Direct limits of Schubert varieties and global sections of line bundles (Q958529)

Let \(G\) be a classical simple ind-group and let \(B\) be a Borel subgroup of \(G\). The main purpose of the paper is to study \(B\)-stable subvarieties of \(G/B\); they are called ind-Schubert varieties. Namely, the following two questions are considered: (1) When is an ind-Schubert variety projective? (2) For a globally generated line bundle \(\mathcal L\) on a projective ind-Schubert variety \(X\), what is the algebraic structure of the \(B\)-module \(\Gamma(X, {\mathcal L})\)? The author proves a necessary and sufficient condition for projectivity of an ind-Schubert variety. This result shows that although there is a rich class of infinite-dimensional projective ind-Schubert varieties, they are essentially subvarieties of products of projective \(G/B\)'s. Regarding the second question, the \(B\)-module \(\Gamma(X, {\mathcal L})\) is canonically isomorphic to the dual of a certain \(B\)-module \(V_X(\mathcal L)\) (explicitly defined in Section 2.5), and the second result obtained in this paper relates the geometry of \(X\) to the algebraic structure of \(V_X(\mathcal L)\). It is proved that when \(X\) is projective, for any globally generated line bundle \(\mathcal L\) on \(X\) the weight spaces of \(V_X(\mathcal L)\) are finite-dimensional and there is a weight \(\lambda\) such that the weight space of a weight \(\mu\) is nonzero only if \(\lambda=\mu\) is a linear combination of positive roots. A theorem is also proved which provides a method of describing weight multiplicities of \(V_X(\mathcal L)\) in terms of weight multiplicities of certain Demazure modules.