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DUALIZING INVOLUTIONS ON THE METAPLECTIC GL(2) à la TUPAN

Part of: Lie groups

Published online by Cambridge University Press:  10 July 2020

KUMAR BALASUBRAMANIAN  and
EKTA TIWARI
KUMAR BALASUBRAMANIAN
Affiliation:
Department of Mathematics, IISER Bhopal, Bhopal, Madhya Pradesh462066, India, e-mails: bkumar@iiserb.ac.in; ekta@iiserb.ac.in
EKTA TIWARI
Affiliation:
Department of Mathematics, IISER Bhopal, Bhopal, Madhya Pradesh462066, India, e-mails: bkumar@iiserb.ac.in; ekta@iiserb.ac.in
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Abstract

Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 426 - 437
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

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