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Multiplicativity of the gamma factors of Rankin-Selberg integrals for \(\mathrm{SO} _{2l} \times\mathrm{GL}_n\). (English) Zbl 1318.11158
Let \(F\) be a \(p\)-adic field. For positive integers \(l\) and \(n\), let \(H\) be a quasisplit form of the special orthogonal group \(\mathrm{SO}_{2l}\), and let \(G = \mathrm{GL}_n\). Let \(\pi\) and \(\tau\) be irreducible smooth generic representations of \(H(F)\) and \(G(F)\), respectively. The article under review studies the Rankin-Selberg integrals attached to the pair \(\pi,\tau\) by S. Gelbart et al. [Explicit constructions of automorphic \(L\)-functions. Berlin: Springer (1987; Zbl 0612.10022)] and D. Ginzburg [J. Reine Angew. Math. 405, 156–180 (1990; Zbl 0684.22009)]. These integrals can be used to define the \(\gamma\)-factor \(\Gamma (\pi\times\tau,\psi,s)\), where \(\psi\) is a fixed additive character of \(F\).
The author proves that the association \((\pi, \tau) \rightsquigarrow\Gamma (\pi\times\tau,\psi,s)\) behaves multiplicatively in the arguments \(\pi\) and \(\tau\) with respect to parabolic induction. Using archimedean results of D. Soudry [Ann. Sci. Éc. Norm. Supér. (4) 28, No. 2, 161–224 (1995; Zbl 0824.11034)] and global arguments, the author then shows that the function \(\Gamma (\pi\times\tau,\psi,s)\) coincides with the standard \(\gamma\)-factor defined by Shahidi. The results of this paper have been used to show that the local \(L\)-factor attached to \(\pi ,\tau\) (in the tempered case) defined via Rankin-Selberg integrals coincides with the \(L\)-function \(L(\pi\times\tau,s)\) defined by Shahidi.

MSC:
11S40 Zeta functions and \(L\)-functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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