For a compact, connected, simply-connected Lie group
, the loop
group
is the infinite-dimensional Hilbert Lie group consisting of
–Sobolev
maps The
geometry of
and its homogeneous spaces is related to representation theory
and has been extensively studied. The space of based loops
is an example of a
homogeneous space of and
has a natural Hamiltonian
action, where is the
maximal torus of . We
study the moment map
for this action, and in particular prove that its regular level sets are connected. This
result is as an infinite-dimensional analogue of a theorem of Atiyah that states that
the preimage of a moment map for a Hamiltonian torus action on a compact
symplectic manifold is connected. In the finite-dimensional case, this connectivity
result is used to prove that the image of the moment map for a compact Hamiltonian
–space
is convex. Thus our theorem can also be viewed as a companion result
to a theorem of Atiyah and Pressley, which states that the image
is convex. We also show that for the energy functional
, which is the
moment map for the
rotation action, each non-empty preimage is connected.