Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity

We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra $\text{Gc}(\text{Om})$ providing a way to measure the $\text{G}$ and the $\text{Ginf}$- regularity in $\text{LL}(\text{Gc}(\text{Om}),\text{wt}\text{C})$. For the smaller family of functionals having a ``basic structure’’ we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic $\text{G}$ and $\text{Ginf}$-regularity.