Rankin-Selberg integrals in two complex variables
The authors formulate some Rankin-Selberg type integrals in two complex variables, which turn out to represent a product of two automorphic \(L\)-functions. In the cases considered in the paper, one of the \(L\)-functions is the Spin \(L\)-function of irreducible cuspidal automorphic representations of \(G=\text{GSp}_{2n}\) for \(n=2,3,4\). The first example of this type was constructed by \textit{D. Bump} and \textit{S. Friedberg} [Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Pt. II: Papers in analysis, number theory and automorphic L-functions, Pap. Workshop L-Funct., Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 3, 47--65 (1990; Zbl 0712.11030)], which represents a product of the second exterior power \(L\)-function and the standard \(L\)-function of irreducible cuspidal automorphic representations of \(\text{GL}_n\). This new type of construction not only extends the usual Rankin-Selberg integrals from one complex variable to two complex variables, but also yields well-determination of poles of \(L\)-functions in the product, based on the various types of Siegel-Weil identities [\textit{S. S. Kudla} and \textit{S. J. Rallis}, Ann. Math. (2) 140, No. 1, 1--80 (1994; Zbl 0818.11024); see also the reviewer, J. Number Theory 70, No. 1, 67--98 (1998; Zbl 0913.11021)]. The information on the poles of \(L\)-functions coming from such integrals may classify cuspidal automorphic representations of \(G\) via Langlands functorial liftings.
scientific article; zbMATH DE number 1336994
scientific article