Langlands-Shahidi method and poles of automorphic \(L\)-functions. II. (English) Zbl 1041.11035
Isr. J. Math. 117, 261-284 (2000); correction 118, 379 (2000).
Summary: We use the Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations to study the Rankin-Selberg \(L\)-functions of \(\text{GL}_k\times\) classical groups. Especially we prove that \(L(s, \sigma\times \tau)\) is holomorphic, except possibly at \(s=0, \frac12, 1\), where \(\sigma\) is a cuspidal representation of \(\text{GL}_k\) which satisfies the weak Ramanujan property in the sense of J. W. Cogdell and I. I. Piatetski-Shapiro [Geom. Funct. Anal. 5, 164–173 (1995; Zbl 0842.11021)] and \(\tau\) is any generic cuspidal representation of \(\text{SO}_{2l+1}\). Also we study the twisted symmetric cube \(L\)-functions, twisted by cuspidal representations of \(\text{GL}_2\).
For Part I, see Can. J. Math. 51, 835–849 (1999; Zbl 0934.11024) and also the author and F. Shahidi, Ann. Math. (2) 150, 645–662 (1999; Zbl 0957.11026). The correction gives eight minor changes.
For Part I, see Can. J. Math. 51, 835–849 (1999; Zbl 0934.11024) and also the author and F. Shahidi, Ann. Math. (2) 150, 645–662 (1999; Zbl 0957.11026). The correction gives eight minor changes.
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
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