Locally algebraic automorphisms of the ${\mathrm{PGL}}_2(F)$-tree and $\mathfrak{o}$-torsion representations

Abstract

For a local field $F$ and an Artinian local coefficient ring $\Lambda$ with the same positive residue characteristic $p$ we define, for any $e\in \mathbb{N}$, a category ${\mathfrak{C}}^{(e)}(\Lambda )$ of ${\mathrm{GL}}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\mathrm{PGL}}_2(F)$. There is an obvious functor from the category of ${\mathrm{GL}}_2(F)$-representations over $\Lambda$ to ${\mathfrak{C}}^{(e)}(\Lambda )$. If $F={\mathbb{Q}}_p$ then ${\mathfrak{C}}^{(1)}(\Lambda )$ is equivalent to the category of smooth ${\mathrm{GL}}_2({\mathbb{Q}}_p)$-representations over $\Lambda$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${\mathfrak{C}}^{(e)}(\Lambda )$ with trivial central character is equivalent to a category of representations of a certain subgroup of $\mathrm{Aut}(X)$ consisting of "locally algebraic automorphisms of level $e$". For $e=1$ there is a functor from this category to that of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is a bijection between irreducible objects. Finally, we present a parallel of Colmez' functor $V\mapsto \mathbf{D}(V)$: to objects in ${\mathfrak{C}}^{(e)}(\Lambda )$ (for any $F$) we assign certain étale $(\phi ,\Gamma )$-modules over an Iwasawa algebra $\mathfrak{o}[[{\hat{N}}_{0,1}^{(1)}]]$ which contains the (usually considered) Iwasawa algebra $\mathfrak{o}[[N_0]]$. This assignment preserves finite generation.