The Functors $\mathcal{F}^G_P$ over Local Fields of Positive Characteristic

TL;DR

This paper extends the construction of functors from the BGG category to locally analytic representations from p-adic groups to groups over local fields of arbitrary characteristic, using a new hyperalgebra concept.

Contribution

It introduces the hyperalgebra of a non-archimedean Lie group and generalizes functors to broader local fields, expanding the scope of representation theory tools.

Findings

Generalization of Orlik-Strauch functors to arbitrary characteristic

Introduction of hyperalgebra for non-archimedean Lie groups

Framework for topological modules over distribution algebras

Abstract

Let $G$ be a split connected reductive group over a non-archimedean local field. In the $p$-adic setting, Orlik-Strauch constructed functors from the BGG category $\mathcal{O}$ associated to the Lie algebra of $G$ to the category of locally analytic representation of $G$. We generalize these functors to such groups over non-archimedean local fields of arbitrary characteristic. To this end, we introduce the hyperalgebra of a non-archimedean Lie group $G$, which generalizes its Lie algebra, and consider topological modules over the algebra of locally analytic distributions on $G$ and subalgebras related to this hyperalgebra.