Unitriangular shape of decomposition matrices of unipotent blocks

We show that the decomposition matrix of unipotent ℓ-blocks of a finite reductive group G(Fq) has a unitriangular shape, assuming q is a power of a good prime and ℓ is very good for G. This was conjectured by Geck [23] in 1990. 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, see [42, 2.4.5].