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

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.