Left and right regular representations on L2(G)

Assume the Axiom of Choice. Let $G$ be a compact Lie group with normalized Haar measure $dg$, and let $L^2(G)=L^2(G,\mathbb{C})$ be the complex Hilbert space of square-integrable classes with inner product $⟨f,h⟩={\int }_{G}f(g)\overline{h(g)}dg$ ($L^2$ with the integral pairing is a Hilbert space).

• The left regular representation is $(L_{x}f)(y):=f(x^{-1}y),x,y\in G,$
• the right regular representation is $(R_{x}f)(y):=f(yx),x,y\in G\mathrm{.}$

Both are well defined on classes: right translation of the argument by $x$ and left translation by $x^{-1}$ are measure-preserving homeomorphisms of $G$, so they preserve null sets and integrability. Each $L_x$ and $R_x$ is a linear isometry and a unitary operator of $L^2(G)$, because invariance of $dg$ under translations gives $\mathrm{\parallel }{L}_{x}f{\mathrm{\parallel }}_2^2={\int }_G\mathrm{\mid }f(x^{-1}y){\mathrm{\mid }}^{2}dy={\int }_G\mathrm{\mid }f(y){\mathrm{\mid }}^{2}dy=\mathrm{\parallel }f{\mathrm{\parallel }}_2^2,$ and likewise for $R_x$. The assignments $x\mapsto L_x$ and $x\mapsto R_x$ are group homomorphisms $G\to U(L^2(G))$: $L_x{L}_{x^{\mathrm{\prime }}}=L_{x{x}^{\mathrm{\prime }}}$ and $R_x{R}_{x^{\mathrm{\prime }}}=R_{x{x}^{\mathrm{\prime }}}$, and they commute with each other, $L_x{R}_{x^{\mathrm{\prime }}}=R_{x^{\mathrm{\prime }}}{L}_x$, factorising the two-sided action $(x,x^{\mathrm{\prime }})\mapsto L_x{R}_{x^{\mathrm{\prime }}}$ of $G\times G$ on $L^2(G)$.

Both homomorphisms are strongly continuous. Indeed, let $f\in L^2(G)$ and $\epsilon >0$. Since normalized Haar measure is Radon and $G$ is compact, C_c(X) is dense in L^p(mu) for a Radon measure gives $c\in C(G)$ with $\mathrm{\parallel }f-c{\mathrm{\parallel }}_2<\epsilon$. Translation is isometric, so $\mathrm{\parallel }{L}_{x}f-f{\mathrm{\parallel }}_2\le 2\epsilon +\mathrm{\parallel }{L}_{x}c-c{\mathrm{\parallel }}_2\mathrm{.}$ Uniform continuity of $c$ on compact $G$ makes the last term tend to $0$ as $x\to e$; the same argument gives $\mathrm{\parallel }{R}_{x}f-f{\mathrm{\parallel }}_2\to 0$. Continuity at an arbitrary group element follows from the homomorphism law and the isometry of the translations.

These are the infinite-dimensional Hilbert-space representations of $G$ referred to in the definition of a representation (Continuous and unitary representations); their decomposition is proved later on this page.

• The two-sided action is unitary for the same inner product and makes $L^2(G)$ a unitary $G\times G$-module, which is how matrix-coefficient spaces of finite-dimensional representations embed into $L^2(G)$.
• The conventions are fixed so that the left action is by $(L_{x}f)(y)=f(x^{-1}y)$ and the right action by $(R_{x}f)(y)=f(yx)$; the convolution convention below is the corresponding right convolution.
• Below, $L^2(G)$ always carries the complex inner product and the normalized Haar measure; the linear functional $f\mapsto {\int }_{G}fdg$ is denoted by the same symbol as the pairing.