  This thesis is dedicated to the study of certain loci of the Higgs bundle
moduli space on a compact Riemann surface. Motivated by mirror symmetry, we
give a detailed description of the fibres of the $G$-Hitchin fibration
containing $G_0$-Higgs bundles for the real forms $G_0 = SU^*(2m)$, $SO^*(4m)$
and $Sp(m,m)$ of $G = SL(2m,\mathbb{C})$, $SO(4m,\mathbb{C})$ and
$Sp(4m,\mathbb{C})$, respectively. The spectral curves associated to these
fibres are examples of ribbons and our description is done in two different
ways, one in term of objects on the reduced scheme associated to the spectral
curve, while the other in terms of the (non-reduced) spectral curve. A link is
provided between the two approaches. We use this description to give a proposal
for the support of the dual BBB-brane inside the moduli space of Higgs bundles
for the Langlands dual group of $G$, corroborating the conjectural picture.
  In the second part of the thesis we discuss Gaiotto Lagrangian subvarieties
inside the moduli spaces of $G$-Higgs bundles, where $G$ is a complex reductive
group. These Lagrangians are obtained from a symplectic representation of $G$
and we discuss some of its general properties. We then focus our attention to
the Gaiotto Lagrangian for the standard representation of the symplectic group.
This is an irreducible component of the nilpotent cone for the symplectic
Hitchin fibration. We describe this component by using the usual Morse function
on the Higgs bundle moduli space restricted to the Lagrangian in question.
Lastly, we discuss natural questions and applications of the ideas developed in
this thesis. In particular, we say a few words about the hyperholomorphic
bundle, how to generalize the Gaiotto Lagrangian to vector bundles which admit
many sections and give an analogue of the Gaiotto Lagrangian for the orthogonal
group.
