Abstract
The Rankin-Selberg method associates, to each local factorL(s, π v × π ′ v ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π v and ′ v . In this paper we show that, if ν is archimedean, and π v and ′ v are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one.
We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.
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Stade, E. ArchimedeanL-factors onGL(n) ×GL(n) and generalized Barnes integrals. Isr. J. Math. 127, 201–219 (2002). https://doi.org/10.1007/BF02784531
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DOI: https://doi.org/10.1007/BF02784531