On the Langlands parameter of a simple supercuspidal representation: odd orthogonal groups

In this work, we explicitly compute a certain family of twisted gamma factors of a simple supercuspidal representation $\pi$ of a $p$-adic odd orthogonal group. These computations, together with analogous computations for general linear groups carried out in previous work with Liu, allow us to give a prediction for the Langlands parameter of $\pi$. If we assume the “depth-preserving conjecture”, we prove that our prediction is correct if $p$ is sufficiently large.