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On the irreducibility of locally analytic principal series representations


Authors: Sascha Orlik and Matthias Strauch
Journal: Represent. Theory 14 (2010), 713-746
MSC (2010): Primary 22E50
DOI: https://doi.org/10.1090/S1088-4165-2010-00387-8
Published electronically: December 1, 2010
MathSciNet review: 2738585
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Abstract: Let $\mathbf {G}$ be a $p$-adic connected reductive group with Lie algebra $\mathfrak {g}$. For a parabolic subgroup $\mathbf {P} \subset \mathbf {G}$ and a finite-dimensional locally analytic representation $V$ of a Levi subgroup of $\mathbf {P}$, we study the induced locally analytic $\mathbf {G}$-representation $W = \operatorname {Ind}_{\mathbf {P}}^{\mathbf {G}}(V)$. Our result is the following criterion concerning the topological irreducibility of $W$: If the Verma module $U(\mathfrak {g}) \otimes _{U(\mathfrak {p})} V’$ associated to the dual representation $V’$ is irreducible, then $W$ is topologically irreducible as well.


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Sascha Orlik
Affiliation: Fachgruppe Mathematik and Informatik, Bergische Universität Wuppertal, Gaußtraße 20, 42097 Wuppertal, Germany
Email: orlik@math.uni-wuppertal.de

Matthias Strauch
Affiliation: Department of Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47401
MR Author ID: 620508
Email: mstrauch@indiana.edu

Received by editor(s): November 26, 2007
Received by editor(s) in revised form: March 16, 2010, and May 23, 2010
Published electronically: December 1, 2010
Additional Notes: M.S. is partially supported by NSF grant DMS-0902103.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.