Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

We show that the decomposition matrix of unipotent $\mathrm{\ell }$-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 $\mathrm{\ell }$ 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.