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.