Dagger groups and $p$-adic distribution algebras

Let $(G,\omega)$ be a $p$-saturated group and $K/\mathbb{Q}_p$ a finite extension. In this paper we introduce the space of $K$-valued overconvergent functions $\mathcal{C}^\dagger(G,K)$. In the process we promote the rigid analytic group attached to $(G,\omega)$ in a previous work of the first two authors to a dagger group. A main result of this article is that under certain assumptions (satisfied for example when $G$ is a uniform pro-$p$ group) the distribution algebra $D^\dagger(G,K)$, i.e. the strong dual of $\mathcal{C}^\dagger(G,K)$, is a Fr\'{e}chet-Stein algebra in the sense of Schneider and Teitelbaum. In the last section we introduce overconvergent representations and show that there is an anti-equivalence of categories between overconvergent $G$-representations of compact type and continuous $D^\dagger(G, K)$-modules on nuclear Fr\'{e}chet spaces. This is analogous to the anti-equivalence between locally analytic representations and modules over the locally analytic distribution algebra as proved by Schneider and Teitelbaum.