In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.

中文翻译：

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通过 $p$-adic 积分实现几何稳定

在本文中，我们给出了 Ng\^o 的几何稳定定理的新证明，其中包含基本引理。这是一个陈述，它将准分裂还原群方案 $G$ 的希钦纤维的上同调与内窥镜组 $H_{\kappa}$ 的希钦纤维的上同调联系起来。我们的证明避免了分解和支持定理，而是基于在 Deligne-Mumford 堆栈的粗模空间上进行 $p$-adic 积分的结果。在此过程中，我们根据内窥镜数据建立了对 $G$-Higgs 丛的（各向异性）模堆的惯性堆的描述，并将朗兰兹对偶群方案的通用希钦纤维的对偶性扩展到准分裂情况。