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On the archimedean theory of Rankin-Selberg convolutions for \(SO_{2\ell+1}\times GL_ n\). (English) Zbl 0824.11034
Let \(F\) denote the field of real or complex numbers. Denote by \(\pi\) and \(\tau\) finitely generated admissible representations of \(SO_{2l+1} (F)\) and \(GL_ n (F)\) respectively, each assumed to be generic, i.e. with a unique Whittaker model. In this paper the author studies a certain bilinear form \((w, \xi_{\tau,s}) \mapsto A(w, \xi_{\tau, s})\), where \(w\) is in the Whittaker model of \(\pi\) and \(\xi_{\tau, s}\) is in the space of a parabolically induced representation of \(SO_{2n} (F)\) given by \(\tau\) and a complex parameter \(s\). The form \(A\) is defined by an integral convergent for \(s\) in some right half plane. The main theorem of the paper asserts that \(A(w, \xi_{\tau, s})\) admits a meromorphic continuation to the entire plane.
The form \(A\) and its analogue in the nonarchimedean case show up as local factors of global Rankin-Selberg convolutions if \(\pi\) and \(\tau\) come from automorphic cuspidal representations. In a previous paper [Rankin- Selberg convolutions for \(SO_{2l +1}\times GL_ n\): Local theory, Mem. Am. Math. Soc. 105 (1993; Zbl 0805.22007)] the author showed the analogous result in the nonarchimedean case. These papers are part of a larger project initiated by I. Piatetski-Shapiro and prove the existence of liftings of automorphic forms from \(SO_{2l +1}\) to \(GL_{2l}\).

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