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Locally analytic distributions and $p$-adic representation theory, with applications to $GL_{2}$

Authors: Peter Schneider and Jeremy Teitelbaum
Journal: J. Amer. Math. Soc. 15 (2002), 443-468
MSC (2000): Primary 11S80, 22E50
Published electronically: October 18, 2001
MathSciNet review: 1887640
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Abstract: In this paper we study continuous representations of locally $L$-analytic groups $G$ in locally convex $K$-vector spaces, where $L$ is a finite extension of $\mathbb {Q}_p$ and $K$ is a spherically complete nonarchimedean extension field of $L$. The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of $G$, along with interesting new objects such as the action of $G$ on global sections of equivariant vector bundles on $p$-adic symmetric spaces. We introduce a restricted category of such representations that we call “strongly admissible” and we show that, when $G$ is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of $G$. As an application we prove the topological irreducibility of generic members of the $p$-adic principal series for $GL_2(\mathbb {Q}_p)$. Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous $K$-valued representations of locally $L$-analytic groups.

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Additional Information

Peter Schneider
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
MR Author ID: 156590

Jeremy Teitelbaum
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607

Received by editor(s): December 16, 1999
Received by editor(s) in revised form: May 16, 2001
Published electronically: October 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society