A note on a Hecke type integral for \(\mathrm{Sp}(2n) \times \mathrm{GL}(1)\). (English) Zbl 07838078
Let \(\mathbb{A}\) be the ring of adeles over a number field \(F\), \(\pi\) be an irreducible generic cuspidal automorphic representation of \(\operatorname{Sp}_{2n}(\mathbb{A})\), and \(\chi:F^\times\backslash\mathbb{A}_F^{\times} \to \mathbb{C}^\times\) be a unitary idele class character.
In this paper, the author presents an integral representation of Hecke type for the twisted standard \(L\)-function \(L(s, \pi \times \chi)\) of \(\operatorname{Sp}_{2n}(\mathbb{A}) \times \operatorname{GL}_1(\mathbb{A})\) of degree \(2n+1\).
The main computation in this paper involves the unfolding of the global integrals. The non-vanishing of the local integrals and their computation in the unramified case are deduced from [D. Ginzburg et al., Bull. Soc. Math. Fr. 126, No. 2, 181–244 (1998; Zbl 0928.11026)].
In this paper, the author presents an integral representation of Hecke type for the twisted standard \(L\)-function \(L(s, \pi \times \chi)\) of \(\operatorname{Sp}_{2n}(\mathbb{A}) \times \operatorname{GL}_1(\mathbb{A})\) of degree \(2n+1\).
The main computation in this paper involves the unfolding of the global integrals. The non-vanishing of the local integrals and their computation in the unramified case are deduced from [D. Ginzburg et al., Bull. Soc. Math. Fr. 126, No. 2, 181–244 (1998; Zbl 0928.11026)].
Reviewer: Akash Yadav (Mumbai)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
Keywords:
Rankin-Selberg integral; cuspidal automorphic generic representation; twisted \(L\)-functionCitations:
Zbl 0928.11026References:
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