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 of the PhD-thesis of the author [20] extended by several results from the articles [1, 3, 11]. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo form="prefix">Sp</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for a Hermitian algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Using this parametrization, we count connected components of the space of maximal framed representations as well as the space of maximal (non-framed) representations.