Classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitiansymmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinitedimensional irreducible Hermitian-symmetric affine coadjoint orbits of L-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (g, k), where g is a simple L-algebra and k a subalgebra of g, to construct an increasing sequence of finite-dimensional subalgebras gn of g together with an increasing sequence of finite-dimensional subalgebras kn of k such that g = ∪gn, k = ∪kn, and such that the pairs (gn, kn) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact type is an affine-coadjoint orbit of an Hilbert Lie group.