What does (not) drive the variation of the low-mass end of the stellar
  initial mass function of early-type galaxies
  The stellar initial mass function (IMF) seems to be variable and not
universal, as argued in the literature in the last three decades. Several
relations among the low-mass end of the IMF slope and other stellar population,
photometric or kinematic parameters of massive early-type galaxies (ETGs) have
been proposed, but a consolidated agreement on a factual cause of the observed
variations has not been reached yet. We investigate the relations between the
IMF and other stellar population parameters in NGC 3311, the central galaxy of
the Hydra I cluster. NGC 3311 is characterized by old and metal-rich stars,
like other massive ETGs, but has unusual increasing stellar velocity dispersion
and [$\alpha/$Fe] profiles. We use spatially resolved MUSE observations to
obtain stellar population properties using Bayesian full-spectrum fitting in
the central part of NGC 3311 to compare the IMF slope against other stellar
parameters with the goal of assessing their relations/dependencies. For NGC
3311, we unambiguously invalidate the previously observed direct correlation
between the IMF slope and the local stellar velocity dispersion, confirming
some doubts already raised in the literature. This relation may arise as a
spatial coincidence only, between the region with the largest stellar velocity
dispersion, with that where the oldest, $\textit{in situ}$ population is found
and dominates. We also show robust evidence that the proposed IMF-metallicity
relation is contaminated by the degeneracy between these two parameters. The
tightest correlations we found are those between stellar age and IMF and
between galactocentric radius and IMF. The variation of the IMF is not due to
kinematical, dynamical, or global properties in NGC 3311. We speculate that IMF
might be dwarf-dominated in the "red-nuggets" formed at high redshifts that
ended up being the central cores of today's giant ellipticals. [Abridged]
