Solution
= Solution
The character of the <dual representation> is $\chi_{V_n^*}(g)=\chi_n(g^{-1})=\overline{\chi_n(g)}$, 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 $\boxed{V_n\cong V_n^*}$. This is complex-linear self-duality; it does not assert that every $V_n$ has an invariant real structure.