  To investigate the relationship between black holes and their host galaxies,
many groups have used the width of the [O III] 5007 line as a substitute for
the stellar velocity dispersion (sigma*) of galaxy bulges. We directly test
this assumption with a large and homogeneous sample of narrow-line active
galactic nuclei from the Sloan Digital Sky Survey. We consider multiple
transitions ([O II] 3727, [O III] 5007, and [S II] 6716, 6731) and various
techniques for quantifying the line width in order to obtain a calibration
between the gas velocity dispersion, sigma_g, and sigma*. We find that sigma_g
of the low-ionization lines traces sigma*, as does sigma_g for the core of [O
III] after its asymmetric blue wing is properly removed, although in all cases
the correlation between sigma_g and sigma* has considerable scatter. While the
gas kinematics of the narrow-line region of active galaxies are primarily
governed by the gravitational potential of the stars, the accretion rate, as
traced by the Eddington luminosity ratio, seems to play an important secondary
role. Departures from virial motions correlate systematically with accretion
rate. We discuss the implications of these results for previous studies that
use [O III] line widths to infer stellar velocity dispersions in quasars and
narrow-line Seyfert 1 galaxies.
