  We consider a model of inflation consisting a triplet of $U(1)$ vector fields
with the parity violating interaction which is non-minimally coupled to
inflaton. The vector field sector enjoys global $O(3)$ symmetry which admits
isotropic configuration and provides not only vector modes but also scalar and
tensor modes. We decompose the scalar perturbations into the adiabatic, entropy
and isocurvature perturbations and compute all power spectra and cross
correlations of the scalar and the tensor sectors. The tensor modes associated
with the vector fields contribute to the power spectrum of gravitational waves
while the parity violating term generates chirality in gravitational power
spectra and bispectra. We study nonlinear scalar and tensor perturbations and
compute all bispectra and three-point cross-correlations. In particular, it is
shown that the non-Gaussianity of curvature perturbations and gravitational
waves are enhanced by the vector field perturbations. We show that
non-Gaussianities put strong constraints on the model parameters such as the
parity violating coupling and the fractional energy of the vector fields.
