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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 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.

These Lie algebras are L * -algebras in the sense that the following property is satisfied : for every x, y and z. In fact, u 2 (H), o 2 (H R ) and sp 2 (H) are the unique infinite-dimensional simple L *algebras of compact type modulo isomorphisms (see below for the corresponding definition and [1], [6], or [17] for the proof of this statement). The group associated to an L * -algebra is called an L * -group. The L * -groups GL 2 (H), U 2 (H) and Sp 2 (H) are connected, but O 2 (H R ) admits two connected components (see Proposition 12.4.2 on page 245 in [12]). The connected component of O 2 (H R ) containing the special orthogonal group will be denoted by O + 2 (H). The aim of this paper is to prove the following statement. In the finite-dimensional case, every Hermitian-symmetric space of compact type is a coadjoint orbit of its connected group of isometries (see Proposition 8.89 in [3]). In the infinite-dimensional setting, the biggest group of isometries of a given Hermitian-symmetric space is not a Hilbert Lie group in general. For example the restricted unitary group U res (H) (see [12] for its definition) is a Banach Lie group acting by isometries on the restricted Grassmannian. It is a non trivial fact that the unitary Hilbert Lie group U 2 (H), strictly contained in U res (H), acts transitively on each connected components of the restricted Grassmannian (see Proposition 5.2 in [2]). Theorem 1.1 above compared to the work of W. Kaup ([8], [9]), leads to the following generalization : Corollary 1.2 Every Hermitian-symmetric space of compact type is an homogeneous space of an Hilbert Lie group. More precisely, every Hermitian-symmetric space of compact type is an affinecoadjoint orbit of an L * -group.
2 Root Theory of complex L * -algebra The root theory of complex L * -algebras has been developed by J. R. Schue in [13] and [14]. Let us first recall that an L * -algebra g over K ∈ {R, C} is a Lie algebra over K, which is also a Hilbert space over K such that for every element x ∈ g, there exists x * ∈ g with the following property for every y, z in g. In the case when K = C, our convention for the Hermitian product · , · is that it is C-skew-linear with respect to the first variable, and C-linear with respect to the second variable. The first example of L * -algebra is a semi-simple finite-dimensional complex Lie algebra g 0 endowed with an involution σ, which defines a compact real form of g 0 . In this example, the involutions * and σ are related by x * = −σ(x) and the Hermitian scalar product is given by x, y = B(x * , y), where B denotes the Killing form of g 0 . An L * -algebra is called of compact type if x * = −x for every x in g. For a given L * -algebra g the subspace k := {x ∈ g | x * = −x} is a real L * -algebra of compact type. Thus an L * -algebra can be thought as an Hilbert Lie algebra together with a distinguished compact real form.
For every subsets A and B of an L * -algebra g, [A, B] will denote the closure of the vector space spanned by {[a, b], a ∈ A, b ∈ B}. With this notation, an L * -algebra is called semi-simple if g = [g, g], and simple if g is non-commutative and if every closed ideal of g is trivial. Every L * -algebra can be decomposed into an orthogonal sum of its center and a semi-simple closed ideal (see [18], 2.2.13.). A Cartan subalgebra of a complex semi-simple L * -algebra g C is defined as a maximal Abelian * -stable subalgebra of g C . Note that the condition of being * -stable is added in comparison to the finitedimensional setting, hence a Cartan subalgebra may not be maximal in the set of Abelian subalgebras. It is noteworthy that a Cartan subalgebra of an L * -algebra is in fact maximal Abelian (see [14], 1.1). Remark that a finite-dimensional Cartan subalgebra of a complex semi-simple Lie algebra g C (for the usual definition) is contained in a compact real form of g C , thus is also a Cartan subalgebra of the corresponding finite-dimensional L * -algebra. The existence of Cartan subalgebras of L * -algebra is guarantied by Zorn's Lemma. Every semi-simple L * -algebra is an Hilbert sum of closed * -stable simple ideals (see Theorem 1 in [13] for the initial proof. Some missing arguments can be found in [18]).
In the sequel, g C will denote a semi-simple complex L * -algebra and h C a Cartan subalgebra of g C . A root of g C with respect to h C is defined, as in the finite dimensional case, as an element α in the dual of h C such that the corresponding "eigenspace" is non-empty. In the following the set of non-zero roots with respect to a given Cartan subalgebra will be denoted by R. Let us remark that a root has operator norm less than 1 and that for a non-zero root α, the vector space V α is one-dimensional (see [13]). The Jacobi identity implies that By relation (1), The main achievement in [14] is to prove that a semi-simple complex L * -algebra g C admits a Cartan Decomposition with respect to a given Cartan subalgebra h C in the sense that g C is the Hilbert sum Let us remark that in a separable L * -algebra, the set of root is countable or finite. By Zorn's Lemma, one can decompose the set R of non-zero roots into two disjoint subsets R + and R − such that α ∈ R + ⇔ −α ∈ R − . Such a decomposition defines a strict partial ordering on R by In the sequel, a decomposition R = R + ∪ R − as before and the induced ordering on the set of non-zero roots will be identified. The elements in R + will be called positive roots.
For every positive root α, one can choose e α ∈ V α such that e α = 1. Then e * α ∈ V −α and e * α = 1. This choice made, we define e α := e * −α for α ∈ R − , in order to have, for every α ∈ R, the following relation e * α = e −α . By (3), the set {e α , α ∈ R} is an Hilbert basis of h C ⊥ , and by (2), [e α , e * α ] belongs to h C . We define the following elements in the Cartan subalgebra h C : A positive root is called simple if it can not be written as the sum of two positive roots. The set of simple roots will be denoted by S. A subset N of the set of non-zero roots R is called a root system, if it satisfies the following conditions : A subset N ⊂ R is called indecomposable if it can not be written as the union of two orthogonal nonempty subsets. As in the classical theory, one has the following facts. The set R of non-zero roots of a simple L * -algebra is indecomposable. If F is an indecomposable subset of the set of non-zero roots R, then it generates a root system N F , which is again indecomposable. The simple L * -algebra generated by {e α , α ∈ N F } will be denoted by g(N F ).
For the classification of Hermitian-symmetric affine coadjoint orbits given in next section, we will need the following results. They were proved by J. R. Schue in [13] in ordering to classify the complex simple infinite-dimensional L * -algebras.

Proposition 2.1 ([13])
For every finite subset F of the set of non-zero roots R of a simple L * -algebra, there exists a finite indecomposable system of non-zero roots containing F .
the set of non-zero roots with respect to a given Cartan subalgebra of g C . For every n ∈ N \ {0}, set F n := {α 1 , . . . , α n }. Then there exists a sequence {N n } n∈N\{0} of finite subsets of R such that . the simple subalgebras g(N n ) generated by N n form a strictly increasing sequence with 5. The simple complex finite-dimensional algebras g(N n ) are of the same Cartan type A, B, C or D.
Given a sequence {N n } n∈N\{0} as in the previous Theorem, it is possible to define a total ordering on the vector space generated by the set of roots such that : 3. If α > 0 and α / ∈ N n then α > β for all β ∈ N n ; 4. the induced ordering on N n is a lexicographical ordering with respect to a basis of roots. 3. the elements in S are linearly independent on the reals and every positive root α ∈ R + is a linear combination of elements in S with non-negative integral coefficients which are almost all zero.

Classification of irreducible Hermitian-symmetric affine coadjoint orbits
Affine coadjoint orbits have been introduced in particular by K.-H. Neeb in [10]. The classification of finite-dimensional Hermitian-symmetric coadjoint orbits using the notion of roots of non-compact type has been carry out by J. A. Wolf in [16]. In the sequel, g will denote an infinite-dimensional separable simple L * -algebra of compact type. According to [1], [6] or [17], it can be realized as a subalgebra of the L * -algebra gl 2 (H) consisting of Hilbert-Schmidt operators on a separable complex Hilbert space H. Let G be the connected L * -group with Lie algebra g, and G C the connected complex L * -group with Lie algebra g C := g ⊕ ig. By the duality g ′ = g given by the trace, we can identify affine adjoint and affine coadjoint orbits of G. Let D be a derivation of g such that the affine (co-)adjoint orbit O of 0 in g associated to the affine adjoint action of G defined by D is strongly Kähler (see [10]). By Theorem 4.4 in [10], there exists D ∈ B(H) satisfying D * = −D such that for all x in g, Dx = [D, x], as well as a Cartan subalgebra h C of g C which is contained in ker D. To emphasize the relation between the orbit and the bounded operator D, we will often write O = O D . Abusing slightly the notation, we will sometimes denote D by ad(D). An alternative definition of O D is and the affine adjoint action of G on g is given by g · a = Ad(g)(a) + gDg −1 − D where g ∈ G and a ∈ g. The subalgebra k of g which fixes 0 is It is an L * -subalgebra of g. Let K be the associated connected L * -group. The affine (co-)adjoint orbit O D is called (locally-)symmetric if the orthogonal m of k in g satisfies The strongly Kähler orbit O D is called Hermitian-symmetric if, for every x in O D , there exists a globally defined isometry s x (the symmetry with respect to x) preserving the complex structure, such that x is a fixed point of s x , and such that the differential of s x at x is minus the identity of T x O D . An Hermitiansymmetric orbit is (locally-)symmetric. In the following, we will be interested in Hermitian-symmetric orbits O D . Let k C and m C denote the complexifications of k and m respectively. Note that g C is the orthogonal sum of k C and m C with respect to the Hermitian product of the L * -algebra g C .
Proposition 3.1 Let h C be a Cartan subalgebra of g C that is contained in ker adD, and let be the associated Cartan decomposition of g C , where R denotes the set of non-zero roots with respect to h C . Suppose that O D is Hermitian-symmetric. Then there exists two subsets A and B of R such that A ∪ B = R and Let v be a non-zero vector in V α , and v = v 0 + v 1 his decomposition with respect to the direct sum But V α is one-dimensional, hence either v 0 = 0, or v 1 = 0. Consequently V α is contained either in k C or in m C . ✷ Let us denote by m + (resp. m − ) the closed subspace of g C generated by the e α 's, where α runs over the set of roots for which c α > 0 (resp. c α < 0). Let B + (resp. B − ) be the set of roots β in B such that V β ∈ m + (resp. V β ∈ m − ). It follows that [k, m ± ] ⊂ m ± and that m ± is stable under the adjoint action of K. Let us suppose that m + decomposes into a sum of two non-zero Ad(K)-modules m 1 and m 2 . Then and it follows that m decomposes also into the sum of two non-zero Ad(K)-modules, namely g∩(m 1 ⊕m * 1 ) and g ∩ (m 2 ⊕ m * 2 ). The orbit O D being irreducible, m is an irreducible Ad(K)-module and this leads to a contradiction. So the irreducibility of m ± is proved. Let e α be an element in m + and set c = c α : The kernel ker(D − ic) being an Ad(K)-module of m + , one has ad(D) |m+ = ic id |m+ . The relation Given an ordering on the set of non-zero roots R of g C , a simple roots φ is called of non-compact type (see [16]) if every root α ∈ R is of the form or of the form Lemma 3.7 Let O D be a Hermitian-symmetric affine adjoint irreducible orbit of a simple L * -algebra g, h C be a Cartan subalgebra of g C contained in ker adD, and be the associated Cartan decomposition of g C with For every ordering R = R + ∪ R − on the set of roots, there exists a unique simple root φ belonging to B.
△ Proof of Lemma 3.7: Let {φ i , Ψ j } i∈I,j∈J be the set of simple roots with φ i in B and Ψ j in A. Let us suppose that I is empty. The relation [k C , k C ] ⊂ k C implies that every positive root belongs to A and consequently m = {0}, which contradicts the hypothesis that m is a non-zero irreducible Ad(K)-module. Let φ be a simple root in B. The closed vector space spanned by the adjoint action of k on e φ is a non-zero irreducible Ad(K)-module of m C . It follows that φ is necessarily unique. △ Lemma 3.8 Under the hypothesis of Lemma 3.7, there exists an increasing sequence of finite indecomposable root systems N n such that 1. R = ∪ n∈N\{0} N n ; 2. all the finite-dimensional subalgebras g(N n ) generated by N n belong to the same type A, B, C, or D and g C is the closure of the union of the subalgebras g(N n ); 3. φ is a simple root of non-compact type for each subalgebra g(N n ) with respect to the ordering on the roots of g(N n ) induced by the ordering on R defined in Proposition 2.3.
△ Proof of Lemma 3.8: Let {α 1 , . . . , α n , . . .} be a numbering of the roots in A. Set F n = {α 1 , . . . , α n }. Let us construct by induction an increasing sequence of finite indecomposable root systems N n as follows. By Proposition 2.1, there exists a finite indecomposable root system N 1 containing {φ} ∪ F 1 . Suppose that N n−1 is constructed, then there exists a finite indecomposable root system N n containing F n ∪N n−1 . Since every root in B is the sum of φ and roots in A, R = ∪ n∈N\{0} N n . The sequence of finite-dimensional simple subalgebras g(N n ) generated by the root systems N n is increasing and such that g C = ∪ n∈N\{0} g(N n ).
Since there exists only 9 types of finite-dimensional simple algebras, at least one type occurs an infinite number of times. Since g C is infinite-dimensional and since only the types A, B, C, or D corresponds to algebras of arbitrary dimension, at least one of the types A, B, C, or D occurs an infinite number of times. It follows that there exists a subsequence N n k of N n such that all the subalgebras g(N n k ) are of the same type A, B, C, or D. Let S n k be the set of simple roots of g(N n k ) with respect to the ordering induced by the ordering on R defined in Proposition 2.3. By Proposition 2.4, S n k = S ∩ g(N n k ), where S is the set of simple roots of g C . For every positive root γ in N n k , there exists a finite sequence {γ i , i = 1, . . . , k} of roots in S n k such that and such that the partial sums γ 1 + · · · + γ j , 1 ≤ j ≤ k are roots (see [4] Since ad(D) preserves k C , m + and m − , it follows that for γ in A ∩ R + , card ({i, γ i = φ}) = 0 and for γ in B ∩ R + , card ({i, γ i = φ}) = 1. Consequently φ is of non-compact type. △ Proposition 3.9 Let O = G/K be a Hermitian-symmetric irreducible affine coadjoint orbit of an L *group G of compact type, and g = k⊕m the associated decomposition of the Lie algebra g of G, where k is the Lie algebra of the isotropy group K. Then there exists an increasing sequence of finite-dimensional subalgebras g n of g, of the same type A, B, C or D, and an increasing sequence of subalgebras k n of k such that 1. g = ∪g n 2. k = ∪k n 3. for every n ∈ N \ {0}, the orthogonal m n of k n in g n satisfies [k n , m n ] ⊂ m n and [m n , m n ] ⊂ k n , hence (g n , k n ) is a symmetric pair.
✷ Proof of Proposition 3.9 : This is a direct consequence of Lemma 3.8, with g n = g ∩ g(N n ) and k n = k ∩ g(N n ). ✷ From the discussion above it follows that the classification of Hermitian-symmetric irreducible affine coadjoint orbits of L * -groups of compact type can be deduced from the knowledge of the simple roots of non-compact type of finite-dimensional simple algebras (see the proof of Theorem 1.1 below). A simple root of a simple finite-dimensional algebra is of non-compact type if and only if it appears with the coefficient +1 in the expression of the greatest root. We recall the list of simple roots of non-compact type in the finite-dimensional Lie algebras of type A, B, C, or D in tabular 1 (see [5] or [16]). Only the root αn is of non-compact type.
Only the root α1 is of non-compact type.
Only the roots α1, α2 and αn are of non-compact type.  = kI 0 . Denote by H C the C-extension of H R and by Z + (resp. Z − ) the eigenspace of the C-linear extension of I 0 with eigenvalue +i (resp. −i). One has H C = Z + ⊕ Z − . The homogeneous space Z(H) injects into the restricted Grassmannian of the polarized Hilbert space H C = Z + ⊕ Z − as a totally geodesic submanifold via the application which maps a complex structure I to the subspace consisting of (1, 0)-type vectors X with respect to I, i.e. satisfying IX = iX.
The Grassmannian L(H) = Sp 2 (H)/U 2 (H + ) of Lagrangian subspaces close to H + is the Sp 2 (H)affine adjoint orbit of 0 for the derivations given by the bounded operators D (∞) l,l = il p + − ilp − . It is a totally geodesic submanifold of the restricted Grassmannian Gr res .