We describe the space of maximal components of the character variety of surface group
representations into
and
.
For every real rank
Lie group of Hermitian type, we construct a mapping class group
invariant complex structure on the maximal components. For the groups
and
,
we give a mapping class group invariant parametrization of each maximal
component as an explicit holomorphic fiber bundle over Teichmüller space.
Special attention is put on the connected components which are singular:
we give a precise local description of the singularities and their geometric
interpretation. We also describe the quotient of the maximal components of
and
by
the action of the mapping class group as a holomorphic submersion over the moduli
space of curves.
These results are proven in two steps: first we use Higgs bundles
to give a nonmapping class group equivariant parametrization,
then we prove an analog of Labourie’s conjecture for maximal
–representations.
Keywords
character varieties, mapping class group, Higgs bundles,
maximal representations