On the standard \(L\)-function for \(G_ 2\)
The problem of determining the analytic continuation and functional equation of general \(L\)-functions associated to automorphic representations of reductive Lie groups and representations of the \(L\)- groups has been pursued by two different methods. The first is the interpretation of the \(L\)-function as arising in the constant term of Eisenstein series. The second is the generalized Rankin-Selberg integral representation of the \(L\)-function. In dealing with the case \(G_ 2\), \textit{I. Piatetski-Shapiro}, \textit{S. Rallis} and \textit{G. Schiffman} [ Am. J. Math. 114, 1269-1315 (1993; reviewed below)] have shown that the latter method gives new examples of \(L\)-functions not possible by the first method. In this paper another Rankin-Selberg integral which represents the standard \(L\)-function of \(G_ 2\) is constructed. It involves an Eisenstein series on the double cover of \(SL_ 2\). Since the poles of this Eisenstein series are known it can be shown that the partial standard \(L\)-function for \(G_ 2\) can have at most one simple pole.
scientific article; zbMATH DE number 179194
scientific article