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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On certain elements in the Bernstein center of $\mathbf {GL}_2$
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by Sandeep Varma PDF
Represent. Theory 17 (2013), 99-119 Request permission


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
  • Email:
  • Received by editor(s): December 23, 2011
  • Received by editor(s) in revised form: August 19, 2012
  • Published electronically: February 6, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 17 (2013), 99-119
  • MSC (2010): Primary 22E50, 22E35
  • DOI:
  • MathSciNet review: 3017263