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

TL;DR

This paper classifies infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of L*-groups of compact type, extending finite-dimensional results to an infinite-dimensional Hilbert manifold setting using simple roots of non-compact type.

Contribution

It provides a classification of infinite-dimensional Hermitian-symmetric affine coadjoint orbits, generalizing finite-dimensional classifications to Hilbert Lie groups.

Findings

Classified infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits.

Established that such spaces are affine coadjoint orbits of Hilbert Lie groups.

Connected the classification to simple roots of non-compact type.

Abstract

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 Hermitian-symmetric 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 infinite-dimensional 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 g_n of g together with an increasing sequence of finite-dimensional subalgebras k_n of k such that g (resp. k) is the closure of the union of g_n (resp. k_n), and such that the pairs (g_n, k_n) 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.