Vladimir Fock
Flat connections and triples of flags

Abstract: We are going to explain that the moduli space of flat
$G$-connections on Riemann surfaces for simple Lie group $G$ and its
versions can be constructed by a so-called amalgamation procedure out of
much simpler space $F$ - configuration space of triples of flags with some
additional data. Namely each moduli space of flat connections can be
represented as a power of $F$ quotiented by a power of the Cartan subgroup.
This construction respects Poisson structure and quantisation. The
presentation is not unique, but in order to pass from one presentation to
another as well as to describe the mapping class group action it is
sufficient to study configurations of four flags. A particular case of such
moduli spaces is the Lie group $G$ as well as the dual group $G^*$.  In this
case the construction gives explicitly the standart Poisson-Lie structure on
them.