Schur orthogonality for compact groups

Remarks

For a linear map $A$ between the two spaces, the average $\int_G \sigma(g) A \pi(g)^{-1} \, dg$ intertwines $\pi$ and $\sigma$. By Schur's lemma it is $0$ when they are not isomorphic and a scalar when $\pi = \sigma$. Taking $A$ to be a matrix unit gives parts 1 and 2, the scalar being found by taking traces; summing over $i = j$ and $k = l$ gives part 3. Example: for $G = \mathbb{T}$ with the characters $\chi_n(z) = z^n$, part 3 reads $\frac{1}{2\pi}\int_0^{2\pi} e^{in\theta} e^{-im\theta} \, d\theta = \delta_{nm}$. For a finite group with Haar measure the counting measure divided by $\abs{G}$, the statement is the Schur orthogonality relations of finite groups. A consequence: a finite-dimensional continuous representation with character $\chi$ is irreducible if and only if $\ip{\chi}{\chi} = 1$.

Going deeper

For a student

These relations give the orthonormality half of the Peter–Weyl theorem; the completeness half is the harder part. For compact connected Lie groups, the Weyl integration formula turns the character relation into an integral over a maximal torus, which is the route to the Weyl character formula. The relations apply to every finite-dimensional continuous representation after a change of basis, since each one is unitary for a suitable inner product by complete reducibility.