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

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.