Let \(k\) be a number field, let \(\mathbb{A}\) be its ring of adeles and let \(W_k\) be its Weil group. Let \(G\) be a quasi-split classical group of rank \(n\) defined over \(k\), i.e., \(G\) is either (a) \({\mathrm{SO}}_{2n+1}\), (b) \({\mathrm{SO}}_{2n}\), (c) \({\mathrm{SO}}^*_{2n}\), (d) \({\mathrm{Sp}}_{2n}\) or (e) \(U_n\). The \(L\)-group \(^LG\) is \({\mathrm{Sp}}_{2n}(\mathbb{C})\times W_k\), \({\mathrm{SO}}_{2n}(\mathbb{C})\times W_k\), \({\mathrm{SO}}_{2n}(\mathbb{C}) \rtimes W_k\), \({\mathrm{SO}}_{2n+1}(\mathbb{C})\times W_k\) or \({\mathrm{GL}}_n(\mathbb{C}) \rtimes W_k\), respectively. In (e), the \(L\)-group has a quotient \({\mathrm{GL}}_n(\mathbb{C})\rtimes\mathrm{Gal}(E/k)\), where \(E\) is a quadratic extension. Hence we have a natural representation \({\mathbb{C}}^N\) of \(^LG\) in (a) to (d) and the representation \({\mathbb{C}}^N \times {\mathbb{C}}^N\) of \(^LG\) in (e). Let \(H= {\mathrm{GL}}_N\) in (a) to (d), and let \(H= {\mathrm{Res}}_{E/k} {\mathrm{GL}}_N\) in (e). The natural representation of \(^LG\) in turn defines an embedding \(\iota : {}^LG \to {}^LH\). Let \(\pi = \otimes' \pi_v\) be an irreducible automorphic representation of \(G(\mathbb{A})\). Suppose \(v\) is a finite place such that \(\pi_v\) is unramified (and \(E_v/k_v\) is unramified in (e)), then \(\pi_v\) is parametrized by an admissible homomorphism \(\phi_v: W_v\to {}^LG\) of the local Weil group \(W_v\) via the Satake isomorphism. Then \(\iota_v\circ \phi_v: W_u\to {}^LH\) corresponds to an irreducible unramified admissible representation \(\Pi_v\) of \(H(k_v)\). A similar consideration applies when \(v\) is an Archimedean place. The representation \(\Pi_v\) is called the local functorial lift of \(\pi_v\) and there is an equality of local \(L\)-functions \[ L(s,\Pi_v)= L(s,\pi_v, \iota_v). \] An automorphic representation \(\Pi= \otimes'\Pi_v\) of \(H(\mathbb{A})\) is called a functorial lift of \(\pi\) if for every Archimedean place and almost all non-Archimedean places \(v\), \(\Pi_v\) is a local functorial lift of \(\pi_v\).
The main theorem of this paper is as follows: Suppose \(\pi\) is an irreducible globally generic cuspidal automorphic representation of \(G(\mathbb{A})\). Then it has a functorial lift \(\Pi\) to \(H(\mathbb{A})\) associated to the embedding t above.
This extends the main results of [the authors and H. K. Kim, Publ. Math., Inst. Hautes Étud. Sci. 93, 5–30 (2001; Zbl 1028.11029) and ibid. 99, 163–233 (2004; Zbl 1090.22010)], where \(G\) is a split odd orthogonal group and a split classical group respectively.
The construction of \(\Pi\) uses the converse theorem for \(\text{GL}_N\).
For the entire collection see [Zbl 1214.11007].