  It is well established that supermassive black holes in nearby elliptical
galaxies correlate tightly with the kinematic property ($\mbhsigma$
correlation) and stellar mass ($\mbhhost$ correlation) of their host spheroids.
However, it is not clear what the relations would be at the low-mass end, and
how they evolve. Here, we investigate these relations in low-mass systems
($\MBH \sim \rm{10^{6}- 10^{8}}\, \Msun$) using the Aquila Simulation, a
high-resolution cosmological hydrodynamic simulation which follows the
formation and evolution of stars and black holes in a Milky Way-size galaxy and
its substructures. We find a number of interesting results on the origin and
evolution of the scaling relations in these systems: (1) there is a strong
redshift evolution in the $\mbhsigma$ relation, but a much weaker one in the
$\mbhhost$ relation; (2) there is a close link between the $\mbhsigma$ relation
and the dynamical state of the system -- the galaxies that fall on the observed
correlation appear to have reached virial equilibrium. (3) the star formation
and black hole growth are self-regulated in galaxies -- the ratio between black
hole accretion rate and star formation rate remains nearly constant in a wide
redshift span $z = 0-6$. These findings suggest that the observed correlations
have different origins: the $\mbhsigma$ relation may be the result of virial
equilibrium, while the $\mbhhost$ relation may the result of self-regulated
star formation and black hole growth in galaxies.
