My talk is a survey of the following joint paper with Ishii and Naito:
M. Ishii, S. Naito, and D. Sagaki, Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras, Adv. Math. (vol.290, pp.967--1009, 2016).
In this paper, we introduced semi-infinite Lakshmibai-Seshadri (LS) paths by using the semi-infinite Bruhat order on affine Weyl groups(in place of the usual Bruhat order), which enable us to give an explicit realization of the crystal basis of the extremal weight module over the quantum affine algebra.
My plan of the talk is as follows:
(1) Basic notation for affine Lie algerbas and quantum affine algebras.
(2) Definitions of the semi-infinite Bruhat order and semi-infinite LS paths.
(3) Crystal structure on the set of semi-infinite LS paths.
(4) Extremal weight modules and their crystal bases.
(5) Isomorphism theorem: "the crystal of semi-infinite LS paths is isomorphic, as an affine crystal, to the crystal basis of the extremal weight module over the quantum affine algebra".
(6) Sketch of the proof of the isomorphism theorem (if I have time)