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Rankin-Selberg convolutions: Archimedean theory. (English) Zbl 0712.22011
Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop L-Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, 125-207 (1990).
[For the entire collection see Zbl 0698.00020.]
For \(r\in {\mathbb{N}}\) let \(G_ r\) denote the group \(GL_ r(F)\), where \(F={\mathbb{R}}\) or \({\mathbb{C}}\). Let W be a function in a Whittaker model W(\(\pi\)) of some irreducible representation \(\pi\) of \(G_ 2\). Let further \(\chi\) denote a character of \(G_ 1\). In [H. Jacquet, R. P. Langlands; Automorphic forms on GL(2). (Lect. Notes Math. 114, 1970; Zbl 0236.12010)] is attached to this data a zeta function \(\zeta\) (S,W,\(\chi\)) where \(s\in {\mathbb{C}}\) with Re \(s\gg 0\). This zeta function satisfies a functional equation when divided by an Euler factor L(s,\(\pi\)), the L-function of the representation \(\chi\otimes \pi\). In the paper under consideration this method is extended to pairs of representations \(\pi\), \(\pi '\) of \(G_ r\), \(G_ t\) in the roles of \(\pi\) and \(\chi\). Let \(W,W'\) be vectors in Whittaker models of \(\pi\) and \(\pi '\) which are assumed to exist. Then a zeta function \(\zeta (S,W,W')\) is defined in an analogous way to the above and it is shown to satisfy a functional equation when divided by the L-function of \(\pi \otimes \pi '\). This is the archimedean counterpart of an earlier paper that treated the p-adic case. Taken together, applications to automorphic representations now seem possible.
Reviewer: A.Deitmar

22E50 Representations of Lie and linear algebraic groups over local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields