Archimedean \(L\)-factors on \(\text{GL}(n)\times\text{GL}(n)\) and generalized Barnes integrals (Q1611540)

To each local factor \(L(s,\pi_\nu\times\pi'_\nu)\) of an automorphic \(L\)-function on \(\text{GL}(n)\times \text{GL}(n)\), the Rankin-Selberg method associates a certain local integral of Whittaker functions for \(\pi_\nu\) and \(\pi'_\nu\). The main Theorem of this paper is that, if \(\nu\) is archimedean and \(\pi_\nu\) and \(\pi'_\nu\) are spherical principal series representations with trivial central character, then the local \(L\)-factor and the local integral are, in fact, equal. This verifies a conjecture of \textit{D. Bump} [The Rankin-Selberg method: a survey, in Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, Academic Press, New York, 49--109 (1989; Zbl 0668.10034)] which was previously only known in the cases \(n=2,3\). The idea of the proof is to express the local integral first as a convolution of Mellin transforms and then, using known formulae for these Mellin transforms, as a (generalized) Barnes integral. Using a new identity for Barnes integrals (which reduces an \(m\)-fold Barnes integral to an \((m-1)\)-fold integral) and some residue formulae for Mellin transforms, the local integral can then be expressed as a convolution of Mellin transforms of \(\text{GL}(n-1)\) Whittaker functions. The Theorem then follows by induction, the case \(n=2\) being equivalent to ``Barnes's (First) Lemma'': \textit{E. W. Barnes} [Proc. Lond. Math. Soc. (2) 6, 141--177 (1908; JFM 39.0506.01)].