# Unitriangular shape of decomposition matrices of unipotent blocks

### 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.

## Authors

Olivier Brunat

Université de Paris, Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France

Olivier Dudas

Université de Paris, Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France

Jay Taylor

University of Southern California, Los Angeles, CA, United States