For a connected reductive algebraic group $G$ defined over a finite field $\mathbb F_q$, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group $G(\mathbb F_q)$ in the case where $q$ is a power of a good prime for $G$. This representation has been widely studied and used in various contexts. Recently, Geck proposed a conjecture, characterizing Lusztig's special unipotent classes in terms of weighted Dynkin diagrams. Based on this conjecture, he gave a guideline for extending the definition of GGGRs to the case where $q$ is a power of a bad prime for $G$. Here, we will give a proof of Geck's conjecture. Combined with Geck's pioneer work, our proof verifies Geck's conjectural characterization of special unipotent classes, and completes his definition of GGGRs in bad characteristics.

中文翻译：

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Geck 猜想和坏特征中的广义 Gelfand-Graev 表示

对于定义在有限域 $\mathbb F_q$ 上的连通还原代数群 $G$，Kawanaka 引入了有限群 $G(\mathbb F_q)$ 的广义 Gelfand-Graev 表示（简称 GGGRs），其中 $ q$ 是 $G$ 的良好质数的幂。这种表示已被广泛研究并在各种情况下使用。最近，Geck 提出了一个猜想，根据加权 Dynkin 图来表征 Lusztig 的特殊单能类。基于这个猜想，他给出了将 GGGR 的定义扩展到 $q$ 是 $G$ 的坏素数的幂的情况的指导方针。在这里，我们将给出 Geck 猜想的证明。结合 Geck 的开创性工作，我们的证明验证了 Geck 对特殊单能类的推测表征，并完成了他对不良特征中 GGGR 的定义。