Schur orthogonality for compact Lie groups

Matrix coefficients of distinct irreducible unitary representations are orthogonal in L²(G), with a sharp normalization.

Schur orthogonality for compact Lie groups

Let $G$ be a compact Lie group . Fix the normalized Haar measure $dg$ on $G$ (so $\int_G 1\,dg=1$ ). Let $(\pi,V)$ and $(\sigma,W)$ be finite-dimensional continuous unitary representations of $G$ (see representation of a Lie group ), with $\pi,\sigma$ irreducible (see irreducible representation ). Choose orthonormal bases so that $\pi(g)$ and $\sigma(g)$ have matrix entries $\pi_{ij}(g)$ and $\sigma_{kl}(g)$ .

Schur orthogonality asserts:

\int_G \pi_{ij}(g)\,\overline{\sigma_{kl}(g)}\,dg \;=\; \begin{cases} \frac{1}{\dim V}\,\delta_{ik}\delta_{jl}, & \text{if }\pi\simeq \sigma,\\[6pt] 0, & \text{if }\pi\not\simeq \sigma. \end{cases}

Equivalently, the matrix coefficients of irreducible unitary representations form an orthogonal family in $L^2(G)$ , and within a fixed irreducible representation they are orthogonal with the explicit scale factor $1/\dim V$ .

This is the analytic avatar of Schur’s lemma and is one of the key inputs in the Peter–Weyl theorem , which decomposes $L^2(G)$ into finite-dimensional isotypic pieces. In practice, Schur orthogonality is the tool that turns representation theory into concrete integral identities on compact groups (compare also complete reducibility ).