Spectral analysis of the free orthogonal matrix

We compute the spectral measure of the standard generators $u_{ij}$ of the Wang algebra $A_o(n)$. We show in particular that this measure has support $[-2\mathrm{/}\sqrt{n+2},2\mathrm{/}\sqrt{n+2}]$, and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra $A_o(F)$, a representation of $S{U}_2^q$ due to Woronowicz, and several calculations with orthogonal polynomials.