Borel--de Siebenthal pairs, Global Weyl modules and Stanley--Reisner rings

Abstract: We develop the theory of integrable representations for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel--de Siebenthal pair of semisimple Lie algebras. We see that although there are similarities with the represenation thery of the standard maximal parabolic subalgebra there are also very interesting and non--trivial differences; including the fact that there are examples of non--trivial global Weyl modules which are irreducible and finite--dimensional. We also give a presentation of the endomorphism ring of the global Weyl module; although these are no longer polynomial algebras we see that for certain parabolics these algebras are Stanley--Reisner rings which are both Koszul and Cohen--Macaualey.