Rigid analytic vectors in locally analytic representations

Abstract

Let H be a uniform pro-p group. Associated to H are rigid analytic affinoid groups \({\mathbb {H}}_n\), and their “wide open” subgroups \({\mathbb {H}}_n^{\circ }\). Denote by \(D^\mathrm{la}(H)= C^\mathrm{la}(H)'_b\) the locally analytic distribution algebra of H and by \(D({\mathbb {H}}_n^{\circ }, H)\) Emerton’s ring of \({\mathbb {H}}_n^{\circ }\)-rigid analytic distributions on H. If V is an admissible locally analytic representation of H, and if \(V_{{\mathbb {H}}_n^\circ -\mathrm{an}}\) denotes the subspace of \({\mathbb {H}}_n^\circ \)-rigid analytic vectors (with its intrinsic topology), then we show that the continuous dual of \(V_{{\mathbb {H}}_n^\circ -\mathrm{an}}\) is canonically isomorphic to \(D({\mathbb {H}}_n^{\circ }, H)\otimes _{D^\mathrm{la}(H)} V'\). From this we deduce the exactness of the functor \(V \rightsquigarrow V_{{\mathbb {H}}_n^\circ -\mathrm{an}}\) on the category of admissible locally analytic representations of H.