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.

中文翻译：

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单能块分解矩阵的单位三角形形状

我们证明了有限还原群 $\mathbf{G}(\mathbb{F}_q)$ 的单能 $\ell$ 块的分解矩阵具有单位三角形状，假设 $q$ 是一个好的素数的幂而 $\ell$ 非常适合 $\mathbf{G}$。这是 Geck 在 1990 年作为他博士论文的一部分推测的。我们通过使用 Kawanaka 引入的广义 Gelfand--Graev 字符的修改构造投影模块来建立这个结果。我们证明每个这样的字符至多有一个单能成分，它以多重性一出现。这建立了川中 30 年前的猜想。