Hilbert flag varieties and their Kähler structure

In this paper we introduce the infinite-dimensional flag varieties associated with integrable systems of the $KdV$- and $Toda$-type and we discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and we conclude by showing that all flag varieties introduced before possess a K\"{a}hler structure