Simple cuspidal representations of symplectic groups: Langlands parameter

TL;DR

This paper computes the Jordan set and Langlands parameter of simple cuspidal representations of symplectic groups over non-archimedean fields, using explicit Hecke algebra calculations.

Contribution

It provides explicit computations of the Jordan set and Langlands parameters for simple cuspidal representations of symplectic groups, advancing understanding of their structure.

Findings

Explicit Jordan set computations for symplectic groups.

Determination of Langlands parameters for these representations.

Use of Hecke algebra generators reflecting parabolic induction.

Abstract

Let $F$ be a non-archimedean local field of odd residual characteristic. We compute the Jordan set of a simple cuspidal representation of a symplectic group over $F$, using explicit computations of generators of the Hecke algebras of covers reflecting the parabolic induction under study. When $F$ is a $p$-adic field we obtain the Langlands parameter of the representation.