodular unless $n=1.$
5. A finite group has Haar measure $\mu \left(f\right)=\frac{1}{\left|G\right|}\sum _{g\in G}f\left(g\right).$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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