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

## TL;DR

This paper develops the theory of integrable representations for maximal parabolic subalgebras of affine Lie algebras, revealing new properties of global Weyl modules and their endomorphism rings, including connections to Stanley--Reisner rings.

## Contribution

It introduces a novel framework for understanding integrable representations of parabolic subalgebras derived from Borel--de Siebenthal pairs, highlighting new finite-dimensional irreducible modules and algebraic structures.

## Key findings

- Global Weyl modules can be irreducible and finite-dimensional.
- Endomorphism rings of global Weyl modules can be Stanley--Reisner rings.
- Certain parabolic subalgebras have endomorphism rings that are Koszul and Cohen--Macaualay.

## 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.