Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

TL;DR

This paper proves that the decomposition matrices of unipotent blocks in finite reductive groups are unitriangular, confirming a 30-year-old conjecture by constructing specific projective modules and analyzing Gelfand--Graev characters.

Contribution

It establishes the unitriangular shape of these matrices assuming good prime conditions, confirming Kawanaka's conjecture from 1990.

Findings

Decomposition matrix of unipotent blocks is unitriangular.

Each Gelfand--Graev character has at most one unipotent constituent.

Confirms Kawanaka's 30-year-old conjecture.

Abstract

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.