Let $G$ be a compact Lie group, and let $\mu_G$ denote the normalized Haar probability measure on $G$. Let $(\pi,V_\pi)$ and $(\sigma,V_\sigma)$ be irreducible finite-dimensional continuous unitary complex representations of $G$. Define their characters $\chi_\pi:G\to\mathbb C$ and $\chi_\sigma:G\to\mathbb C$ by

\begin{align*} \chi_\pi(g)=\operatorname{tr}(\pi(g)) \end{align*}

and

\begin{align*} \chi_\sigma(g)=\operatorname{tr}(\sigma(g)). \end{align*}

If $\pi$ and $\sigma$ are unitarily equivalent, then

\begin{align*} \int_G \chi_\pi(g)\overline{\chi_\sigma(g)}\,d\mu_G(g)=1. \end{align*}

If $\pi$ and $\sigma$ are not unitarily equivalent, then

\begin{align*} \int_G \chi_\pi(g)\overline{\chi_\sigma(g)}\,d\mu_G(g)=0. \end{align*}

Equivalently, after choosing one irreducible unitary representative from each unitary equivalence class,

\begin{align*} \int_G \chi_\pi(g)\overline{\chi_\sigma(g)}\,d\mu_G(g)=\delta_{\pi\sigma}. \end{align*}

Moreover, the set of irreducible characters, one chosen from each unitary equivalence class of irreducible finite-dimensional continuous unitary complex representations of $G$, is an orthonormal Hilbert basis of the closed subspace $L^2(G,\mu_G)^{\operatorname{class}}$ of $L^2(G,\mu_G)$ consisting of conjugation-invariant classes. Equivalently, $L^2(G,\mu_G)^{\operatorname{class}}$ is the fixed-point subspace for the unitary conjugation action of $G$ on $L^2(G,\mu_G)$.