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On the nonvanishing of the central value of the Rankin-Selberg $$L$$-functions. (English) Zbl 1057.11029
The authors characterize the nonvanishing of the central value of the Rankin-Selberg $$L$$-functions in terms of periods of Fourier-Jacobi type. Let $$\mathbb A$$ denote the adele ring of a number field $$k$$. Let $$\pi$$ be an irreducible unitary cuspidal automorphic representation of $$\text{GL}_n(\mathbb A)$$. It is known that the Rankin-Selberg $$L$$-function $$L(s, \pi\times \pi)$$ has a simple pole at $$s=1$$ if and only if $$\pi^\vee\simeq \pi$$. If $$\pi$$ is self-dual, then it follows from $L(s,\pi\times \pi)=L(s,\pi, \Lambda^2)L(s, \pi, \text{Sym}^2)$ that either $$L(s,\pi, \Lambda^2)$$, the exterior square $$L$$-function, has a simple pole at $$s=1$$, or $$L(s, \pi, \text{Sym}^2)$$, the symmetric square $$L$$-function, has simple pole at $$s=1$$. In the first case, $$n$$ is even and $$\pi$$ is called symplectic; $$\pi$$ is called orthogonal in the latter case. The terminology is suggested from the Langlands principle of functoriality. The following theorem is known, due to D. Ginzburg, S. Rallis and D. Soudry for almost everywhere [Ann. Math. (2) 150, No. 3, 807–866 (1999; Zbl 0949.22019), Int. Math. Res. Not. 2001, No. 14, 729–764 (2001; Zbl 1060.11031)], and J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi everywhere [Publ. Math., Inst. Hautes Étud. Sci. 99, 163–233 (2004; Zbl 1090.22010)]. If $$\pi$$ is symplectic, the $$n=2r$$ and $$\pi$$ is the functorial lift from an irreducible generic cuspidal automorphic representation $$\sigma$$ of $$\text{SO}_{2r+1}(\mathbb A)$$. If $$\pi$$ is orthogonal, then if $$n=2\ell$$ is even, $$\pi$$ is the functorial lift from an irreducible generic cuspidal automorphic representation $$\sigma$$ of $$\text{SO}_{2\ell}(\mathbb A)$$; and if $$n=2\ell+1$$ is odd, $$\pi$$ is the functorial lift from an irreducible generic cuspidal automorphic representation $$\sigma$$ of $$\text{Sp}_{2\ell}(\mathbb A)$$.
One of the main results in the paper is the following
Theorem. Let $$\pi_1$$ be an irreducible unitary cuspidal automorphic orthogonal representation of $$\text{GL}_{2\ell+1} (\mathbb A)$$, and let $$\pi_2$$ be an irreducible unitary cuspidal automorphic symplectic representation of $$\text{GL}_{2r} (\mathbb A)$$. Assume that the standard $$L$$-function $$L(\frac{1}{2}, \pi_2)\not =0$$. Let $$\sigma$$ be an irreducible unitary generic cuspidal automorphic representation of $$\text{Sp}_{2\ell}(\mathbb A)$$ which lifts functorially to $$\pi_1$$, and let $$\widetilde \tau$$ be an irreducible unitary generic cuspidal automorphic representation $$\sigma$$ of $$\widetilde {\text{Sp}}_{2r }(\mathbb A)$$ which has the $$\psi$$-transfer $$\pi_2$$. If the period integrals $\mathcal P_{r,r-\ell} (\phi_\ell, \widetilde \phi_r,\varphi_\ell)\;(r\geq \ell), \quad \widetilde {\mathcal P}_{\ell,\ell-r} (\widetilde \phi_r, \phi_\ell, \varphi_r) \;(r\leq \ell)$ attached to $$(\sigma,\widetilde \tau, \psi)$$ is nonzero for some choice of data, then $$L(\frac{1}{2}, \pi_1\times \pi_2)\not= 0$$. $$\tau$$ is an irreducible unitary generic cuspidal automorphic representation $$\sigma$$ of $$\text{SO}_{2r+1}(\mathbb A)$$ which lifts functorially to $$\pi_2$$. By the global theta correspondence, we know that if the standard $$L$$-function $$L(\frac{1}{2},\tau)\not=0$$, then $$\tau$$ is a global theta lift (with respect to a given character $$\psi$$) from an irreducible unitary generic cuspidal automorphic representation $$\widetilde \tau$$ of $$\widetilde {\text{Sp}}_{2r}(\mathbb A)$$, where $$\widetilde {\text{Sp}}_{2r}$$ is the metaplectic double cover of $$\text{Sp}_{2r}$$. In this case, $$\pi_2$$ is called a $$\psi$$-transfer of $$\widetilde \tau$$ from $$\widetilde {\text{Sp}}_{2r}$$ to $$\text{GL}_{2r}$$. The authors also show that under certain assumption $$L(\frac{1}{2}, \pi_1\times \pi_2)\not= 0$$ implies the non-vanishing of certain period integrals of Fourier-Jacobi type.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E46 Semisimple Lie groups and their representations 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
##### Keywords:
special value; L-function; period; automorphic form
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