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

Abstract 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 $$\mathfrak {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> . We provide infinitely many $$\mathfrak {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -subrepresentations of the basic representation and we prove that these are all the finite dimensional $$\mathfrak {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -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.