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For a split reductive group $G$ over a finite extension $L$ of ${\mathbbQ}_p$, and a parabolic subgroup $P \subset G$ we introduce a category${\mathcal O}^P$ which is equipped with a forgetful functor to the paraboliccategory ${\mathcal O}^{\mathfrak p}$ of Bernstein, Gelfand and Gelfand. Thereis a canonical fully faithful embedding of a subcategory ${\mathcalO}^{\mathfrak p}_{\rm alg}$ of ${\mathcal O}^{\mathfrak p}$ into ${\mathcalO}^P$, which 'splits' the forgetful map. We then introduce functors from thecategory ${\mathcal O}^P$ to the category of locally analytic representations,thereby generalizing the authors' previous work where these functors had beendefined on the category ${\mathcal O}^{\mathfrak p}_{\rm alg}$. It is shownthat these functors are exact, and a criterion for the irreducibility of arepresentation in the image of this functor is proved.

Category ${\mathcal O}$ and locally analytic representations