$S^1=\{z\in\mathbb C:|z|=1\}$. 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 $\int_{S^1}z^n\overline{z^m}\,dz=δ_{nm}$.


Solution

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

 Solution (source code)