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

TL;DR

This paper introduces a new category ${ m f O}^P$ for split reductive groups over $p$-adic fields, extending previous work to connect algebraic and locally analytic representations, with results on exactness and irreducibility criteria.

Contribution

The authors define the category ${ m f O}^P$, embed a subcategory of algebraic modules, and develop functors to locally analytic representations, generalizing prior constructions.

Findings

Functor from ${ m f O}^P$ to locally analytic representations is exact.

Established a criterion for irreducibility of representations in the functor's image.

Embedded algebraic subcategory splits the forgetful functor.

Abstract

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