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, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75013, Paris, France

Olivier Dudas

Université de Paris, Sorbonne Université CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75013, Paris, France

Jay Taylor

Department of Mathematics, University of Southern California, 3620 South Vermont Avenuke, Los Angeles, CA 90089, United States