  Calculations are performed for the circular polarization of maser radiation
from a turbulent, Keplerian disk that is intended to represent the sub-parsec
disk at the nucleus of the galaxy NGC4258. The polarization in the calculations
is a result of the Zeeman effect in the regime in which the Zeeman splitting is
much less than the spectral linebreadth. Plausible configurations for turbulent
magnetic and velocity fields in the disk are created by statistical methods.
This turbulence, along with the Keplerian velocity gradients and the blending
of the three hyperfine components to form the $6_{16} - 5_{23}$ masing
transition of water, are key ingredients in determining the appearance of the
polarized spectra that are calculated. These spectra are quite different from
the polarized spectra that would be expected for a two-level transition where
there is no hyperfine structure. The effect of the hyperfine structure on the
polarization is most striking in the calculations for the maser emission that
represents the central (or systemic) features of NGC4258. Information about
magnetic fields is inferred from observations for polarized maser radiation and
bears on the structure of accretion disks.
