  This article presents a new relation between the basic representation of
split real simply-laced affine Kac-Moody algebras and finite dimensional
representations of its maximal compact subalgebra $\mathfrak{k}$. We provide
infinitely many $\mathfrak{k}$-subrepresentations of the basic representation
and we prove that these are all the finite dimensional
$\mathfrak{k}$-subrepresentations of the basic representation such that the
quotient of the basic representation by the subrepresentation is a finite
dimensional representation of a certain parabolic algebra and of the maximal
compact subalgebra. By this result we provide an infinite composition series
with a cosocle filtration of the basic representation. Finally, we present
examples of the results and applications to supergravity.
