Direct limits of Schubert varieties and global sections of line bundles

We study the space of global sections Γ(X,L) of a line bundle L 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 L is globally generated, then Γ(X,L) is the dual of a B-module VX(L) with finite-dimensional weight multiplicities. Finally, we describe the structure of these multiplicities in terms of Demazure modules.