On explicit lifts of cusp forms from $GL_m$ to classical groups

David Ginzburg, Stephen Rallis and David Soudry

Review from Zentralblatt MATH:

Fix a number field \$F\$ and let \$\Bbb A\$ denote the adeles of \$F\$. Let \$\sigma\$ be an automorphic representation of either a split special orthogonal group \$SO_r({\Bbb A})\$ or a symplectic group \$Sp_{2k}({\Bbb A})\$. Then the Langlands functoriality conjecture predicts the existence of an automorphic representation \$\pi(\sigma)\$ on a suitable general linear group \$GL_m({\Bbb A})\$. If instead \$\sigma\$ is on the metaplectic double cover \$\widetilde{Sp}_{2k}({\Bbb A})\$, one expects a lift \$\pi_\psi(\sigma)\$, not canonical but depending on a choice of additive character \$\psi\$ on \$F\backslash {\Bbb A}\$. Conversely, given \$\pi\$ on \$GL_m({\Bbb 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 GL_m({\Bbb 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{Sp}_{2k}({\Bbb A}) \times GL_{2n}({\Bbb 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 \$GL_{2k}\otimes GL_{2n}\$. To establish this, the authors construct a series of candidates for the inverse lifts \$\sigma_{\psi,\ell}(\tau)\$ on \$\widetilde{Sp}_{2\ell}({\Bbb 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 \$SO_{4n}({\Bbb A})\$ induced from the parabolic subgroup with Levi factor \$GL_{2n}\$ and the function \$\tau\otimes |\text{ 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 \$Sp_{2n}({\Bbb A}) \times Sp_{2n}({\Bbb A})\$.

Reviewed by Solomon Friedberg

Classification (MSC2000): 22E50 11F70 11F50 11F55

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