On the Archimedean Local Gamma Factors for Adjoint Representation of ${\mathrm{G}\mathrm{L}}_3$, Part I

Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been 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 to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg’s global integral for the adjoint representation of ${\mathrm{G}\mathrm{L}}_3$. Via the local functional equation, the local gamma factor $\mathrm{\Gamma }(s,\pi ,\mathrm{A}\mathrm{d},\psi )$ can be defined. In a forthcoming paper, we will compute the local gamma factor $\mathrm{\Gamma }(s,\pi ,\mathrm{A}\mathrm{d},\psi )$ explicitly, which fills in some blanks in the archimedean local theory of Ginzburg’s global integral.