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

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- by David Ginzburg, Stephen Rallis and David Soudry
- J. Amer. Math. Soc.
**12**(1999), 849-907 - DOI: https://doi.org/10.1090/S0894-0347-99-00300-8
- Published electronically: April 26, 1999
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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.## References

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## Bibliographic 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**- MR Author ID: 205346
- Email: soudry@math.tau.ac.il
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 849-907 - MSC (1991): Primary 11F27, 11F70, 11F85
- DOI: https://doi.org/10.1090/S0894-0347-99-00300-8
- MathSciNet review: 1671452