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On CAP representations for even orthogonal groups. I: A correspondence of unramified representations. (English) Zbl 1381.11047

An automorphic cuspidal representation is said to be CAP (cuspidal associated to parabolic) if it is nearly equivalent to a parabolically induced representation. In this paper and its sequel, the authors use a generalized automorphic descent method to construct CAP representations of split even orthogonal groups.
Let \(\pi\) be an irreducible cuspidal representation of \(\mathrm{SO}_{4n+2k}(\mathbb{A})\). Assume that there exists an irreducible cuspidal representation \(\tau\) of \(\mathrm{GL}_{2n}(\mathbb{A})\) such that the partial \(L\)-function \(L^S(\pi \times \tau,s)\) has a pole at \(s=\frac{3}{2}\) and is holomorphic for \(\mathrm{Re}(s) > \frac{3}{2}\). Langlands functoriality predicts that \(\pi\) lifts to an irreducible automorphic representation \(\Pi\) of \(\mathrm{GL}_{4n+2k}(\mathbb{A})\). The conjecture is that \(\Pi\) is nearly equivalent to a representation parabolically induced from \(\Delta(\tau,2) \otimes \Pi'\), where \(\Delta(\tau,2)\) is the Speh block of length two, and \(\Pi'\) is an irreducible automorphic representation of \(\mathrm{GL}_{2k}(\mathbb{A})\) lifted from an irreducible cuspidal representation \(\sigma\) of \(\mathrm{SO}_{2k}(\mathbb{A})\).
In the sequel to this paper, the authors use the endoscopy descent to construct \(\sigma\), starting with \(\pi\) and \(\tau\) as above (with an additional condition on \(\pi\)). In this paper, the authors prove that certain conditions on \(\tau\), \(\pi\) and \(\sigma\) imply that \(\pi\) is a CAP representation, up to an outer conjugation, with respect to the parabolic induction from \(\tau |\det|^{1/2} \otimes \sigma\). The result follows from the corresponding local theorem, which gives the local unramified correspondence. The proof is long and technical, based on the Mackey theory and twisted Jacquet modules.

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
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