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

摘要

We develop the theory of integrable representations for an arbitrary standard 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 representation theory 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-Macaulay.