  The existence of hadrons containing gluons as constituent particles (like
hybrid mesons) will involve the necessity to admit an hadronic spectroscopy
beyond the naive quark model. In the QGC model, the lowest J^{PC}= 1^{-+}
hybrid mesons may be built in two different modes: the GE-mode corresponding to
an angular momentum between the gluon and qqbar system, while the QE-mode
corresponding to an angular momentum between q and qbar. In this paper, we give
an analysis of the possible 1^{-+} hybrid mesons in the 1.32-2.04 GeV mass
range, and predictions on their decay. We discuss on the inconsistency between
some theoretical results and experimental data.We find that the 1^{-+}(1400
MeV), dominated by the QE-mode will decay in the favourite rho-pi channel,
while the GE-1^{-+}(1600 MeV), and the 1^{-+}(2.0 GeV), pure GE-hybrids, will
decay preferably into b1-pi channel (never observed); we indicate problems
concerning the observation of decay channels which are forbidden by theoretical
present models.The total decay widths of the 1^{-+}(2.0 GeV) and the
GE-1^{-+}(1600 MeV) being very large, this leads to a problem of their
observability as resonances. We may conclude that at this time, there is no
absolute confirmation of their existence, but some asked questions remain not
solved.
