Vol. 304, No. 1, 2020

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

Fangyang Tian

Vol. 304 (2020), No. 1, 303–345

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 GL3. Via the local functional equation, the local gamma factor Γ(s,π,Ad,ψ) can be defined. In a forthcoming paper, we will compute the local gamma factor Γ(s,π,Ad,ψ) explicitly, which fill in some blanks in the archimedean local theory of Ginzburg’s global integral.

archimedean local integral, Rankin–Selberg integral, adjoint L-function for general linear group
Mathematical Subject Classification 2010
Primary: 22E45, 22E50
Secondary: 11F70
Received: 8 January 2018
Revised: 29 November 2018
Accepted: 13 March 2019
Published: 18 January 2020
Fangyang Tian
Department of Mathematics
National University of Singapore