Geck–Hiss conjectures on decomposition matrices of finite reductive groups

Let $G$ be a finite reductive group with Weyl group $W$, defined over a field of characteristic $p$, and let $k$ be a field of characteristic $ell$ with $ell eq p$. Let $d$ denote the decomposition map from ordinary characters to the Grothendieck group of $kG$-modules, and call a character or module unipotent when it belongs to the unipotent part. Assume that $ell$ is large with respect to $|W|$. Geck–Hiss conjectures. (i) The decomposition matrix has a unitriangular shape. (ii) If $rho$ is unipotent and cuspidal, then $d(rho)$ is irreducible, that is, $d(rho)=[S]$ for some simple $kG$-module $S$. (iii) The unipotent part of the decomposition matrix is independent of $q$ and depends only on the order of $q$ in $\mathbb{F}_ell^\times$. These conjectures organize expected structural properties of modular decomposition matrices. They were known in several cases, including type $A$ and ${}^2A$ and some classical groups; the source also records a proof of part (ii) under the additional assumption that $p$ is good, while the full collection is not presented as settled.