  Crystal electric field states in rare earth intermetallics show an intricate
entanglement with the many-body physics that occurs in these systems and that
is known to lead to a plethora of electronic phases. Here, we attempt to trace
different contributions to the crystal electric field (CEF) splittings in
CeIrIn$_5$, a heavy-fermion compound and member of the Ce$M$In$_5$ ($M$= Co,
Rh, Ir) family. To this end, we utilize high-resolution resonant angle-resolved
photoemission spectroscopy (ARPES) and present a spectroscopic study of the
electronic structure of this unconventional superconductor over a wide
temperature range. As a result, we show how ARPES can be used in combination
with thermodynamic measurements or neutron scattering to disentangle different
contributions to the CEF splitting in rare earth intermetallics. We also find
that the hybridization is stronger in CeIrIn$_5$ than CeCoIn$_5$ and the
effects of the hybridization on the Fermi volume increase is much smaller than
predicted. By providing the first experimental evidence for $4f_{7/2}^{1}$
splittings which, in CeIrIn$_5$, split the octet into four doublets, we clearly
demonstrate the many-body origin of the so-called $4f_{7/2}^{1}$ state.
