Distinguished representations and poles of twisted tensor -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 | References | Similar Articles | Additional Information

Abstract: Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . 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

Email:
anand@math.tifr.res.in

**Anthony C. Kable**

Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
akable@math.okstate.edu

**R. Tandon**

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

Email:
rtsm@uohyd.ernet.in

DOI:
https://doi.org/10.1090/S0002-9939-04-07424-6

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