  Gamma Doradus stars present an incredibly rich pulsation spectra, with
gravito-inertial modes, in some cases supplemented with delta Scuti-like
pressure modes and in numerous cases with Rossby modes. The present paper aims
at showing that, in addition to these modes established in the radiative
envelope, pure inertial modes, trapped in the convective core, can be detected
in Kepler observations of gamma Doradus stars, thanks to their resonance with
the gravito-inertial modes.
  We start by using a simplified model of perturbations in a full sphere of
uniform density. Under these conditions, the spectrum of pure inertial modes is
known from analytical solutions of the so-called Poincare equation. We then
compute coupling factors which help select the pure inertial modes which
interact best with the surrounding dipolar gravito-inertial modes. Using
complete calculations of gravito-inertial modes in realistic models of gamma
Doradus stars, we are able to show that the pure inertial/gravito-inertial
resonances appear as dips in the gravito-inertial mode period spacing series at
spin parameters close to those predicted by the simple model. We find the first
evidence of such dips in the Kepler gamma Doradus star KIC5608334. Finally,
using complete calculations in isolated convective cores, we find that the spin
parameters of the pure inertial/gravito-inertial resonances are also sensitive
to the density stratification of the convective core.
  In conclusion, we have discovered that certain dips in gravito-inertial mode
period spacings observed in some Kepler stars are in fact the signatures of
resonances with pure-inertial modes that are trapped in the convective core.
  This holds the promise to finally access the central conditions , i.e.
rotation and density stratification, of intermediate-mass stars on the main
sequence.
