If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $ \chi^{\lambda} *\chi^{\mu} = \delta_{\lambda, \mu} \frac{|G|}{\dim V^{\lambda}} \,\chi^{\lambda}$ where $\chi^{\lambda} *\chi^{\mu}(\sigma)=\sum_{g\in G}\chi^{\lambda}(\sigma g^{-1})\chi^{\mu}(g)$ is the product of convoluition and $\delta_{\lambda, \mu}$ equals 1 if $\lambda=\mu$ and 0 if $\lambda\neq\mu$.
I'm actually interested in representations of the symetric group $\mathcal{S}_n$, and this relation was instrumental in the definition of an isomorphism between the center of $\mathbb{C}[\mathcal{S}_n]$ and the complex functions on the conjugacy classes of $\mathcal{S}_n$.
I found a demonstration on this page http://drexel28.wordpress.com/2011/03/02/representation-theory-using-orthogonality-relations-to-compute-convolutions-of-characters-and-matrix-entry-functions/:
\begin{aligned}\left(\chi^{(\alpha)}\ast\chi^{(\beta)}\right)(x) &= \sum_{g\in G}\chi^{(\alpha)}\left(xg^{-1}\right)\chi^{(\beta)}(g)\\ &=\sum_{g\in G} \sum_{p=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,p}\left(xg^{-1}\right)D^{(\beta)}_{q,q}(g)\\ &=\sum_{p,s=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,s}(x)\sum_{g\in G}\overline{D^{(\alpha)}_{p,s}(g)}D^{(\beta)}_{q,q}(g)\\ &= \sum_{p,s=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,s}(x)\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\delta_{p,q}\delta_{q,s}\\ &=\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\sum_{p,s=1}^{d_\alpha}D^{(\alpha)}_{p,s}(x)\delta_{p,q}\delta_{p,s}^2\\ &= \frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\sum_{p=1}^{d_\alpha}D^{(\alpha)}_{p,p}(x)\\ &= \frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\chi^{(\alpha)}(x)\end{aligned}
but it is not clear for me why $\sum_{g\in G}\overline{D^{(\alpha)}_{p,s}(g)}D^{(\beta)}_{q,q}(g)$ is equal to $\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\delta_{p,q}\delta_{q,s} $. It seems to me that it is a not-so-trivial consequence of Schur's lemma. So my question is : Is there a simpler way of proving this relation about convolution of characters ?