Combinatorics of level 0 representations

Recent work of Kato and Kato-Naito-Sagaki has provided an improved understanding of the combinatorics of integrable level 0 representations of the affine Lie algebra. In particular, there is a connection to the Schubert calculus of the semi-infinite flag variety from which Kato-Naito-Sagaki prove an analogue of the Pieri-Chevalley formula for the semi-infinite flag variety (line bundle multiplied with a Schubert class). The connection to crystals in this new formula (analogous to that in the Pieri-Chevalley formula of Pittie-Ram) provides a geometric interpretation of (a part of) the path model which was used by Ram-Yip to give a formula for Macdonald polynomials.