Unimodular locally compact group - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group. It is unimodular if its modular function is identically one:

\Delta(g)=1\qquad\text{for every }g\in G.

Equivalently, any left Haar measure on $G$ is also right invariant. The property is independent of the scaling of the Haar measure. Thus, for a unimodular group, left and right translation have the same measure-theoretic behavior, and group inversion preserves Haar measure. Unimodularity is a property of the topological group, not an extra choice of measure.

Equivalent characterizations

For a fixed left Haar measure $\mu$ , unimodularity is equivalent to each of the identities

\mu(Eg)=\mu(E),\qquad \int_G f(x^{-1})\,d\mu(x)=\int_G f(x)\,d\mu(x)

whenever the expressions are defined. These equivalences follow from the change-of-variables formulas for Haar measure; see Folland, chapter 2.

Examples and non-examples

Every abelian, compact, or discrete locally compact group is unimodular. Connected semisimple Lie groups are also unimodular. The affine group of the real line, consisting of transformations $x\mapsto ax+b$ with $a>0$ , is a standard nonunimodular example.

Consequences for harmonic analysis

On a unimodular group, the convolution involution simplifies to $f^*(x)=\overline{f(x^{-1})}$ , and the right regular representation needs no modular correction. These simplifications do not imply that the group is abelian or compact.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: chapter 2 on Haar measure and the modular function.