Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space

Abstract

We study the equivalence classes of slice-regular functions $f:\hat{I}\text{c◯}\to \mathbb{H}$ on a symmetric slice domain $\hat{I}\text{c◯}$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.