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 \(GL_{2l}\).