Schur orthogonality for compact Lie groups - Imported Knowlpedia Content

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}

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).

Equivalent characterizations

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 $1/\dim V$ .