An algebraic-geometric construction of ind-varieties of generalized flags (Q2082722)

The authors study the class of admissible linear embeddings of flag varieties in the context of algebraic geometry. They prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra, showing that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags. That is, generalized flags in a countable-dimensional vector space are in a natural one-to-one correspondence with splitting parabolic subgroups \(\mathsf{P}\) of the ind-group \(\mathsf{GL}(\infty)\), and hence the points of homogeneous ind-spaces of the form \(\mathsf{GL}(\infty)/\mathsf{P}\) can be thought of as generalized flags. Ind-varieties are discussed as the ind-group \(\mathsf{SL}(\infty)\), respectively, \(\mathsf{O}(\infty)\) or \(\mathsf{Sp}(\infty)\) for isotropic generalized flags, where the authors construct them in purely algebro-geometric terms.