The Langlands parameter of a simple supercuspidal representation: symplectic groups

Abstract

Let $$\pi $$ be a simple supercuspidal representation of the symplectic group $${\mathrm {Sp}}_{2l}(F)$$, over a p-adic field F. In this work, we explicitly compute the Rankin–Selberg $$\gamma $$-factor of rank-1 twists of $$\pi $$. We then completely determine the Langlands parameter of $$\pi $$, if $$p \ne 2$$. In the case that $$F = \mathbb {Q}_2$$, we give a conjectural description of the functorial lift of $$\pi $$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.