Let $G$ be a connected compact Lie group. If $f : G \to \mathbb C$ is a class function ($f(ghg^{-1})=f(h)$ for any $g,h \in G$), the integral of $f$ can be computed in terms of an integral over a maximal torus $T \subset G$ (Weyl integration formula): $$ \int_G f(g) dg = \int_T f(t)w(t) dt $$ for a certain known and explicit function $w$ ($dg,dt$ are Haar measures on $G$ and $T$). It is not surprising that such formula exists, since $f$ is uniquely determined by its restriction to $T$.
Similarly, I would like to compute the convolution of two class functions $f_1,f_2$ in terms of their restriction to $T$: $$ f_1 * f_2 (t) = \int_{T} \int_T K(t;t_1,t_2) f_1(t_1) f(t_2) dt_1 dt_1 $$ with some known distribution $K$. Using the fact that characters $\chi_\alpha$ of irreducible representations form an orthonormal basis of class functions and $\chi_\alpha * \chi_\alpha = \frac{1}{d_{\alpha}} \chi_\alpha$ ($d_{\alpha}$ being the dimension of representation $\alpha$) I got $$ K(t; t_1, t_2 ) = \sum_{\alpha} \frac{1}{d_{\alpha}} \chi_{\alpha}(t) \overline{\chi_{\alpha}(t_1)} \overline{\chi_{\alpha}(t_2)} w(t_1) w(t_2) .$$ This formula is not terribly explicit and the summation converges only in some rather weak sense.
Has the distribution $K$ (or some equivalent) been studied? Is there an explicit formula, similar to Weyl integration formula? I would be interested even in the case $G= SU(2)$.