On the formal degree conjecture for simple supercuspidal representations

TL;DR

This paper proves the formal degree conjecture for simple supercuspidal representations of certain p-adic groups by computing Swan conductors using Kloosterman sheaves, under the assumption that p is odd.

Contribution

It establishes the formal degree conjecture for simple supercuspidal representations of symplectic and orthogonal groups, utilizing Swan conductor calculations and Kloosterman sheaves.

Findings

Confirmed the formal degree conjecture for specified groups

Computed Swan conductors using geometric methods

Extended results to cases with p odd

Abstract

We prove the formal degree conjecture for simple supercuspidal representations of symplectic groups and quasi-split even special orthogonal groups over a p-adic field, under the assumption that p is odd. The essential part is to compute the Swan conductor of the exterior square of an irreducible local Galois representation with Swan conductor 1. It is carried out by passing to the equal characteristic local field and using the theory of Kloosterman sheaves.