Let $G$ be a compact group (e.g. $SU(2)$) and $\rho: G \mapsto GL(n)$ a representation of it. Then we can define the delta function
$\delta(g-1)=\sum_{l}\chi_l(g)\chi_l(1) = \sum_l\dim_l(G)\chi_l(g)$
for any group element $g \in G$, where $l$ is the index that runs over all irreducible representation of $G$ and $\chi_l(g)=Tr(\rho_l(g))$ is the character in $l$-th representation. Above equation is Peter-Weil theorem.
What is the $m$-th derivative of the delta function $\delta^{(m)}(g-1)$? Is there also an expression in terms of the characters?
Maybe I would derive the representation operators $\rho_l(g)$ as follows:
$\frac{d}{dg}\rho_l(g) = \lim_{\lambda \mapsto 0} Tr(\frac{\rho_l(\delta g_{\lambda}g)-\rho_l(g)}{\lambda})$
Here I have an infinitesimal group displacement $\delta g_{\lambda}$ for an infinitesimal parameter $\lambda$. Is there a recipe for $m$-fold derivatives?
