What does (not) drive the variation of the low-mass end of the stellar initial mass function of early-type galaxies

Abstract: 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]