These modules naturally divide themselves into three categories: positive level, negative level and level 0. The positive level modules are highest weight, the negative level ones are lowest weight, and the level 0 ones are neither. But all three classes of modules have some nice character formulas, a good crystal theory in the sense of Kashiwara-Lusztig-Littelmann, and Borel-Weil-Bott type geometric constructions. The geometric constructions use, respectively, the thin affine flag variety (for positive level), the thick affine flag variety (for negative level), and the semi-infinite flag variety (for level 0).
This video is part of the University of Georgia‘s Algebra seminar.
