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Distinguished representations and poles of twisted tensor $L$-functions

Authors: U. K. Anandavardhanan, Anthony C. Kable and R. Tandon
Journal: Proc. Amer. Math. Soc. 132 (2004), 2875-2883
MSC (2000): Primary 11F70, 11F85
Published electronically: May 12, 2004
MathSciNet review: 2063106
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Abstract: Let $E/F$ be a quadratic extension of $p$-adic fields. If $\pi$ is an admissible representation of $GL_n(E)$ that is parabolically induced from discrete series representations, then we prove that the space of $P_n(F)$-invariant linear functionals on $\pi$ has dimension one, where $P_n(F)$ is the mirabolic subgroup. As a corollary, it is deduced that if $\pi$ is distinguished by $GL_n(F)$, then the twisted tensor $L$-function associated to $\pi$ has a pole at $s=0$. It follows that if $\pi$ is a discrete series representation, then at most one of the representations $\pi$ and $\pi \otimes \chi$ is distinguished, where $\chi$ is an extension of the local class field theory character associated to $E/F$. This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

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Additional Information

U. K. Anandavardhanan
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005, India

Anthony C. Kable
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

R. Tandon
Affiliation: Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India

Keywords: Distinguished representations, local twisted tensor $L$-function, Asai $L$-function, Bernstein-Zelevinsky derivatives
Received by editor(s): September 11, 2002
Received by editor(s) in revised form: June 3, 2003
Published electronically: May 12, 2004
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2004 American Mathematical Society

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