  It has been shown that a self-gravitating system of massive keV fermions in
thermodynamic equilibrium correctly describes the dark matter (DM) distribution
in galactic halos and predicts a denser quantum core towards the center of the
configuration. Such a quantum core, for a fermion mass in the range of $50$ keV
$\lesssim m c^2 \lesssim 345$ keV, can be an alternative interpretation of the
central compact object in Sgr A*. We present in this work the gravitational
lensing properties of this novel DM model in Milky Way-like spiral galaxies. We
describe the lensing effects of the pure DM component both on halo scales,
where we compare them to the effects of the Navarro-Frenk-White and the
Non-Singular Isothermal Sphere DM models, and near the galaxy center, where we
compare them with the effects of a Schwarzschild BH. For the particle mass
leading to the most compact DM core, $m c^2\approx 10^{2}$ keV, we draw the
following conclusions. At distances $r\gtrsim 20$ pc from the center of the
lens the effect of the central object on the lensing properties is negligible.
However, we show that measurements of the deflection angle produced by the DM
distribution in the outer region at a few kpc, together with rotation curve
data, could help to discriminate between different DM models. We show that at
distances $\sim 10^{-4}$ pc strong lensing effects, such as multiple images and
Einstein rings, may occur. Large differences in the deflection angle produced
by a DM central core and a central BH appear at distances $r\lesssim 10^{-6}$
pc; in this regime the weak-field formalism is no longer applicable and the
exact general-relativistic formula has to be used. We find that quantum DM
cores do not show a photon sphere what implies that they do not cast a shadow.
Similar conclusions apply to the other DM distributions for other fermion
masses in the above specified range and for other galaxy types.
