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Archimedean distinguished representations and exceptional poles

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Abstract

Let F be an archimedean local field and let E be \(F\times F\) (resp. a quadratic extension of F). We prove that an irreducible generic (resp. nearly tempered) representation of \(\textrm{GL}_n(E)\) is \(\textrm{GL}_n(F)\) distinguished if and only if its Rankin-Selberg (resp. Asai) L-function has an exceptional pole of level zero at 0. Further, we deduce a necessary condition for the ramification of such representations using the theory of weak test vectors developed by Humphries and Jo.

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Acknowledgements

I would like to thank my advisor Professor Ravi Raghunathan for his constant encouragement, invaluable comments, and for carefully reading several preliminary versions of this paper. I also wish to thank Professor Dipendra Prasad and Professor U.K. Anandavardhanan for several useful remarks.

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Correspondence to Akash Yadav.

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Yadav, A. Archimedean distinguished representations and exceptional poles. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01568-w

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