Abstract:
Symplectic implosion was introduced by Guillemin, Jeffrey, and Sjamaar in [GJS02], and is an
abelianisation of symplectic reduction. In implosion, one constructs the imploded cross-section
Mimpl of a Hamiltonian G-space M. It is characterised by the property that the reduction of M
by the whole group G is isomorphic to the reduction of the imploded cross-section Mimpl by a
maximal torus T of G.
On the other hand, a real structure on a symplectic manifold is an anti-symplectic involution on
the manifold, i.e. an anti-isomorphism in the symplectic category which squares to the identity
function. The fixed point set of such an involution is either empty or a Lagrangian submanifold.
These structures were generalised to Hamiltonian G-spaces by Duistermaat in [Dui83], in the
case that G is an abelian Lie group. The non-abelian case was studied by O’Shea and Sjamaar in
[OS00].
In this thesis, we find conditions under which the imploded cross-section of a real Hamiltonian
G-space inherits a real Hamiltonian T-structure, where T ⊆ G is a certain maximal torus of G.
The fixed point set of the induced real structure on the imploded cross-section is then either empty
or a Lagrangian submanifold. Hence we have a method of constructing Lagrangian submanifolds
in imploded cross-sections.