  The central part of a dark matter halo reacts to the presence and evolution
of a bar. Not only does the halo absorb angular momentum from the disk, it can
also be compressed and have its shape modified. We study these issues in a
series of cosmologically motivated, highly resolved N-body simulations of
barred galaxies run under different initial conditions. In all models we find
that the inner halo's central density increases. We model this density increase
using the standard adiabatic approximation and the modified formula by Gnedin
et al. and find that halo mass profiles are better reproduced by this latter.
In models with a strong bar, the dark matter in the central region forms a
bar-like structure (``dark matter bar''), which rotates together with the
normal bar formed by the stellar component (``stellar bar''). The
minor-to-major axial ratio of a halo bar changes with radius with a typical
value 0.7 in the central disk region. DM bar amplitude is mostly a function of
the stellar bar strength. Models in which the bar amplitude increases or stays
roughly constant with time, initially large (40%-60%) misalignment between the
halo and disk bars quickly decreases with time as the bar grows. The halo bar
is nearly aligned with the stellar bar (~10 degrees lag for the halo) after ~2
Gyr. The torque, which the halo bar exerts on the stellar bar, can serve as a
mechanism to regulate the angular momentum transfer from the disk to the halo.
