The (Schottky) uniformization of Riemann surfaces motivated the search of a parametrization of holomorphic bundles. Semistable bundles over a fixed Riemann surface X are obtained from a unitary representation of the fundamental group of X, as was proven by Narasimhan and Seshadri for vector bundles, and by Ramanathan for principal bundles (for reductive groups over C). In the spirit of Schottky uniformization, Florentino and later with Ludsteck intruduced the notion of Schottky vector and principal bundles (for a general complex reductive algebraic group), which are isomorphic to holomorphic bundles associated to a particular (Schottky) representation of the fundamental group of X.
    
In our work, we first prove that Schottky principal bundles have trivial topological type. Next, we study the Schottky map, which associates to each Schottky representation a semistable principal bundle, proving that it is a local submersion. We give two special classes of examples: principal bundles with diagonalizable group, and principal bundles over an elliptic curve.  On the other hand, Schottky representations also appear in a different context, related to non-abelian Hodge theory. We prove that some of them are included in the (A,B,A)-brane, in the sense of Baraglia and Schaposnik.
    
This is a joint work with Susana Ferreira and Carlos Florentino.