On the nonvanishing of the central value of the Rankin-Selberg \(L\)-functions. II.

*(English)*Zbl 1115.11027
Cogdell, James W. (ed.) et al., Automorphic representations, \(L\)-functions and applications. Progress and prospects. Proceedings of a conference honoring Steve Rallis on the occasion of his 60th birthday, Columbus, OH, USA, March 27–30, 2003. Berlin: de Gruyter (ISBN 3-11-017939-3/hbk). Ohio State Univ. Math. Res. Inst. Publ. 11, 157-191 (2005).

In this well-written paper, the authors continue their earlier paper [J. Am. Math. Soc. 17, No. 3, 679–722 (2004; Zbl 1057.11029)] on the characterization of nonvanishing of the central value of the Rankin-Selberg \(L\)-functions in terms of periods of Fourier-Jacobi type and of Gross-Prasad 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 a 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. Soundry 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. Codgell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi everywhere [Publ. Math., Inst. Hautes Etud. Sci. 99, 163–233 (2004; Zbl 1090.22010)].

If \(\pi\) is symplectic, then \(n=2r\) and \(\pi\) is the functorial lift from an irreducible generic cuspidal automorphic representation \(\tau\) 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 \(\text{Sp}_{2\ell}(\mathbb A)\)

Let \(\pi_1\) and \(\pi_2\) be irreducible unitary cuspidal automorphic representations of \(\text{GL}_n(\mathbb A)\) and \(\text{GL}_m(\mathbb A)\), respectively. The authors treat the case where \(\pi_1\) is symplectic (\(n=2r\)), and \(\pi_2\) is orthogonal with \(m=2\ell\). Let \(\sigma\) be an irreducible unitary generic cuspidal automorphic representation of \(\text{SO}_{2\ell}(\mathbb A)\) which lifts functorially to \(\pi_2\), and let \(\tau\) be an irreducible unitary generic cuspidal automorphic representation of \(\text{SO}_{2r+1}(\mathbb A)\) which lifts functorially to \(\pi_1\).

One of the main results states as follows. If the period of Gross-Prasad type \[ \mathcal P_{2r+1,2\ell}(\phi_\tau,\phi_\sigma) (r\geq \ell) \quad \text{or} \quad \mathcal P_{2\ell,2r+1}(\phi_\sigma,\phi_\tau) (r\leq \ell) \] is non-zero, then \(L(\frac{1}{2},\pi_1\times \pi_2)\neq 0\).

Conversely, under certain assumption, the authors show that if \(L(\frac{1}{2},\pi_1\times \pi_2)\neq 0\), then the period \[ \mathcal P_{2r+1,2\ell}(\phi_{\tau'},\phi_{\sigma'}) (r\geq \ell) \quad \text{or} \quad \mathcal P_{2\ell,2r+1}(\phi_{\sigma'},\phi_{\tau'}) (r\leq \ell) \] is non-zero for a pair \((\tau',\sigma')\) nearly equivalent to \((\tau,\sigma)\).

For the entire collection see [Zbl 1076.11002].

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 a 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. Soundry 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. Codgell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi everywhere [Publ. Math., Inst. Hautes Etud. Sci. 99, 163–233 (2004; Zbl 1090.22010)].

If \(\pi\) is symplectic, then \(n=2r\) and \(\pi\) is the functorial lift from an irreducible generic cuspidal automorphic representation \(\tau\) 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 \(\text{Sp}_{2\ell}(\mathbb A)\)

Let \(\pi_1\) and \(\pi_2\) be irreducible unitary cuspidal automorphic representations of \(\text{GL}_n(\mathbb A)\) and \(\text{GL}_m(\mathbb A)\), respectively. The authors treat the case where \(\pi_1\) is symplectic (\(n=2r\)), and \(\pi_2\) is orthogonal with \(m=2\ell\). Let \(\sigma\) be an irreducible unitary generic cuspidal automorphic representation of \(\text{SO}_{2\ell}(\mathbb A)\) which lifts functorially to \(\pi_2\), and let \(\tau\) be an irreducible unitary generic cuspidal automorphic representation of \(\text{SO}_{2r+1}(\mathbb A)\) which lifts functorially to \(\pi_1\).

One of the main results states as follows. If the period of Gross-Prasad type \[ \mathcal P_{2r+1,2\ell}(\phi_\tau,\phi_\sigma) (r\geq \ell) \quad \text{or} \quad \mathcal P_{2\ell,2r+1}(\phi_\sigma,\phi_\tau) (r\leq \ell) \] is non-zero, then \(L(\frac{1}{2},\pi_1\times \pi_2)\neq 0\).

Conversely, under certain assumption, the authors show that if \(L(\frac{1}{2},\pi_1\times \pi_2)\neq 0\), then the period \[ \mathcal P_{2r+1,2\ell}(\phi_{\tau'},\phi_{\sigma'}) (r\geq \ell) \quad \text{or} \quad \mathcal P_{2\ell,2r+1}(\phi_{\sigma'},\phi_{\tau'}) (r\leq \ell) \] is non-zero for a pair \((\tau',\sigma')\) nearly equivalent to \((\tau,\sigma)\).

For the entire collection see [Zbl 1076.11002].

Reviewer: Chia-Fu Yu (Taipei)

##### 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 |