  Cryogenic mirrors have been introduced to the KAGRA gravitational wave
telescope in Japan, and are also planned to be used in next-generation
gravitational wave telescopes to further improve their sensitivity. Molecular
gases inside vacuum chambers adhere to cold mirror surfaces because they lose
their kinetic energy when they hit cryogenic surfaces. Finally, a number of
adsorbed molecules form an adlayer, which will grow with time. The growing
adlayer functions as an optical coating and changes the properties of the
underlying mirror, such as reflectance, transmittance, and absorption, which
are carefully chosen to maximize the detector sensitivity. The adlayer possibly
affects the gravitational wave detector sensitivity. In order to characterize
these changes, a high-finesse Fabry--Perot cavity was introduced to a KAGRA
cryostat and the finesse of the cavity was monitored for 35 days under
cryogenic conditions. We confirmed that the molecular adlayer was formed on a
cold mirror and caused an oscillation in the finesse. The real and imaginary
parts of the refractive index of the adlayer were $1.26 \pm 0.073$ and $2.2
\times 10^{-7} \pm 1.3 \times 10^{-7} $, respectively. These are considered to
be that of $\mathrm{H_2O}$ molecules. The formation rate of the molecular
adlayer was 27 $\pm$ 1.9 $\mathrm{nm/day}$. In this paper, we describe
theoretical and experimental studies of the formation of a molecular adlayer on
cryogenic mirrors. Furthermore, the effects of a molecular adlayer on the
quantum noise and the input heat to the test mass are also discussed.
