  Observational studies of nearby galaxies have demonstrated correlations
between the mass of the central supermassive black holes (BHs) and properties
of the host galaxies, notably the stellar bulge mass or central stellar
velocity dispersion. Motivated by these correlations, the theoretical paradigm
has emerged, in which BHs and bulges co-evolve. However, this picture was
challenged by observational and theoretical studies, which hinted that the
fundamental connection may be between BHs and dark matter halos, and not
necessarily with their host galaxies. Based on a study of 3130 elliptical
galaxies $-$ selected from the Sloan Digital and ROSAT All Sky Surveys $-$ we
demonstrate that the central stellar velocity dispersion exhibits a
significantly tighter correlation with the total gravitating mass, traced by
the X-ray luminosity of the hot gas, than with the stellar mass. This hints
that the central stellar velocity dispersion, and hence the central
gravitational potential, may be the fundamental property of elliptical galaxies
that is most tightly connected to the larger-scale dark matter halo.
Furthermore, using the central stellar velocity dispersion as a surrogate for
the BH mass, we find that in elliptical galaxies the inferred BH mass and
inferred total gravitating mass within the virial radius (or within five
effective radii) can be expressed as $M_{\rm{BH}} \propto M_{\rm
tot}^{1.6^{+0.6}_{-0.4}} $ (or $M_{\rm{BH}} \propto
M_{\rm{5r_{eff}}}^{1.8^{+0.7}_{-0.6}}$). These results are consistent with a
picture in which the BH mass is directly set by the central stellar velocity
dispersion, which, in turn, is determined by the total gravitating mass of the
system.
