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Category ${\mathcal O}$ and locally analytic representations (1412.5270v1)

Published 17 Dec 2014 in math.RT and math.NT

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.

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