Representation theoretic realization of non-symmetric MacDonald polynomials at infinity

We study the non-symmetric MacDonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric MacDonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules for simply-laced Dynkin types except for type E 8 {\mathrm{E}_{8}}.