On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras

Let $\mathcal{G}\simeq H⋊\mathrm{\Gamma }$ be the semidirect product of a finite group $H$ and $\mathrm{\Gamma }\simeq {\mathbb{Z}}_p$. Let $F\mathrm{/}{\mathbb{Q}}_p$ be a finite extension with ring of integers ${\mathcal{O}}_F$. Then the total ring of quotients ${\mathcal{Q}}^F(\mathcal{G})$ of the completed group ring ${\mathcal{O}}_F[[\mathcal{G}]]$ is a semisimple ring. We determine its Wedderburn decomposition under a ramification hypothesis by relating it to the Wedderburn decomposition of the group ring $F[H]$.