The Rest-frame $H$-band Luminosity Function of Red Sequence Galaxies in
  Clusters at $1.0 < z < 1.3$
  We present results on the rest-frame $H$-band luminosity functions (LF) of
red sequence galaxies in seven clusters at 1.0 < z < 1.3 from the Gemini
Observations of Galaxies in Rich Early Environments Survey (GOGREEN). Using
deep GMOS-z' and IRAC $3.6 \mu$m imaging, we identify red sequence galaxies and
measure their LFs down to $M_{H} \sim M_{H}^{*} + (2.0 - 3.0)$. By stacking the
entire sample, we derive a shallow faint end slope of $ \alpha \sim
-0.35^{+0.15}_{-0.15} $ and $ M_{H}^{*} \sim -23.52^{+0.15}_{-0.17} $,
suggesting that there is a deficit of faint red sequence galaxies in clusters
at high redshift. By comparing the stacked red sequence LF of our sample with a
sample of clusters at z~0.6, we find an evolution in the faint end of the red
sequence over the ~2.6 Gyr between the two samples, with the mean faint end red
sequence luminosity growing by more than a factor of two. The faint-to-luminous
ratio of our sample ($0.78^{+0.19}_{-0.15}$) is consistent with the trend of
decreasing ratio with increasing redshift as proposed in previous studies. A
comparison with the field shows that the faint-to-luminous ratios in clusters
are consistent with the field at z~1.15 and exhibit a stronger redshift
dependence. Our results support the picture that the build up of the faint red
sequence galaxies occurs gradually over time and suggest that faint cluster
galaxies, similar to bright cluster galaxies, experience the quenching effect
induced by environment already at z~1.15.
