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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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 PDF
J. Amer. Math. Soc. 12 (1999), 849-907 Request permission

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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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.
  • © 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