  The space $\Omega(G)$ of all based loops in a compact semisimple simply
connected Lie group $G$ has an action of the maximal torus $T\subset G$ (by
pointwise conjugation) and of the circle $S^1$ (by rotation of loops). Let $\mu
: \Omega(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting
$T\times S^1$ action. We show that all levels (that is, pre-images of points)
of $\mu$ are connected subspaces of $\Omega(G)$ (or empty). The result holds if
in the definition of $\Omega(G)$ loops are of class $C^{\infty}$ or of any
Sobolev class $H^s$, with $s\ge 1$ (for loops of class $H^1$, connectedness of
regular levels has been proved by Harada, Holm, Jeffrey, and the author).
