Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

Olivier Brunat (1) , Olivier Dudas (2) , Jay Taylor

1 ICJ - Institut Camille Jordan
2 MI - Mathematical Institute [Oxford]

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. 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.

Domaines: Théorie des représentations [math.RT]

Olivier Brunat, Olivier Dudas, Jay Taylor. Unitriangular Shape of Decomposition Matrices of Unipotent Blocks. 2019. ⟨hal-02334862⟩