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**On explicit lifts of cusp forms from \(\text{GL}_m\) to classical groups.**
*(English)*
Zbl 0949.22019

Fix a number field \(F\) and let \(\mathbb A\) denote the adeles of \(F\). Let \(\sigma\) be an automorphic representation of either a split special orthogonal group \(\text{SO}_r({\mathbb A})\) or a symplectic group \(\text{Sp}_{2k}({\mathbb A})\). Then the Langlands functoriality conjecture predicts the existence of an automorphic representation \(\pi(\sigma)\) on a suitable general linear group \(\text{GL}_m({\mathbb A})\). If instead \(\sigma\) is on the metaplectic double cover \(\widetilde{\text{Sp}}_{2k}({\mathbb A})\), one expects a lift \(\pi_\psi(\sigma)\), not canonical but depending on a choice of additive character \(\psi\) on \(F\backslash {\mathbb A}\). Conversely, given \(\pi\) on \(\text{GL}_m({\mathbb A})\), self-dual and cuspidal, then from the factorization of the partial Rankin-Selberg convolution of \(\pi\) with itself \(L^S(s,\pi\otimes \pi)=L^S(s,\pi,\wedge^2)L^S(s,\pi,\vee^2)\) and the pole of the left-hand side at \(s=1\), one expects that \(\pi\) is the “backwards lift” of some \(\sigma(\pi)\) on an orthogonal or symplectic group. In the orthogonal case, one may moreover move \(\sigma(\pi)\) to a symplectic or metaplectic group using the theta correspondence, under suitable (non-vanishing) hypotheses.

The authors combine this motivation with the theory of global Rankin-Selberg integrals for \(G\times\text{GL}_m({\mathbb A})\), \(G\) one of the groups mentioned above, to create such a backwards lift, and to prove additional related results. Let now \(\sigma\otimes\tau\) be a cuspidal irreducible automorphic representation of \(\widetilde{\text{Sp}}_{2k}({\mathbb A}) \times\text{GL}_{2n}({\mathbb A})\), \(1\leq k<n\), which is generic. Suppose that the standard \(L\)-function for \(\tau\) satisfies \(L^S(1/2,\tau)\neq 0\) and that the exterior square \(L\)-function \(L^S(s,\tau,\wedge^2)\) has a pole at \(s=1\). Then the authors prove that the partial standard \(L\)-function which should be attached to \(\pi_\psi(\sigma)\otimes\tau\) is holomorphic at \(s=1\). This is consistent with existence of the lift \(\pi_\psi(\sigma)\) and with the Rankin-Selberg theory on \(\text{GL}_{2k}\otimes\text{GL}_{2n}\). To establish this, the authors construct a series of candidates for the inverse lifts \(\sigma_{\psi,\ell}(\tau)\) on \(\widetilde{\text{Sp}}_{2\ell}({\mathbb A})\), \(0\leq\ell\leq 2n-1\). These are obtained by spaces of Fourier coefficients of residues of Eisenstein series; this construction is motivated by a global Rankin-Selberg integral. They show that these lifts have the tower property: for the first index \(\ell\) for which the inverse lift \(\sigma_{\psi,\ell}(\tau)\) is nonzero, it is cuspidal, and then for higher indices the lift is noncuspidal. They also show that this first occurrence must satisfy \(\ell\geq n\). They establish similar results with odd orthogonal, even orthogonal, and symplectic towers. The full story, which is obtained in a subsequent paper by these authors, is that the first occurrence is precisely when \(\ell=n\) and that these inverse lifts give precisely the generic member of the corresponding \(L\)-packet. To prove these results, the authors study certain periods of Eisenstein series, about which they prove additional theorems. For example, let \(\tau\) satisfy the nonvanishing hypotheses above, and let \(E_{\tau,s}(h)\) be the Eisenstein series on \(\text{SO}_{4n}({\mathbb A})\) induced from the parabolic subgroup with Levi factor \(\text{GL}_{2n}\) and the function \(\tau\otimes |\det(\cdot)|^{s-1/2}\). Then the authors show that the residue of \(E_{\tau,s}(h)\) at \(s=1\) has a nontrivial period along the subgroup \(\text{Sp}_{2n}({\mathbb A}) \times\text{Sp}_{2n}({\mathbb A})\).

The authors combine this motivation with the theory of global Rankin-Selberg integrals for \(G\times\text{GL}_m({\mathbb A})\), \(G\) one of the groups mentioned above, to create such a backwards lift, and to prove additional related results. Let now \(\sigma\otimes\tau\) be a cuspidal irreducible automorphic representation of \(\widetilde{\text{Sp}}_{2k}({\mathbb A}) \times\text{GL}_{2n}({\mathbb A})\), \(1\leq k<n\), which is generic. Suppose that the standard \(L\)-function for \(\tau\) satisfies \(L^S(1/2,\tau)\neq 0\) and that the exterior square \(L\)-function \(L^S(s,\tau,\wedge^2)\) has a pole at \(s=1\). Then the authors prove that the partial standard \(L\)-function which should be attached to \(\pi_\psi(\sigma)\otimes\tau\) is holomorphic at \(s=1\). This is consistent with existence of the lift \(\pi_\psi(\sigma)\) and with the Rankin-Selberg theory on \(\text{GL}_{2k}\otimes\text{GL}_{2n}\). To establish this, the authors construct a series of candidates for the inverse lifts \(\sigma_{\psi,\ell}(\tau)\) on \(\widetilde{\text{Sp}}_{2\ell}({\mathbb A})\), \(0\leq\ell\leq 2n-1\). These are obtained by spaces of Fourier coefficients of residues of Eisenstein series; this construction is motivated by a global Rankin-Selberg integral. They show that these lifts have the tower property: for the first index \(\ell\) for which the inverse lift \(\sigma_{\psi,\ell}(\tau)\) is nonzero, it is cuspidal, and then for higher indices the lift is noncuspidal. They also show that this first occurrence must satisfy \(\ell\geq n\). They establish similar results with odd orthogonal, even orthogonal, and symplectic towers. The full story, which is obtained in a subsequent paper by these authors, is that the first occurrence is precisely when \(\ell=n\) and that these inverse lifts give precisely the generic member of the corresponding \(L\)-packet. To prove these results, the authors study certain periods of Eisenstein series, about which they prove additional theorems. For example, let \(\tau\) satisfy the nonvanishing hypotheses above, and let \(E_{\tau,s}(h)\) be the Eisenstein series on \(\text{SO}_{4n}({\mathbb A})\) induced from the parabolic subgroup with Levi factor \(\text{GL}_{2n}\) and the function \(\tau\otimes |\det(\cdot)|^{s-1/2}\). Then the authors show that the residue of \(E_{\tau,s}(h)\) at \(s=1\) has a nontrivial period along the subgroup \(\text{Sp}_{2n}({\mathbb A}) \times\text{Sp}_{2n}({\mathbb A})\).

Reviewer: Solomon Friedberg (Chestnut Hill)

### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F50 | Jacobi forms |

11F55 | Other groups and their modular and automorphic forms (several variables) |