Left and right regular representations of a locally compact group

Definition

Let $G$ be a locally compact group with left Haar measure $\mu$ and modular function $\Delta$ , normalized by $\int_G h(xg)\,d\mu(x)=\Delta(g)^{-1}\int_G h(x)\,d\mu(x)$ . On the $L^2$ -space $L^2(G,\mu)$ , the left and right regular representations are

(\lambda(g)\xi)(x)=\xi(g^{-1}x),\qquad (\rho(g)\xi)(x)=\Delta(g)^{1/2}\xi(xg).

Both maps are strongly continuous unitary representations, and their images commute: $\lambda(g)\rho(h)=\rho(h)\lambda(g)$ . The modular factor is precisely what makes right translation unitary for a left Haar measure; it equals $1$ when $G$ is unimodular. With the displayed conventions, both $g\mapsto\lambda(g)$ and $g\mapsto\rho(g)$ are group homomorphisms.

Unitarity and continuity

Left invariance of $\mu$ gives $\lVert\lambda(g)\xi\rVert_2=\lVert\xi\rVert_2$ . Right translation changes the squared $L^2$ -norm by $\Delta(g)^{-1}$ , so the factor $\Delta(g)^{1/2}$ makes $\rho(g)$ unitary. Strong continuity follows first for compactly supported continuous functions and then for all of $L^2(G)$ by density, as in Folland, §§2.4 and 3.1.

Relation to convolution

For suitable $f$ , the integrated operator $\lambda(f)=\int_G f(g)\lambda(g)\,d\mu(g)$ acts by left convolution. Its adjoint is $\lambda(f^*)$ , where $f^*$ is the convolution involution. This is the analytic regular representation used to define reduced group $C^*$ -algebras.

Conventions and scope

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §§2.4 and 3.1 on Haar translation and regular unitary representations.