  For a compact, connected, simply-connected Lie group G, the loop group LG is
the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps
S^1-->G. The geometry of LG and its homogeneous spaces is related to
representation theory and has been extensively studied. The space of based
loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural
Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the
moment map mu 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 T-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 mu(Omega(G)) is convex. We also show that
for the energy functional E, which is the moment map for the S^1 rotation
action, each non-empty preimage is connected.
