Tower equivalence and Lusztig's truncated Fourier transform

TL;DR

This paper proves key results about Lusztig's truncated Fourier transform for spetsial reflection groups, linking it to Deligne-Lusztig combinatorics and reflection representations.

Contribution

It provides a new proof of existing results, connecting the truncated Fourier transform with exterior powers of reflection representations for spetsial groups.

Findings

The image of the characteristic function of a Coxeter element under the transform is an alternating sum of exterior powers.

A class function is tower equivalent to its image under the transform.

The proof is based on Deligne-Lusztig combinatorics.

Abstract

We give a proof of the results of Chapuy and Douvropoulos [3] for irreducible spetsial reflection groups based on Deligne-Lusztig combinatorics. In particular, if f denotes the truncated Lusztig Fourier transform, we show that the image by f of the normalized characteristic function of a Coxeter element is the alternate sum of the exterior powers of the reflection representation, and that a class function is tower equivalent to its image by f .