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The aim of the talk is to outline a classical study starting from
Weil-Seshadri. We study representations of certain Fuchsian groups
(discrete subgroups of SL(2,R) plus some conditions) in unitary groups and
interpret the equivalence class of such Representations in terms of
geometric objects on the Riemann surface obtained by quotient the upper
half-space modulo the Fuchsian group. Generalizations of this question to
representations into maximal compacts of semisimple algebraic groups lead
to surprising consequences.
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