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

Authors:
David Ginzburg, Stephen Rallis and David Soudry

Journal:
J. Amer. Math. Soc. **12** (1999), 849-907

MSC (1991):
Primary 11F27, 11F70, 11F85

DOI:
https://doi.org/10.1090/S0894-0347-99-00300-8

Published electronically:
April 26, 1999

MathSciNet review:
1671452

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Abstract | References | Similar Articles | Additional Information

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**

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.

Article copyright:
© Copyright 1999
American Mathematical Society