Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal{F}(X; \mathbf{k})$ if and only if it represents a non-zero object of that summand (slightly more general results are also provided). Our result is based on: an explicit understanding of the wrapped Fukaya category $\mathcal{W}(T^*G; \mathbf{k})$ through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor $D^b \mathcal{W}(T^*G; \mathbf{k}) \to D^b\mathcal{F}(X^{-} \times X; \mathbf{k})$ canonically associated to the Hamiltonian $G$-action on $X$. We explore several examples which can be studied in a uniform manner including toric Fano varieties and certain Grassmannians.

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生成哈密顿量 $G$-流形的 Fukaya 范畴

令 $G$ 是一个紧李群，$\mathbf{k}$ 是一个特征域 $p \geq 0$ 使得 $H^* (G)$ 没有 $p$-torsion。我们证明了哈密顿量 $G$ 作用在紧凑、单调、辛流形 $X$ 上的自由拉格朗日轨道分裂生成单调 Fukaya 范畴 $\mathcal{F}(X; \mathbf{k })$ 当且仅当它表示该被加数的非零对象（还提供了更一般的结果）。我们的结果基于：通过涉及零截面和余切纤维的 Koszul 扭曲复合物对包裹的 Fukaya 类别 $\mathcal{W}(T^*G; \mathbf{k})$ 的明确理解；和一个函子 $D^b \mathcal{W}(T^*G; \mathbf{k}) \to D^b\mathcal{F}(X^{-} \times X; \mathbf{k}) $ 与 $X$ 上的哈密顿量 $G$-action 规范相关。