Finite-group Schur orthogonality

Give $G$ the discrete topology and singleton charts to ${\mathbb{R}}^0$. It is Hausdorff and second countable, its finite underlying space is compact, and multiplication and inversion are smooth in these zero-dimensional charts. Thus it is a compact Lie group. Since $G$ contains its identity, $\mathrm{\mid }G\mathrm{\mid }>0$ and $\mu (E)=\mathrm{\mid }E\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ is a probability. Every subset is open and compact, so this Borel measure is regular. Left translations, right translations and inversion permute $G$ and hence preserve cardinality and $\mu$. All hypotheses of [L1] hold, proving that $\mu$ is normalized Haar measure.

For a function $f$ on the finite group the Haar integral is therefore ${\int }_{G}fd\mu =\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}f(g)$, and substituting this into the compact orthogonality relations of [L2] gives the displayed finite-group formula. If $\pi$ and $\sigma$ are equivalent they are the same chosen representative and use the same fixed orthonormal basis, so the delta case is licensed exactly. Otherwise the inequivalent case applies. For the trivial group the sole irreducible is one-dimensional and the formula is $1=1$.