On the Rankin-Selberg $L$-factors for ${\rm SO}_{5}\times{\rm GL}_2$

TL;DR

This paper proves the compatibility of Rankin-Selberg $L$- and $\, ext{-}$-factors with Weil-Deligne representations for generic representations of ${ m SO}_5$ and ${ m GSp}_4$ when paired with ${ m GL}_2$, over non-archimedean fields.

Contribution

It establishes the equality of $L$- and $\, ext{-}$-factors from Rankin-Selberg integrals and Weil-Deligne representations for specific classical groups, extending known compatibility results.

Findings

Compatibility of $L$- and $\, ext{-}$-factors for ${ m SO}_5 imes { m GL}_2$

Compatibility of local factors for ${ m GSp}_4 imes { m GL}_2$

Extension of factor compatibility to induced representations

Abstract

Let $\pi$ and $\tau$ be a smooth generic representation of ${\rm SO}_5$ and ${\rm GL}_2$ 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 ${\rm GSp}_4\times{\rm GL}_2$.