The Structure of Hilbert Flag Varieties

Abstract

In this paper we present a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over them. In general the action of the restricted linear group can not be lifted to the line bundles and thus leads to central extensions of this group. It is determined exactly when these extensions are non-trivial. These representations are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how the flag varieties occur in the latter context.

Infinite-dimensional flag manifolds in integrable systems

“…Moreover Gl res ( H ) acts on F through its natural action on subspaces of H and this action is transitive, see [8]. Hence, we have F = Gl res ( H )/ P, where P is the parabolic subgroup stabilizing the basic flag.…”

“…From [8] we know then that the homogeneous space F = GL res ( H )/ P carries an analytic E-manifold structure for which τ is a submersion and for which the natural action of…”

“…It is shown in [8] that the group of unitary operators in Gl res ( H ) acts transitively on F . Hence, to make F into a hermitian manifold it suffices to give on the tangent space E of the basic flag F (0) a hermitian form, which is invariant under the stabilizer of F (0) .…”

“…By direct computation or by using the Lie algebra 2-cocycles for B res ( H ) from [8], one shows that ω is closed: dω = 0. Hence F is a symplectic manifold and even a Kähler manifold, another property that carries over from the finite dimensional…”

“…For a proof we refer to [8]. An important operator in Gl res ( H ) is the shift operator defined by ( n∈‫ޚ‬ a n z n ) = n∈‫ޚ‬ a n z n+1 .…”