  The Riemann-Hilbert correspondence is an isomorphism between the de Rham
moduli space and the Betti moduli space, defined by associating to each
Fuchsian system its monodromy representation class. In 1997 Hitchin proved that
this map is a symplectomorphism. In this paper, we address the question of what
happens to this theory if we extend the de Rham moduli space by allowing
connections with higher order poles. In our previous paper arXiv:1511.03851,
based on the idea of interpreting higher order poles in the connection as
boundary components with bordered cusps (vertices of ideal triangles in the
Poincar\'e metric) on the Riemann surface, we introduced the notion of
decorated character variety to generalize the Betti moduli space. This
decorated character variety is the quotient of the space of representations of
the fundamental groupid of arcs by a product of unipotent Borel sub-groups (one
per bordered cusp). Here we prove that this representation space is endowed
with a Poisson structure induced by the Fock--Rosly bracket and show that the
quotient by unipotent Borel subgroups giving rise to the decorated character
variety is a Poisson reduction. We deal with the Poisson bracket and its
quantization simultaneously, thus providing a quantisation of the decorated
character variety. In the case of dimension 2, we also endow the representation
space with explicit Darboux coordinates. We conclude with a conjecture on the
extended Riemann-Hilbert correspondence in the case of higher order poles.
