On a correspondence between cuspidal representations of <IMG ALIGN=MIDDLE HEIGHT=42 WIDTH=39>期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On a correspondence between cuspidal representations of

Authors:	David Ginzburg Stephen Rallis David Soudry

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

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 . Assume that $L^S(\eta,\Lambda^2,s)$ has a pole at 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 is a -adic field. Assume that $L(\tau,\Lambda^2,s)$ has a pole at . Given a nontrivial character $\psi$ of , 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 , and we prove that $\sigma _\psi(\tau)$ is the unique representation of $\widetilde{\operatorname{Sp}}_{2n}(F)$ satisfying these properties.

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