Extension of a theorem of Kostant for affine algebras (Q1379075)

Let \({\mathfrak g}\) be a semisimple Lie algebra over an algebraically closed field of characteristic zero. Let \({\mathfrak p}={\mathfrak z}\oplus{\mathfrak s}\oplus{\mathfrak m}\) be a parabolic subalgebra of \({\mathfrak g}\) where \({\mathfrak z}\oplus{\mathfrak s}\) is reductive with center \({\mathfrak z}\) and \({\mathfrak m}\) is nilpotent. Kostant's theorem says the \({\mathfrak z}\) isotypic components of \({\mathfrak m}\) are simple \({\mathfrak s}\) modules. This paper generalizes this theorem to the case of affine algebras. The theorem remains true in the affine algebra setting for \({\mathfrak z}\) isotypic components of \({\mathfrak m}\) in which root spaces are one-dimensional, but not in general. In all cases it is possible to give a complete description of the \({\mathfrak s}\) module structure of the \({\mathfrak z}\) isotypic components.