Abstract: Let $ G$ be a compact Lie group, and let $ k$ be a field of characteristic $ p \geq 0$ such that $ H^*(G)$ has no $ p$-torsion if $ p>0$. 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; k)$ if and only if it represents a nonzero 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; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $ D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; 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.