We present new high-resolution near-IR spectroscopy and OH maser observations to investigate the population of cool luminous stars of the young massive Galactic cluster RSGC1. Using the 2.293 μm CO band-head feature, we make high-precision radial velocity measurements of 16 of the 17 candidate red supergiants (RSGs) identified by Figer et al. We show that F16 and F17 are foreground stars, while we confirm that the rest are indeed physically associated RSGs. We determine that star F15, also associated with the cluster, is a yellow hypergiant based on its luminosity and spectroscopic similarity to ρ Cas. Using the cluster's radial velocity, we have derived the kinematic distance to the cluster and revisited the stars' temperatures and luminosities. We find a larger spread of luminosities than in the discovery paper, consistent with a cluster age 30% older than previously thought (12+/-2 Myr), and a total initial mass of (3+/-1)×104 Msolar. The spatial coincidence of the OH maser with F13, combined with similar radial velocities, is compelling evidence that the two are related. Combining our results with recent SiO and H2O maser observations, we find that those stars with maser emission are the most luminous in the cluster. From this we suggest that the maser active phase is associated with the end of the RSG stage, when the luminosity-mass ratios are at their highest.