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Representation Theory

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On certain elements in the Bernstein center of $ \mathbf{GL}_2$

Author: Sandeep Varma
Journal: Represent. Theory 17 (2013), 99-119
MSC (2010): Primary 22E50, 22E35
Published electronically: February 6, 2013
MathSciNet review: 3017263
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Abstract: Let $ F$ be a nonarchimedean local field of residue characteristic $ p$, and let $ r$ be an odd natural number less than $ p$. Using the work of Moy and Tadić, we find an element $ z$ of the Bernstein center of $ G = \mathbf {GL}_2(F)$ that acts on any representation $ \pi $ of $ G$ by the scalar $ z(\pi ) = \operatorname {tr} \left (\operatorname {Frob} ; \left ( \operatorname {Sym}^r \circ \varphi _{\pi }\right )^{I_F} \right )$, the trace of any geometric Frobenius element $ \operatorname {Frob}$ of the absolute Weil group $ W_F$ of $ F$, acting on the inertia-fixed points of the representation $ \operatorname {Sym}^r \circ \varphi _{\pi }$ of $ W_F$, where $ \varphi _{\pi } : W_F \rightarrow \hat {G}$ is the restriction to $ W_F$ of the Langlands parameter of $ \pi $. This element $ z$ is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of $ G$, that includes all the groups of both the congruence and the Iwahori filtrations of $ G$ having depth at least one.

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Additional Information

Sandeep Varma
Affiliation: Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai - 400 005, India

Keywords: Bernstein center, stable Bernstein center
Received by editor(s): December 23, 2011
Received by editor(s) in revised form: August 19, 2012
Published electronically: February 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.