The structure of Hilbert flag varieties
Let \(H\) be a complex Hilbert space. If \(H\) is finite dimensional, then it is a classical result that the finite dimensional irreducible representations of the general linear group \(GL(H)\) can be realized geometrically as the natural action of the group \(GL(H)\) on the space of global holomorphic sections of a holomorphic line bundle over a space of flags in \(H\). By choosing a basis of \(H\), one can identify this space of holomorphic sections with a space of holomorphic functions on \(GL(H)\) that are certain polynomial expressions in minors of the matrices corresponding to the elements of \(GL(H)\). Infinite dimensional analogues of some of these representations occur in quantum field theory, infinite dimensional Grassmann manifolds play an important role in the framework of integrable systems. The first person to realize this was Sato. In the paper under review the authors give an infinite dimensional analogue of all these representations. In a separable Hilbert space \(H\) they consider a collection of flags that generalizes the Grassmannian from \textit{A. Pressley's} and \textit{G. Segal's} book [Loop groups, Oxford: Clarendon Press (1986; Zbl 0618.22011)]. This flag variety carries a natural Hilbert space structure and there exist line bundles over it that are similar to the finite dimensional ones. This includes the determinant bundle and its dual form. In the ``dominant'' case the space of global holomorphic sections of such a line bundle turns out to be non-trivial. However, the action of the analogue of the general linear group can, in general, not be lifted to the line bundle under consideration and one has to pass to a central extension of this group. The paper under review consists of three sections. In the first section the authors give the definition of the flag variety and treat some properties of the flag variety. The second section is devoted to the construction of the holomorphic line bundles, to a description of the corresponding central extensions and to the analysis of the space of global holomorphic sections. As an application they show in the final section what role geometry plays in the context of some integrable systems.
finite dimensional irreducible representations
scientific article; zbMATH DE number 716437
scientific article