The question is from Donaldson's paper "scalar curvature and projective embeddings I (MR1916953)".

Let (M, $\omega$) be a compact symplectic manifold, $(L, h)\to (M,\omega)$ be an Hermitian line bundle with curvature $\sqrt{-1}\omega$. Consider the group $\mathcal{G}$ of Hermitian bundle maps from $L$ to $L$ which preserves the connection. Then this group action induces a Lie algebra $C^\infty(M)$ action on the space of sections $\Gamma(L^k)$, which is given by for $s\in \Gamma(L^k)$ $$ R_f(s)=\nabla_{\xi_f}(s)-\sqrt{-1}kfs, $$ where $\xi_f$ is the Hamiltonian vector field, such that $i_{\xi_f}\omega=df$.

My question is how $R_f$ comes from? I know it's from the linearization of the group action, but I could not figure out the correct one so that after the linearization, one can obtain $R_f$. Any help will be appreciated.