Higher level cusp forms on exceptional group of type \(E_7\). (English) Zbl 1531.11048
Let \(N \in \mathbb{Z}_{\geq 1}\). Let \(S_k(\Gamma_0(N), \chi)^{\mathrm{new}}\) be the space of elliptic newforms of integral weight \(k \geq 1\) and level \(\Gamma_0(N)\) with character \(\chi\) such that \(\chi(-1) = (-1)^k\), and let \(S_k(\Gamma_0(N), \chi)^{\mathrm{ns}}\) be the subspace of forms whose associated representation \(\pi\) has no supercuspidal components. The main result (Theorem 1.1) of this paper constructs from every \(h \in S_k(\Gamma_0(N), \chi)^{\mathrm{ns}}\) a cuspidal automorphic representation \(\Pi\) of \(\mathbf{G} = \mathrm{GE}_{7, 3}\) with central character \(\chi^3\). Here \(\mathrm{GE}_{7, 3}\) is the exceptional similitude group of type \(\mathrm{E}_{7, 3}\).
More concretely and classically, one constructs a complex linear map \(S_k(\Gamma_0(N), \chi)^{\mathrm{ns}} \to S_{k+8}(\Gamma^{\mathbf{G}}_0(N), \chi^3)\) that preserves Hecke eigenforms off \(N\). The standard \(L\)-function of degree \(56\) of \(\Pi\) (Theorem 1.2) and the adjoint \(L\)-function of degree \(133\) (Theorem 1.3) are also expressed explicitly in terms of \(L\)-functions of \(\pi\).
The proof is based on techniques with degenerate Whittaker functions developed by Ikeda and Yamana. When \(N=1\), the earlier result by the same authors [Compos. Math. 152, No.2, 223–254 (2016; Zbl 1337.11031)] are recovered.
More concretely and classically, one constructs a complex linear map \(S_k(\Gamma_0(N), \chi)^{\mathrm{ns}} \to S_{k+8}(\Gamma^{\mathbf{G}}_0(N), \chi^3)\) that preserves Hecke eigenforms off \(N\). The standard \(L\)-function of degree \(56\) of \(\Pi\) (Theorem 1.2) and the adjoint \(L\)-function of degree \(133\) (Theorem 1.3) are also expressed explicitly in terms of \(L\)-functions of \(\pi\).
The proof is based on techniques with degenerate Whittaker functions developed by Ikeda and Yamana. When \(N=1\), the earlier result by the same authors [Compos. Math. 152, No.2, 223–254 (2016; Zbl 1337.11031)] are recovered.
Reviewer: Wen-Wei Li (Beijing)
MSC:
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 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 |
| 20G41 | Exceptional groups |
Citations:
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