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arXiv · 2401.09063

Archimedean Distinguished Representations and Exceptional Poles

Akash Yadav
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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 $\operatorname{GL}_n(E)$ is $\operatorname{GL}_n(F)$ 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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Akash Yadav. 2024-01-17. Archimedean Distinguished Representations and Exceptional Poles. https://doi.org/10.1007/s00229-024-01568-w

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