Direct limits of Schubert varieties and global sections of line bundles

We study the space of global sections of a line bundle on a B-stable ind-subvariety X of G/B, where G is a classical simple ind-group and B is an arbitrary Borel subgroup of G. We give a necessary and sufficient condition for projectivity of X, and use it to prove that if X is projective and is globally generated, then is the dual of a B-module with finite-dimensional weight multiplicities. Finally, we describe the structure of these multiplicities in terms of Demazure modules.