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LOCALLY ANALYTIC REPRESENTATIONS OF $\text{GL}(2,L)$ VIA SEMISTABLE MODELS OF $\mathbb{P}^{1}$

Part of: Linear algebraic groups and related topics Lie groups Differential algebra Analytic spaces

Published online by Cambridge University Press:  12 January 2017

Deepam Patel ,
Tobias Schmidt  and
Matthias Strauch
Deepam Patel
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA (deeppatel1981@gmail.com)
Tobias Schmidt
Affiliation:
Institut de Recherche Mathématiques de Rennes, Campus Beaulieu, Universite de Rennes 1, 35042 Rennes Cedex, France (tobias.schmidt@univ-rennes1.fr)
Matthias Strauch
Affiliation:
Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405, USA (mstrauch@indiana.edu)
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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 the work of M. Emerton, we then describe admissible representations of $\text{GL}_{2}(L)$ in terms of sheaves on the projective limit of these formal schemes. As an application, we show that representations coming from certain equivariant line bundles on Drinfeld’s first étale covering of the $p$-adic upper half plane are admissible.

Keywords

representations of Lie and linear algebraic groups over local fieldsrepresentation theorysheaves of differential operators and their modulesD-modulesrings of differential operators and their modules

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 20G05: Representation theory 32C38: Sheaves of differential operators and their modules, $D$-modules 13N10: Rings of differential operators and their modules
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 1 , January 2019 , pp. 125 - 187
Copyright
© Cambridge University Press 2017 

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Footnotes

M. S. would like to acknowledge the support of the National Science Foundation (award DMS-1202303). T. S. would like to acknowledge support of the Heisenberg Programme of Deutsche Forschungsgemeinschaft.

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