In this paper, we introduce the Newton decomposition on a connected reductive $p$-adic group $G$. Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on $G$ and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori–Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe’s conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description of the structure of the rigid cocenter.

中文翻译：

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-ADIC 群的中心，I：牛顿分解

在本文中，我们介绍了连接约简上的牛顿分解$p$-adic 组$G$. 在此基础上，我们对其 Hecke 代数的重心进行了很好的分解。在这里，我们考虑与通常的共轭作用相关的普通共心$G$以及由扭曲内窥镜理论产生的扭曲同心。我们在同心的牛顿分量上给出了 Iwahori-Matsumoto 型发生器。在此基础上，我们证明了 Howe 猜想关于（普通和扭曲）不变分布约束的推广。最后，我们对刚性重心的结构进行了明确的描述。