Generating the Fukaya categories of Hamiltonian G-manifolds

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.