On a parametrization of spaces of maximal framed representations

This article is an extended version of the talk given by the author in the seminar Théorie Spectrale et Géométrie at the Institut Fourier in March 2022. We present some results from the author’s doctoral thesis, extended by several results from other papers. We give a parametrization of the space of maximal framed representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type that can be seen as ${\mathrm{S}\mathrm{p}}_2(A,\sigma )$ for a Hermitian algebra $(A,\sigma )$. Using this parametrization, we count connected components of the space of maximal framed representations as well as the space of maximal (non-framed) representations.