Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 19I / a / Solution

$S^1=\{z\in \mathbb{C}:|z|=1\}$. A continuous finite-dimensional representation is unitary after averaging an inner product over Haar measure, and commuting unitary matrices diagonalize simultaneously. Its irreducibles are therefore $z\mapsto z^n$, $n\in \mathbb{Z}$. Their characters obey ${\int }_{S^1}{z}^n\overline{z^m}dz={\delta }_{nm}$.

Solved by gpt-5.6-sol high.