  Using subvarieties, which we call Demazure quiver varieties, of the quiver
varieties of Nakajima, we give a geometric realization of Demazure modules of
Kac-Moody algebras with symmetric Cartan data. We give a natural geometric
characterization of the extremal weights of a representation and show that
Lusztig's semicanonical basis is compatible with the filtration of a
representation by Demazure modules. For the case of affine sl_2, we give a
characterization of the Demazure quiver variety in terms of a nilpotency
condition on quiver representations and an explicit combinatorial description
of the Demazure crystal in terms of Young pyramids.
