Duality for admissible locally analytic representations

We study the problem of constructing a contragredient functor on the category of admissible locally analytic representations of $p$-adic analytic group $G$. A naive contragredient does not exist. As a best approximation, we construct an involutive ``duality'' functor from the bounded derived category of modules over the distribution algebra of $G$ with coadmissible cohomology to itself. on the subcategory corresponding to complexes of smooth representations, this functor induces the usual smooth contragredient (with a degree shift). Although we construct our functor in general we obtain its involutivity, for technical reasons, only in the case of locally $\mathbb{Q}_p$-analytic groups.