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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

##### MSC:
 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