Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

In this paper, we give a characterization of the crystal bases , , of Demazure submodules , , of a level-zero extremal weight module over a quantum affine algebra , where is an arbitrary level-zero dominant integral weight, and denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis of the level-zero extremal weight module and the crystal of semi-infinite Lakshmibai-Seshadri paths of shape , which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule in terms of the specialization at of the symmetric Macdonald polynomial P-lambda(x; q, t).