Let $k$ be a non-archimedean local field with residual characteristic $p$. Let $G$ be a connected reductive group over $k$ that splits over a tamely ramified field extension of $k$. Suppose $p$ does not divide the order of the Weyl group of $G$. Then we show that every smooth irreducible complex representation of $G(k)$ contains an $\mathfrak {s}$-type of the form constructed by Kim–Yu and that every irreducible supercuspidal representation arises from Yu’s construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on $p$. By contrast, our bound on $p$ is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu’s construction from a given representation.

假设$ k $是具有残差特征$ p $的非档案本地字段。假设$ G $是$ k $上的一个连通的归约组，它分裂为$ k $的分枝后的字段扩展。假设$ p $不除以$ G $的Weyl组的顺序。然后，我们证明，每个$ G（k）$的光滑不可约复杂表示都包含由Kim–Yu构造的$ \ mathfrak {s} $类型的形式，并且每个不可约超尖峰表示都来自于'的构造。这改善了Kim的早期结果，后者仅保持特征零，并且对$ p $具有非常大且无效的界限。相比之下，我们对$ p $的约束是明确且严格的，我们的结果也具有积极的特征。此外，我们的方法在从给定表示中提取Yu的构造的输入时更为明确。