On the Rankin-Selberg $L$-factors for ${\mathrm{SO}}_5\times {\mathrm{GL}}_2$

Abstract

Let $\pi$ and $\tau$ be a smooth generic representation of ${\mathrm{SO}}_5$ and ${\mathrm{GL}}_2$ respectively over a non-archimedean local field. Assume that $\pi$ is irreducible and $\tau$ is irreducible or induced of Langlands' type. We show that the $L$- and $\epsilon$-factors attached to $\pi \times \tau$ defined by the Rankin-Selberg integrals and the associated Weil-Deligne representation coincide. Similar compatibility results are also obtained for the local factors defined by the Novodvorsky's local zeta integrals attached to generic representations of ${\mathrm{GSp}}_4\times {\mathrm{GL}}_2$.

22 pages, assumptions are lifted, arguments are simplified