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Langlands-Shahidi Method and Poles of Automorphic L-Functions: Application to Exterior Square L-Functions

Published online by Cambridge University Press:  20 November 2018

Henry H. Kim
Henry H. Kim*
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA email: [email protected]
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Abstract

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In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several completed automorphic $L$-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$-functions of $\text{G}{{\text{L}}_{\text{n}}},\,n$ odd, the Rankin-Selberg $L$-functions of $\text{G}{{\text{L}}_{n}}\times \,\text{G}{{\text{L}}_{m}},\,n\,\ne \,m$, and $L$-functions $L\left( s,\,\sigma ,\,r \right)$, where $\sigma $ is a generic cuspidal representation of $\text{S}{{\text{O}}_{10}}$ and $r$ is the half-spin representation of GSpin $\left( 10,\,\mathbb{C} \right)$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

Keywords

11F22E
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 4 , 01 August 1999 , pp. 835 - 849
Copyright
Copyright © Canadian Mathematical Society 1999

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