\(\mathfrak {k}\)-Structure of Basic Representation of Affine Algebras

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.