  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.

Complex Variables;Algebraic Geometry;Group Theory