Duality for admissible locally analytic representations

We study the problem of constructing a contragredient functor on the category of admissible locally analytic representations ofpp-adic analytic groupGG. 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 ofGGwith 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 locallyQpQ_p-analytic groups.