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

Abstract

We develop the local theory of the generalized doubling method for the $m$-fold central extension $S{p}_{2n}^{(m)}$ of Matsumoto of the symplectic group. We define local $\gamma$-, $L$- and $\epsilon$-factors for pairs of genuine representations of $S{p}_{2n}^{(m)}\times {\widetilde{GL}}_k$ and prove their fundamental properties, in the sense of Shahidi. Here ${\widetilde{GL}}_k$ is the central extension of $G{L}_k$ arising in the context of the Langlands--Shahidi method for covering groups of $S{p}_{2n}\times G{L}_k$. 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 $m=2$).

Generalized the Rankin-Selberg integrals of Sect. 3 to $G{L}_c^{(m)}\times G{L}_k^{(m)}$ for all $c$

References in corpus (6)

• The Langlands-Shahidi Method for the metaplectic group and applications
• Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group
• Doubling constructions: Global functoriality for non-generic cuspidal representations
• Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition
• A Godement-Jacquet type integral and the metaplectic Shalika model
• Tensor Product $L$-Functions On Metaplectic Covering Groups of $G{L}_r$

Cited by in corpus (1)

• Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group