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

## TL;DR

This paper computes the Langlands parameter of simple supercuspidal representations of split even special orthogonal groups by analyzing Rankin-Selberg gamma factors and provides explicit descriptions in the case of _2.

## Contribution

It determines the Langlands parameter of simple supercuspidal representations of even orthogonal groups using gamma factors, extending previous work to this class of representations.

## Key findings

- Computed Rankin-Selberg gamma factors for twisted representations.
- Determined the Langlands parameter up to wild inertia.
- Fully described the parameter over _2.

## 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 $F^*$. 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 $SO_{2l}$. In the particular case of the field $\mathbb{Q}_2$, we are able to describe the parameter completely.