Let $G$ be a compact Lie group, and let $\mu$ denote the normalized Haar probability measure on $G$. Let $V$ and $W$ be non-zero finite-dimensional complex vector spaces equipped with Hermitian inner products $(\cdot,\cdot)_V$ and $(\cdot,\cdot)_W$, linear in the first argument. Let $\rho:G\to U(V)$ and $\sigma:G\to U(W)$ be continuous irreducible unitary complex representations of $G$. Let $(v_i)_{i=1}^{m}$ be an [orthonormal basis](/page/Orthonormal%20Basis) of $V$, and let $(w_a)_{a=1}^{n}$ be an orthonormal basis of $W$.

For $1\leq i,j\leq m$, define the matrix coefficient

\begin{align*} \rho_{ij}:G\to \mathbb C,\qquad g\mapsto (\rho(g)v_j,v_i)_V. \end{align*}

For $1\leq a,b\leq n$, define the matrix coefficient

\begin{align*} \sigma_{ab}:G\to \mathbb C,\qquad g\mapsto (\sigma(g)w_b,w_a)_W. \end{align*}

Let $\delta_{pq}$ denote the Kronecker delta, equal to $1$ when $p=q$ and equal to $0$ when $p\neq q$.

If $(\rho,V)$ and $(\sigma,W)$ are not isomorphic as continuous representations of $G$, then for all $1\leq i,j\leq m$ and all $1\leq a,b\leq n$,

\begin{align*} \int_G \rho_{ij}(g)\overline{\sigma_{ab}(g)}\,d\mu(g)=0. \end{align*}

If $(\sigma,W)=(\rho,V)$ and the same orthonormal basis is used, then for all $1\leq i,j,a,b\leq m$,

\begin{align*} \int_G \rho_{ij}(g)\overline{\rho_{ab}(g)}\,d\mu(g)=\frac{1}{\dim V}\delta_{ia}\delta_{jb}. \end{align*}