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arXiv 2001.081862020-01math.NTmath.RT

Doubling Constructions: the complete L-function for coverings of the symplectic group

Eyal Kaplan

备注：: Generalized the Rankin-Selberg integrals of Sect. 3 to $GL_c^{(m)} \times GL_k^{(m)}$ for all $c$

英文摘要：

We develop the local theory of the generalized doubling method for the $m$-fold central extension $Sp_{2n}^{(m)}$ of Matsumoto of the symplectic group. We define local $γ$-, $L$- and $ε$-factors for pairs of genuine representations of $Sp_{2n}^{(m)}\times\widetilde{GL}_k$ and prove their fundamental properties, in the sense of Shahidi. Here $\widetilde{GL}_k$ is the central extension of $GL_k$ arising in the context of the Langlands--Shahidi method for covering groups of $Sp_{2n}\times GL_k$. We then construct the complete $L$-function for cuspidal representations and prove its global functional equation. Possible applications include classification results and a Shimura type lift of representations from covering groups to general linear groups (a global lift is sketched here for $m=2$).

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