Lusztig constants and endoscopy

TL;DR

This paper establishes a precise relation between invariant functions and their Fourier transforms supported on the nilpotent cone of a semisimple Lie algebra over a large finite field, revealing the Lusztig constant and confirming a conjecture on Fourier transform compatibility.

Contribution

It explicitly determines the Lusztig constant as a fourth root of unity and proves a conjecture relating Fourier transform and Deligne--Lusztig induction.

Findings

Identifies the Fourier transform relation for functions supported on the nilpotent cone.

Determines the Lusztig constant as a specific fourth root of unity.

Confirms Letellier's conjecture on Fourier transform compatibility with Deligne--Lusztig induction.

Abstract

We prove that on a semisimple Lie algebra $\mathfrak{g}$ over a finite field of large characteristic, if a complex-valued invariant function $f$ and its Fourier transform $\hat f$ are both supported in the nilpotent cone of $\mathfrak{g}$, then $\hat f = \gamma^{-1}f$ for an explicit quadratic Gauss sum $\gamma$. Consequently, we determine a fourth root of unity appearing in various formulae of generalised Gel'fand--Graev characters, known as Lusztig constant, previously known in special cases due to works of Kawanaka, Digne--Lehrer--Michel, Waldspurger and Geck. As consequence, we show the validity of a conjecture of Letellier on the compatibility of Fourier transform with Deligne--Lusztig induction.