  We have been studying the index theory for some special infinite-dimensional
manifolds with a "proper cocompact" actions of the loop group LT of the circle
T, from the viewpoint of the noncommutative geometry. In this paper, we will
introduce the LT-equivariant KK-theory and we will construct three KK-elements:
the index element, the Clifford symbol element and the Dirac element. These
elements satisfy a certain relation, which should be called the
(KK-theoretical) index theorem, or the KK-theoretical Poincar\'e duality for
infinite-dimensional manifolds. We will also discuss the assembly maps.
