Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types' automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.