Recent stellar population analysis of early-type galaxy spectra has demonstrated that the low-mass galaxies in cluster centers have high [$alpha/rm Fe$] and old ages characteristic of massive galaxies and unlike the low-mass galaxy population in the outskirts of clusters and fields. This phenomenon has been termed coordinated assembly to highlight the fact that the building blocks of massive cluster central galaxies are drawn from a special subset of the overall low-mass galaxy population. Here we explore this idea in the IllustrisTNG simulations, particularly the TNG300 run, in order to understand how environment, especially cluster centers, shape the star formation histories of quiescent satellite galaxies in groups and clusters ($M_{200c,z=0}geq10^{13} M_{odot}$). Tracing histories of quenched satellite galaxies with $M_{star,z=0}geq10^{10} M_{odot}$, we find that those in more massive dark matter halos, and located closer to the primary galaxies, are quenched earlier, have shorter star formation timescales, and older stellar ages. The star formation timescale-$M_{star}$ and stellar age-$M_{star}$ scaling relations are in good agreement with observations, and are predicted to vary with halo mass and cluster-centric distance. The dependence on environment arises due to the infall histories of satellite galaxies: galaxies that are located closer to cluster centers in more massive dark matter halos at $z=0$ were accreted earlier on average. The delay between infall and quenching time is shorter for galaxies in more massive halos, and depends on the halo mass at its first accretion, showing that group pre-processing is a crucial aspect in satellite quenching.
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