Equivariant K\"ahlerian extensions of contact manifolds

For contact manifolds $(M, \eta)$ a complexification $M^c$ is constructed to which the contact form $\eta$ extends such that the exterior derivative of the extended form is K\"ahlerian. In the case of a proper action of an extendable Lie group this construction is realized in an equivariant way. In a simultaneous stratification of $M$ and $M^c$ according to the istropy type, it is shown that the K\"ahlerian reduction of the complexification can be seen as the complexification of the contact reduction.