Semi-infinite Lakshmibai–Seshadri path model for level-zero extremal weight modules over quantum affine algebras

Abstract

We introduce semi-infinite Lakshmibai–Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of the previous result due to Naito and Sagaki (which uses Littelmann's Lakshmibai–Seshadri paths), in which the level-zero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight.