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

Abstract

Let $\pi$ be a simple supercuspidal representation of the split even special orthogonal group. We compute the Rankin-Selberg $\gamma$-factors for rank 1-twists of $\pi$ by quadratic tamely ramified characters of $F^{\ast }$. We then use our results to determine the Langlands parameter of $\pi$ up to its restriction to the wild inertia subgroup, subject to an analogue of a work of Blondel, Henniart, and Stevens for $S{O}_{2l}$. In the particular case of the field ${\mathbb{Q}}_2$, we are able to describe the parameter completely.

19 pages