## 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 $$\text{Sp}_{2n}({\mathbb A}) \times\text{Sp}_{2n}({\mathbb A})$$.

### 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)
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