On the Rankin-Selberg gamma factor of simple supercuspidal representations of the unitary group for $p$-adic local fields

TL;DR

This paper computes the Rankin-Selberg gamma factor for simple supercuspidal representations of unramified unitary groups over p-adic fields, extending known results to dyadic cases and connecting to Langlands parameters.

Contribution

It provides the first computation of the gamma factor in dyadic cases and links the gamma factor to the Langlands parameter for simple supercuspidal representations.

Findings

Gamma factor computed for non-dyadic cases.

Original gamma factor result for dyadic cases.

Connection established between gamma factor and Langlands parameter.

Abstract

Let $\pi$ be a simple supercuspidal representation of the quasi-split unramified even unitary group with respect to an unramified quadratic extension $E/F$ of $p$-adic fields. We compute the Rankin-Selberg gamma factor for rank-$1$ twists of $\pi$ by a tamely ramified character of $E^\times$. For non-dyadic cases, the gamma factor can also be recovered by considering the endoscopic lift of the representation. For the dyadic case, the result is original and we expect to extend the results on the Langlands parameter of the simple supercuspidal representations to the dyadic case with our computation.