Complementary results on the Rankin–Selberg gamma factors of classical groups

Let G be an orthogonal or symplectic group, defined over a local field, or the metaplectic group. We study the γ  -factor for a pair of irreducible generic representations of G×GLnG×GLn, defined using the Rankin–Selberg method. In the metaplectic case we use Shimura type integrals. We prove that the γ-factor satisfies a list of fundamental properties, stated by Shahidi, which define it uniquely. In particular, we show full multiplicativity for symplectic and metaplectic groups. It is important for applications to relate this γ-factor to the one arising from the Langlands–Shahidi method. As a corollary of our results, these factors coincide. This is a refinement of previous works on orthogonal groups, showing such an equality up to certain normalization factors.