Let $G$ be a compact Lie group with identity element $e$, let $\mathcal B(G)$ be its Borel $\sigma$-algebra, and let $\mu:\mathcal B(G)\to[0,1]$ be the normalized left Haar measure on $G$. Regard $L^2(G,\mu)$ as the complex [Hilbert space](/page/Hilbert%20Space) of $\mu$-equivalence classes of square-integrable Borel [measurable functions](/page/Measurable%20Functions) $G\to\mathbb C$, and let $GL(L^2(G,\mu))$ denote the group of bounded invertible complex-linear operators on this Hilbert space. For each $g,h\in G$, define operators $L_g,R_h:L^2(G,\mu)\to L^2(G,\mu)$ by $L_g[f]=[x\mapsto f(g^{-1}x)]$ and $R_h[f]=[x\mapsto f(xh)]$. Then $L_gR_h=R_hL_g$ for all $g,h\in G$. Consequently, the formula $\rho:G\times G\to GL(L^2(G,\mu))$, $\rho(g,h)[f]=[x\mapsto f(g^{-1}xh)]$, defines a strongly continuous unitary representation of the Lie group $G\times G$ on $L^2(G,\mu)$, where strong continuity means that $\rho(g_k,h_k)f\to\rho(g,h)f$ in $L^2(G,\mu)$ whenever $(g_k,h_k)\to(g,h)$ in $G\times G$ and $f\in L^2(G,\mu)$.