On the formal degree conjecture for non-singular supercuspidal representations

TL;DR

This paper proves the formal degree conjecture for non-singular supercuspidal representations, extending Schwein's results for regular cases by addressing reducibility and non-abelian S-groups.

Contribution

It establishes the formal degree conjecture for non-singular supercuspidal representations, handling reducible Deligne--Lusztig representations and non-abelian S-groups.

Findings

Proved the formal degree conjecture for non-singular supercuspidal representations.

Extended previous work to cases with reducible Deligne--Lusztig representations.

Addressed non-abelian S-groups in the context of the conjecture.

Abstract

We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessary abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.