On the archimedean local gamma factors for adjoint representation of GL$_3$, part I

Fangyang Tian

Vol. 304 (2020), No. 1, 303–345

DOI: 10.2140/pjm.2020.304.303

Abstract

Studying the analytic properties of the partial Langlands $L$-function via Rankin–Selberg method has proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap in the completion of the analytic theory of the complete $L$-function. We establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg’s global integral (1991) for the adjoint representation of ${GL}_3$. Via the local functional equation, the local gamma factor $\Gamma \left(s,\pi ,Ad,\psi \right)$ can be defined. In a forthcoming paper, we will compute the local gamma factor $\Gamma \left(s,\pi ,Ad,\psi \right)$ explicitly, which fill in some blanks in the archimedean local theory of Ginzburg’s global integral.

Keywords

archimedean local integral, Rankin–Selberg integral, adjoint L-function for general linear group

Mathematical Subject Classification 2010

Primary: 22E45, 22E50

Secondary: 11F70

Milestones

Received: 8 January 2018

Revised: 29 November 2018

Accepted: 13 March 2019

Published: 18 January 2020