Jointly with G. F. Helminck I studied a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This required 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. These representations are of importance in quantum field theory and in the framework of integrable systems. We discuss a number of applications of these flag varieties to integrable systems and mathematical physics. Presently we are looking at a Kahler structure on these Hilbert flag varieties.