Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand–Graev representations of the finite group \( G\left({\mathbbm{F}}_q\right) \), assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka’s construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig’s concept of special unipotent classes of G in terms of weighted Dynkin diagrams.

令G为在具有q个元素的有限域上定义的连通的还原代数群。在1980年代，Kawanaka引入了有限群\（G \ left（{\ mathbbm {F}} _ q \ right）\）的广义Gelfand–Graev表示，假设q是G的质数幂。这些表示在各种情况下都非常有用。在这里，我们研究了在将q的假设舍弃时，Kawanaka的构造可以进行到何种程度。作为好奇的副产品，我们用加权的Dynkin图获得了Lusztig对G的特殊单能类的概念的新的猜想刻画。