  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.
