Locally analytic representations of ${\rm GL}(2,L)$ via semistable models of ${\mathbb P}^1$

TL;DR

This paper explores the structure of locally analytic representations of ${ m GL}(2,L)$ by examining sheaves of differential operators on semistable models of ${ m P}^1$, establishing an equivalence with modules over distribution algebras.

Contribution

It introduces a novel approach linking sheaves on semistable models to analytic distribution algebras, advancing the understanding of ${ m GL}(2,L)$ representations.

Findings

Global sections of sheaves correspond to analytic distribution algebras.

Equivalence between coherent sheaves and finitely presented modules established.

Description of admissible ${ m GL}(2,L)$ representations via sheaves on formal schemes.

Abstract

In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of ${\mathbb Q}_p$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using work of M. Emerton, we then describe admissible representations of ${\rm GL}(2,L)$ in terms of sheaves on the projective limit of these formal schemes.