  Early-type galaxies provide unique tests for the predictions of the cold dark
matter cosmology and the baryonic physics assumptions entering models for
galaxy formation. In this work, we use the Illustris simulation to study
correlations of three main properties of early-type galaxies, namely, the
stellar orbital anisotropies, the central dark matter fractions and the central
radial density slopes, as well as their redshift evolution since $z=1.0$. We
find that lower-mass galaxies or galaxies at higher redshift tend to be bluer
in rest-frame colour, have higher central gas fractions, and feature more
tangentially anisotropic orbits and steeper central density slopes than their
higher-mass or lower-redshift counterparts, respectively. The projected central
dark matter fraction within the effective radius shows a very mild mass
dependence but positively correlates with galaxy effective radii due to the
aperture effect. The central density slopes obtained by combining strong
lensing measurements with single aperture kinematics are found to differ from
the true density slopes. We identify systematic biases in this measurement to
be due to two common modelling assumptions, isotropic stellar orbital
distributions and power-law density profiles. We also compare the properties of
early-type galaxies in Illustris to those from existing galaxy and strong
lensing surveys, we find in general broad agreement but also some tension,
which poses a potential challenge to the stellar formation and feedback models
adopted by the simulation.
