On certain supercuspidal representations of symplectic groups associated with tamely ramified extensions : the formal degree conjecture and the root number conjecture

TL;DR

This paper verifies the formal degree and root number conjectures for certain supercuspidal representations of symplectic groups over local fields, constructed via tamely ramified extensions and explicit induction methods.

Contribution

It explicitly constructs supercuspidal representations and their L-parameters for symplectic groups associated with tamely ramified extensions, confirming key conjectures.

Findings

Verification of the formal degree conjecture.

Verification of the root number conjecture.

Explicit construction of supercuspidal representations and L-parameters.

Abstract

The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $\theta$ of the multiplicative group $K^{\times}$. $L$-parameter is constructed by the data $\{K,\theta\}$ by means of the local Langlands correspondence of tori and Langlands-Schelstad procedure.