AI 中文总结
本文证明Fargues的猜想，将Fargues与Scholze构造的谱作用扩展到Kaletha挠子群的G挠子栈，为非拟分裂群的内形式提供范畴化局部朗兰兹猜想，还对环面证明该猜想并归约中心条件下的扩展版本。
AI 中文摘要
设G为非阿基米德局部域上的连通约化群，Fargues与Scholze在文献[FS21]中关于局部朗兰兹对应几何化的主要结果，是在Fargues-Fontaine曲线上的G挠子栈Bun_G上构造了一个“谱作用”，该栈对应ℓ进层范畴。本文的目标是证明Fargues提出的一个猜想，即可以将该构造扩展到曲线上Kaletha挠子群上的G挠子栈Bun_G^e，这一栈由Fargues在文献[Far22]中引入。这种“扩展谱作用”为任意连通约化群G提供了范畴化局部朗兰兹猜想的一个版本，且是针对那些非拟分裂群的扩展纯内形式的G（如SL_n的非平凡内形式）的首个此类结论。最后，本文遵循Zou在文献[Zou24]中的原始论证，证明了环面的该猜想；当G的中心Z_G连通且H¹(F,Z_G)=0时，本文将范畴化猜想的“扩展”版本归约为Fargues–Scholze中的版本。
英文摘要
Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a ``spectral action'' on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This ``extended spectral action'' allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of [Zou24] and, under the same assumptions as [Zou26] (including connected center), we reduce the ``extended'' version of the categorical conjecture to the one in Fargues--Scholze.
Commentsv2: Fixed typos and other minor expository changes, especially to introduction. 52 pages
