Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 19F / iii / Solution

The character of the dual representation is ${\chi }_{V_n^{\ast }}(g)={\chi }_n(g^{-1})=\overline{{\chi }_n(g)}$, using unitarizability. The displayed sine quotient is real, so this character equals ${\chi }_n$. The theorem that characters determine finite-dimensional semisimple complex representations of a compact group therefore gives $\boxed{V_n\cong V_n^{\ast }}$. This is complex-linear self-duality; it does not assert that every $V_n$ has an invariant real structure.