Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 19H / a / Solution

Every element of $SU(2)$ is unitarily conjugate to an element of the maximal torusSince characters are class functions, ${\chi }_V$ is determined on $T$. The restriction decomposes into finitely many torus weights, soA Weyl-group element conjugates $\mathrm{diag}(z,z^{-1})$ to $\mathrm{diag}(z^{-1},z)$, so ${\chi }_V(z)={\chi }_V(z^{-1})$ and $a_n=a_{-n}$.

Solved by gpt-5.6-sol high.