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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

On a correspondence between cuspidal
representations of $\operatorname{GL}_{2n}$ and $\widetilde {\operatorname{Sp}}_{2n}$


Authors: David Ginzburg, Stephen Rallis and David Soudry
Journal: J. Amer. Math. Soc. 12 (1999), 849-907
MSC (1991): Primary 11F27, 11F70, 11F85
Published electronically: April 26, 1999
MathSciNet review: 1671452
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Abstract: Let $\eta$ be an irreducible, automorphic, self-dual, cuspidal representation of $\operatorname{GL}_{2n}(\mathbb A)$, where $\mathbb A$ is the adele ring of a number field $K$. Assume that $L^S(\eta,\Lambda^2,s)$ has a pole at $s=1$ and that $L(\eta, \frac 12)\neq 0$. Given a nontrivial character $\psi$ of $K\backslash\mathbb A$, we construct a nontrivial space of genuine and globally $\psi^{-1}$-generic cusp forms $V_{\sigma _{\psi}(\eta)}$ on $\widetilde{\operatorname{Sp}}_{2n}(\mathbb A)$-the metaplectic cover of ${\operatorname{Sp}}_{2n}(\mathbb A)$. $V_{\sigma _{\psi}(\eta)}$ is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and $\psi^{-1}$-generic representations of $\widetilde{\operatorname{Sp}}_{2n}(\mathbb A)$, which lift (``functorially, with respect to $\psi$") to $\eta$. We also present a local counterpart. Let $\tau$ be an irreducible, self-dual, supercuspidal representation of $\operatorname{GL}_{2n}(F)$, where $F$ is a $p$-adic field. Assume that $L(\tau,\Lambda^2,s)$ has a pole at $s=0$. Given a nontrivial character $\psi$ of $F$, we construct an irreducible, supercuspidal (genuine) $\psi^{-1}$-generic representation $\sigma _\psi(\tau)$ of $\widetilde{\operatorname{Sp}}_{2n}(F)$, such that $\gamma(\sigma _\psi(\tau)\otimes\tau,s,\psi)$ has a pole at $s=1$, and we prove that $\sigma _\psi(\tau)$ is the unique representation of $\widetilde{\operatorname{Sp}}_{2n}(F)$ satisfying these properties.


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Additional Information

David Ginzburg
Affiliation: School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: ginzburg@math.tau.ac.il

Stephen Rallis
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: haar@math.ohio-state.edu

David Soudry
Email: soudry@math.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00300-8
PII: S 0894-0347(99)00300-8
Received by editor(s): July 22, 1998
Received by editor(s) in revised form: March 1, 1999
Published electronically: April 26, 1999
Additional Notes: The first and third authors’ research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.
Article copyright: © Copyright 1999 American Mathematical Society