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

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

Published electronically:
May 12, 2004

MathSciNet review:
2063106

Full-text PDF

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.

**1.**U.K. Anandavardhanan and R. Tandon: On distinguishedness,*Pacific J. Math.*206, No. 2 (2002), 269-286. MR**2003g:22017****2.**I.N. Bernstein and A.V. Zelevinsky: Representations of the group where is a non-archimedean local field,*Russian Math. Surveys*31:3 (1976), 1-68. MR**54:12988****3.**I.N. Bernstein and A.V. Zelevinsky: Induced representations of reductive -adic groups, I,*Ann. Scient. Ec. Norm. Sup.*(4) 10 (1977), 441-472. MR**58:28310****4.**J. Cogdell and I. Piatetski-Shapiro: Derivatives and -functions for ,*preprint, http://www.math.okstate.edu/cogdell/***5.**Y. Flicker: Twisted tensors and Euler products,*Bull. Soc. Math. France*116 (1988), 295-313. MR**89m:11049****6.**Y. Flicker: On distinguished representations,*J. Reine Angew. Math.*418 (1991), 139-172. MR**92i:22019****7.**Y. Flicker: ``On the local twisted -function.'' Appendix to his On zeroes of the twisted tensor -function,*Math. Ann.*297 (1993), 199-219. MR**95c:11065****8.**Y. Flicker and D. Zinoviev: On poles of twisted tensor -functions,*Proc. Japan Acad. Ser.*A 71 (1995), 114-116. MR**96f:11075****9.**J. Hakim: Distinguished -adic representations,*Duke Math J.*62 (1991), 1-22. MR**92c:22037****10.**G. Harder, R.P. Langlands and M. Rapoport: Algebraische Zyklen auf Hilbert-Blumenthal-Flächen,*J. Reine Angew. Math*366 (1986), 53-120. MR**87k:11066****11.**A. Kable: Asai -functions and Jacquet's conjecture, To appear in*Amer. J. Math.***12.**D. Prasad: Distinguished representations for quadratic extensions,*Compositio Math.*119 (1999), 335-345. MR**2001b:22016****13.**D. Prasad: On a conjecture of Jacquet about distinguished representations of ,*Duke Math J.*109 (2001), 67-78. MR**2002g:22036****14.**G. Warner: Harmonic analysis on semi-simple Lie groups I,*Springer-Verlag*(1972) MR**58:16979**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11F70,
11F85

Retrieve articles in all journals with MSC (2000): 11F70, 11F85

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