On a correspondence between cuspidal representations of 𝐺𝐿_{2𝑛} and 𝑆𝑝̃_{2𝑛}

Ginzburg, David¹ ; Rallis, Stephen² ; Soudry, David

¹ School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
² Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Journal of the American Mathematical Society, Tome 12 (1999) no. 3, pp. 849-907

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-99-00300-8

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