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Hyperkaehler structures and infinite-dimensional Grassmannians

In this paper, we describe an example of a hyperkaehler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can furthermore be identified either with the cotangent space of a connected component of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkaehler manifolds. In addition, Kaehler potentials are computed using Kostant-Souriau's theory of prequantization.


Introduction
The restricted Grassmannian in a Hilbert manifold closely related to Loop groups (see [38], [41]) and equations of the KdV type (see [39], [37], [40], [43]). It is also a Hermitian symmetric space. Therefore it is natural to ask (see below) whether its complexification or its cotangent space admit hyperkähler structures.
Recall that a Kähler manifold of finite dimension is a Riemannian manifold endowed with a complex structure that is parallel for the Levi-Civita connection, or equivalently, a manifold equipped with a closed symplectic real form, called the Kähler form, and a compatible integrable complex structure. A hyperkähler manifold of finite dimension is a manifold endowed with a Riemannian metric g and three complex structures I, J, K such that : IJK = −1, and g is Kähler with respect to each complex structure. Hence a hyperkähler manifold admits three Kähler forms ω 1 , ω 2 , et ω 3 , and the choice of one complex structure, for instance I, allows to build a holomorphic symplectic form, namely Ω = ω 2 + iω 3 .
D. Kaledin and B. Feix have found independently in [16] and [12] that, given a finite-dimensional manifold N endowed with a Kähler metric g N , there exists a hyperkähler metric g defined on a neighborhood of the zero section of the cotangent space T * N of N , compatible with the natural holomorphic symplectic structure of the cotangent space, and such that the restriction of g to N is g N . In addition, g is unique if one requires S 1 -invariance. D. Kaledin uses for his proof the theory of Hodge manifolds, whereas B. Feix uses twistor spaces. As far as we know, this result has not been extended to the infinitedimensional Banach setting. Moreover this existence result does not lead to an explicit expression of the metric and examples of explicit hyperkähler metrics are rare.
O. Biquard and P. Gauduchon provided in [4] a construction of hyperkähler metrics on cotangent bundles of Hermitian symmetric spaces, and in [5] hyperkähler metrics on coadjoint orbits of symmetric type of a complex semi-simple Lie group. Furthermore they established formulas for Kähler potentials that allow explicit expressions of these metrics. In [6] the same authors identified these hyperkähler structures, showing that the cotangent space and the complexified coadjoint orbit are, in the case of Hermitian symmetric spaces of finite dimension, two aspects of the same hyperkähler object, which appear in accordance with the chosen complex structure within the 2-sphere of complex structures.
In the case of the cotangent space of the Grassmannian Gr(p, n) of subspaces of dimension p in C n , the aforementioned hyperkähler structure can be obtained by a hyperkähler quotient, and the corresponding Kähler potential can be computed via the theory of Kostant-Souriau's prequantization. It is the study of this particular example of Hermitian symmetric space that leads to the theory of O. Biquard and P. Gauduchon as developed in [4]. In the present work, we show that each connected component Gr j res (j ∈ Z) of the restricted Grassmannian Gr res introduced by Pressley and Segal in [38], gives an example of an infinite-dimensional Hermitian symmetric space whose cotangent space can be obtained by an infinite-dimensional hyperkähler quotient of a Banach manifold by a Banach Lie group. Moreover, we show that the resulting quotient space can also be identified with a natural complexification O C j of Gr j res , also called complexified orbit, consisting of pairs (P, Q) of elements of Gr j res such that P ∩ Q ⊥ = {0}. In this way, the study of this particular example provides a first step towards the generalization of the aforementioned results of O. Biquard and P. Gauduchon to the infinite-dimensional setting. The full generalization of these results has been carried out in [49] by the construction of hyperkähler metrics on complexifications of Hermitian-symmetric affine coadjoint orbits of semi-simple L * -groups of compact type, and by the identification of these complexifications with the cotangent spaces of the orbits under consideration. This generalization is based on Mostow's Decomposition Theorem (see [48]) and on the notion of strongly orthogonal roots of a L * -algebra (see the Appendix in [49]).
The theory of symplectic quotients was initiated by J. E. Marsden and A. Weinstein in [28]. In finite dimension it was used in particular to construct new examples of symplectic manifolds. In [27] J. E. Marsden and T. Ratiu applied this theory to infinite-dimensional manifolds and obtained new developments of V. I. Arnold's idea that fluid motion equations are the equations of geodesics on a suitable infinite-dimensional Lie group. Another example of infinite-dimensional symplectic reduction is given by J. E. Marsden and A. Weinstein in [30] in relation with the Maxwell-Vlasov equation. (For an overview of applications of the symplectic reduction see [29].) Kähler and hyperkähler reductions are refinements of this theory. An infinite-dimensional version based on the study of Nahm's equations was used by P. B. Kronheimer in [22] and [23] in order to construct hyperkähler structures on maximal semisimple and nilpotent coadjoint orbits of semi-simple complex (finite-dimensional) Lie groups. These results were generalized to all orbits by O. Biquard in [3] and A. G. Kovalev in [21]. In [24], the study of the same Nahm's equations but with different boundary conditions leads P.B. Kronheimer to prove that the cotangent space of any complex Lie group carries a hyperähler structure.
In [15], V. G. Kac classifies infinite-dimensional Lie groups and algebras into four (overlapping) categories: 1) Groups of diffeomorphisms of manifolds and the corresponding Lie algebras of vector fields; 2) Lie groups (resp. Lie algebras) of maps from a finite-dimensional manifold to a finite-dimensional Lie group (resp. Lie algebra); 3) Classical Lie groups and algebras of operators on Hilbert and Banach spaces; 4) Kac-Moody algebras. The examples of infinite-dimensional reduction mentioned above concern only the first two classes of groups. In this work we construct an example of hyperkähler reduction involving the third class of groups.
The structure of the paper is as follows. In Section 2, we introduce the necessary background on Kähler and hyperkähler quotients of Banach manifolds as well as the theory of Kähler potential on a quotient induced by Kostant-Souriau's theory of prequantization. In the infinite-dimensional setting, the definition of a hyperkähler manifold has to be specified so as to avoid problems such as the possible non-existence of a Levi-Civita connection for weak Riemannian metrics. The conditions needed to get a smooth Kähler structure on a Kähler quotient of a Banach manifold by a Banach Lie group have to be strengthened in comparison to the finite-dimensional case. In this Section, a basic definition of the notion of stable manifold associated with a level set and an holomorphic action of a complex Banach Lie group on a Kähler manifold is given. It is the most adapted to our purpose, but can be related to more sophisticated definitions as the one appearing in the Hilbert-Mumford Geometric Invariant Theory (see [33]) or the one appearing in Donaldson's work (see [11] for an exposition of the circle of ideas around this notion). In Subsection 2.2, the existence of a smooth projection of the stable manifold to the level set is proved and used in the identification of a smooth Kähler quotient with the complex quotient of the associated stable manifold by the complexification of the group. In Subsection 2.4, we give a survey of the theory of Kähler potential on a Kähler quotient which includes a natural generalization to the Banach setting of the formula for the Kähler potential proved by O. Biquard and P. Gauduchon in Theorem 3.1 in [4].
In Section 3, we construct a smooth hyperkähler quotient of the tangent bundle T M k of a flat non-reflexive Banach space M k (indexed by k ∈ R * ) by a Banach Lie group G. The key point in the proof of this result is the existence of a G-equivariant slice of the tangent space to the level set, which is orthogonal to the G-orbits and allows one to define a structure of smooth Riemannian manifold on the quotient. As far as we know, the general procedure for finding closed complements to closed subspaces of a Banach space recently developed by D. Beltiţȃ and B. Prunaru in [2] does not apply in our cases, so that the existence of closed complements has to be worked out by hand. For this purpose, the properties of Schatten ideals (see [44]) are extensively used.
In Section 4, we show that the quotient space obtained in Section 3 can be identified with the cotangent bundle of a connected component Gr j res of the restricted Grassmannian , which is therefore endowed with a (1-parameter family of strong) hyperkähler structure(s). To prove this identification, we use the stable manifold associated with one of the complex structures of the quotient space and the general results of Section 2. At the end of Section 4, we compute the Kähler potential associated with this complex structure using the theory explained in Subsection 2.4, and we give an expression of this potential using the curvature of Gr res . The formulas obtained by this method are analogous to the ones proved by O. Biquard and P. Gauduchon in the finite-dimensional setting. By restriction to the zero section, the theorems proved in Section 4 realize each connected component of the restricted Grassmannian as a Kähler quotient and provide the expression of the Kähler potential of the restricted Grassmannian induced by Plücker's embedding (for a description of Plücker embedding, see [38] and [45]). The realization of the restricted Grassmannian as a symplectic quotient was independently obtained by T. Wurzbacher in unplublished work and explained in numerous talks (see [51]).
In Section 5, we show that the hyperkähler quotient constructed in Section 3 can also be identified with a natural complexification O C j of Gr j res . For this purpose, we use various equivalent definitions of the complexified orbit O C j . To give an explicit formula of the Kähler potential associated with this complex structure, we use an invariant of the G C -orbits. The expression of the potential as a function of the curvature of Gr res obtained by this method is again analogous to the one given by O. Biquard and P. Gauduchon in [4]. An equivalent expression, in terms of characteristic angles of a pair of subspaces (P, Q) ∈ O C j is also given.
2 Background on Kähler and hyperkähler quotients of Banach manifolds

Kähler quotient
Let M be a smooth Banach manifold over the field K = R or C, endowed with a smooth action of a Banach Lie group G (over K), whose Lie algebra will be denoted by g. For a Banach space B over K, we will denote by B ′ the topological dual space of B, i.e. the Banach space of continuous linear applications from B to K.
is an injection.
Definition 2.2 A moment map for a G-action on a weakly symplectic Banach manifold M is a map for all x in M and for all a in g, where X a denotes the vector field on M generated by the infinitesimal action of a ∈ g. The G-action is called Hamiltonian if there exists a G-equivariant moment map µ, i.e. a moment map satisfying the following condition : µ(g · x) = Ad * (g) (µ(x)) .

Definition 2.3
A regular value of the moment map is an element ξ ∈ g ′ such that, for every x in the level set µ −1 (ξ), the map dµ x : T x M → g ′ is surjective and its kernel admits a closed complement in T x M.

Remark 2.4
If ξ is a regular value of µ, then µ −1 (ξ) is a submanifold of M. If µ is G-equivariant and if ξ is an Ad * (G)-invariant element of g ′ , then the manifold µ −1 (ξ) is globally G-stable, and one can consider the quotient space µ −1 (ξ)/G.
In the following, we consider an Hamiltonian action of G on M and a regular Ad * (G)-invariant element ξ of g ′ . We recall some classical results on the topology and geometry of the quotient space.  8]) Assume that G acts freely and properly on N . If, for every x ∈ N , the tangent space T x G·x to the orbit G·x at x is closed in T x N and admits a closed complement, then the quotient space N /G has a unique structure of Banach manifold such that the projection π : N → N /G is a submersion.
Remark 2.8 Let b be a continuous bilinear form on a Banach vector space B. Suppose that b realizes an injection of B into its topological dual B ′ byb(X) := b(X, · ), for X ∈ B. For any linear subspace A of B, we have the inclusionĀ but not necessarily the equality. The equality means that any continuous linear form vanishing on A ⊥ b is of the formb(X) for some X ∈ A, which is a particular property of the subspace A. Along the same lines, if b is a positive definite symmetric bilinear form on B, we have : but in general we do not have : even if A is closed, since the right hand side may not be closed. Furthermore J. Lindenstrauss and L. Tzafriri have proved in [26] that a Banach space in which every closed subspace is complemented is isomorphic to a Hilbert space. This implies in particular that for a non-reflexive Banach space B endowed with a weak Riemannian metric, equality (1) is certainly not fulfilled by every closed subspace A.
Proposition 2.9 If µ −1 (ξ)/G has a Banach manifold structure such that the quotient map is a submersion, and if G acts by symplectomorphisms, the condition implies that µ −1 (ξ)/G is a weakly symplectic manifold.
✷ Proof of Proposition 2.9: Denote by π the quotient map π : µ −1 (ξ) → µ −1 (ξ)/G. Let us show that the following expression where X, Y are in T [x] (µ −1 (ξ)/G) and where π * X = X and π * Ỹ = Y , defines a weak symplectic structure on the quotient. Note that for all x ∈ µ −1 (ξ), the tangent space T x (µ −1 (ξ)) is precisely the kernel of the differential dµ x , so that for all a ∈ g, the 1-form i X a ω vanishes on µ −1 (ξ). This implies that the right-hand side of (2) does not depend on the choice ofX andỸ . Since G acts by symplectomorphisms, it does not depend on the choice of the element x in the class [x] either. It follows that π * ω red = ω |µ −1 (ξ) . Since ω is closed, so is ω red . The kernel of ω red at a point [x] in the quotient space is : Note that the tangent space T x G·x to the G-orbit of x is spanned by {X a (x), a ∈ g}, hence we have and For ω red to be symplectic, one needs T x µ −1 (ξ) ⊥ω = T x G·x, which is precisely the hypothesis. ✷ Recall the following definition : , H x is a closed complement to the tangent space T x G·x of the G-orbit G·x, and such that for all x in µ −1 (ξ) and for all g in G.
Remark 2.11 Suppose that the manifold M is endowed with a weakly Riemannian G-invariant metric g. Then the existence of a G-invariant slice H of µ −1 (ξ) allows one to define a weakly Riemannian metric g red on the quotient µ −1 (ξ)/G, as follows. For every x ∈ µ −1 (ξ), we set whereX andỸ are the unique elements of H x such that π * (X) = X and π * (Ỹ ) = Y .
Definition 2.12 A weak Kähler manifold is a Banach manifold M endowed with a weak symplectic form ω and a weak Riemannian metric g (i.e. such that at every x in M, g defines an injection of T x M into its dual), satisfying the following compatibility condition : (C) the endomorphism I of the tangent bundle of M defined by g(IX, Y ) = ω(X, Y ) satisfies I 2 = −1 and the Nijenhuis tensor N of I vanishes.
Recall that the Nijenhuis tensor has the following expression at x in M : where X and Y belong to T x M.
Theorem 2.13 Let M be a smooth Kähler Banach manifold endowed with a free and proper Hamiltonian action of a Banach Lie group G preserving the Kähler structure. If, for every x in the preimage µ −1 (ξ) of an Ad * (G)-invariant regular value of the moment map µ, the tangent space T x G·x of the orbit G·x satisfies the direct sum condition then the quotient space M//G := µ −1 (ξ)/G is a smooth Kähler manifold.
Proof of Theorem 2.13: Let us denote by (g, ω, I) the Kähler structure of M. For x in µ −1 (ξ), the condition implies that the tangent space of the orbit G·x satisfies the following property : (the converse may not be true). In particular, T x G·x is closed and splits, so one is able to define on the quotient space a Banach manifold structure by use of Proposition 2.7. Now equality (3) is equivalent to since I is orthogonal with respect to g. So the condition needed for the definition of the symplectic structure on the quotient in Proposition 2.9 is fulfilled. Moreover the orthogonal H x := (T x G·x) ⊥g of T x G·x in T x µ −1 (ξ) defines a G-equivariant slice for the manifold µ −1 (ξ), and, by Remark 2.10, allows one to define a Riemannian metric on the quotient. It remains to define a compatible complex structure I red on µ −1 (ξ)/G. For this purpose, let us remark that since ω(X a , IU ) = g(IX a , IU ) = g(X a , U ) = 0 for all a in g and U in H x . In addition, IH x is orthogonal to T x G·x with respect to g since g(IU, X a ) = ω(U, X a ) = 0 for U ∈ T x µ −1 (ξ) = (T x G·x) ⊥ω . This implies that H x is stable under I and one can define a complex structure I red on the quotient space by: where x is in [x] and whereX is the unique element of H x whose projection on T [x] µ −1 (ξ)/G is X. Hence one has : I red π * X = π * IX.
The application π being a submersion, given two vector fields X and Y on µ −1 (ξ)/G, one has : where againX satisfiesX(x) ∈ H x and π * (X) = X, and similar conditions forỸ . Therefore the formal integrability condition on I implies the formal integrability condition on I red since the Nijenhuis tensor of I red has the following expression: Remark 2.14 In contrast to the finite-dimensional case illustrated by the Newlander-Nirenberg Theorem (see [34]), the formal integrability condition of an almost complex structure I on a Banach manifold M given by a vanishing Nijenhuis tensor is not sufficient for M to admit a system of holomorphic charts. An example of a formally integrable complex structure on a real Banach manifold which does not admit any open subset biholomorphic to an open subset of a complex Banach manifold was recently constructed by I. Patyi in [35]. However, if M is a real analytic manifold and I a formally integrable analytic complex structure, then M can be endowed with a holomorphic atlas (see [36], and [1] for the details of this result). Note also that, in the Fréchet context, L. Lempert showed in [25] that the complex structure defined in [31] by J. E. Marsden and A. Weinstein on the space of knots does not lead to the existence of holomorphic charts, although this structure was shown to be formally integrable by J. L. Brylinski in [10]. In the context of formally integrable complex structures, we will call a map f between two complex manifolds (M, I M ) and (N,

Stable manifold
Let (M, ω, g, I) be a smooth Kähler Banach manifold endowed with a smooth Hamiltonian action of a Banach Lie group G preserving the Kähler structure. Let ξ be an Ad * (G)-invariant regular value of the moment map µ. Assume that there exists a complex Lie group G C with Lie algebra g C := g ⊕ ig which acts holomorphically and smoothly on M extending the action of G, and that the following assumption holds : (Note that, by definition of the moment map, one has T x µ −1 (ξ) = (I T x G·x) ⊥g , so (H) states that the direct sum of I T x G·x and its orthogonal is closed in T x M, which is not always the case as mentioned in Remark 2.8, but nevertheless a natural assumption to make.) The action of G C on M allows one to define a notion of stable manifold associated with the level set µ −1 (ξ) :

Remark 2.16
The assumption (H) implies that M s is open in M since T x M s = T x M for every element x in µ −1 (ξ), hence, by translation by an element of G C , for every x in M s .

Proposition 2.17
If G C admits a polar decomposition G C = exp ig·G , then for every x in µ −1 (ξ), one has: ✷ Proof of Proposition 2.17: The argument below has already been used by F. Kirwan (see [17], [18], [19]). It goes like this. Clearly Suppose that there exists g ∈ G C such that g · x ∈ µ −1 (ξ). Since µ −1 (ξ) is G-invariant and since G C = exp ig·G, it is sufficient to consider the case when g = exp ia , a ∈ g. Define the function h : R → R by h(t) = µ ((exp ita) · x) (a). One has h(0) = h(1) = ξ(a), hence there exists t 0 ∈ (0, 1) such that : Hence a · y = 0 and exp(iaR) fixes y, thus also x. It follows that : ✷ From now on and till the end of Section 2, it will be assumed that G C admits a polar decomposition.

Corollary 2.18
If G acts freely on µ −1 (ξ), then G C acts freely on M s . ✷ Proof of Corollary 2.18: Let x be an element of µ −1 (ξ) and let g ∈ G C be such that g · x = x. Since G C = exp ig·G, there exists u ∈ G and a ∈ g such that g = exp(ia)u, and one has exp(ia)u · x = x. From the proof of the previous Proposition , one has: It follows that ux = x, thus u = e, since G acts freely on µ −1 (ξ). Now, the condition exp(iaR) · x = x implies that a fixes x, hence a = 0. ✷ Proposition 2.19 Assume that G acts freely on µ −1 (ξ). Then, for every y in M s , there is a unique element g(y) in exp ig such that g(y) maps y to the level set. The resulting application is smooth and the projection q q : is smooth and G-equivariant.
✷ Proof of Proposition 2.19 : Let y be in M s and suppose that there exist two elements a and b in g such that both exp ia · y and exp ib · y belong to µ −1 (ξ). Since exp ia · y and exp ib · y are in the same G C -orbit, by Proposition 2.17, there exists u in G such that exp ia · y = u · exp ib · y. Since G acts freely on the level set, Corollary 2.18 implies that G C acts freely on M s . It follows that exp ia = u · exp ib, hence, by uniqueness of the polar decomposition, u is the unit element of G and exp ia = exp ib. Therefore g(y) = exp ia is well defined, and so is the projection q.
Let us show that the application : is smooth. Since G C acts smoothly on M s , it will imply that the projection q is smooth also. Consider the following map : which maps exp ig × M s onto M s . (Recall that exp ig inherits a Banach manifold structure from its identification with the homogeneous space G C /G.) We will prove that φ is transversal to µ −1 (ξ) (see §5.11.6 and §5.11.7 of [9] for a definition of this notion), so that the subset is a smooth submanifold of exp ig × M s . The smoothness of the application g will therefore follow from the smoothness of the projection p 1 : exp ig × M s → exp ig on the first factor. We will denote by R y the right translation by y on G C . The differential of φ at a point (exp ia, y) in exp ig × M s reads : Note that, for every element (exp ia, y) in φ −1 µ −1 (ξ) , one has : is complemented. For this purpose recall that, by assumption (H), the tangent space so that the tangent space × g by the following isomorphism : . Since G acts freely on M, it follows that the subspace : is a closed complement to it. Let us check the G-equivariance of q. Since µ is G-equivariant and ξ is Ad * (G)-invariant, one has : for all u in G, and y in M s . We can write g(y) = exp ia for some a ∈ g. Now the equality u · exp ia = exp (Ad(u)(ia)) · u and the uniqueness of the element g(u · y) satisfying g(u · y) · (u · y) ∈ µ −1 (ξ) proved above, imply that : Hence q satisfies the G-equivariant condition : for all u ∈ G and y ∈ M s . ✷ Let us show thatC is closed in M s × M s . Denote by C the graph of the equivalence relation defined by the action of G on µ −1 (ξ). Let {(y n , v n · y n )} n∈N be a sequence inC, where y n ∈ M s and v n ∈ G C , which converges to an element (y ∞ , z ∞ ) in M s × M s . From Proposition 2.19 and from the continuity of the projection q, the sequence {(q(y n ), q(v n · y n ))} n∈N belongs to C and converges to (q(y ∞ ), q(z ∞ )).
Let us show that the canonical mapι fromC to G C is continuous. Denote by ι the canonical map from C to G. One has :ι (y, z) → g(z) −1 • ι(q(y), q(z)) • g(y), and the continuity ofι follows from the continuity of the applications ι, g and q. ✷ We conclude this Subsection with Theorem 2.21 Let M be a Banach Kähler manifold endowed with a smooth, free and proper Hamiltonian action of a Banach Lie group G, which preserves the Kähler structure and extends to a smooth holomorphic action of a complex Lie group G C = exp ig·G. Let ξ be an Ad * (G)-invariant regular value of the moment map µ. If for every satisfies the following direct sum condition then the quotient space M s /G C is a smooth complex manifold isomorphic to the smooth Kähler quotient M//G := µ −1 (ξ)/G as complex smooth manifold. Moreover the full integrability of the complex structure on M implies the full integrability of the complex structure on M s /G C , hence on M//G.
Proof of Theorem 2.21 : By Proposition 2.18 and Proposition 2.20, G C acts freely and properly on M s . The smoothness of the application g and the G-equivariant slice H of µ −1 (ξ) given by H x := (T x G·x) ⊥g , allows one to define a G-equivariant slice on M s , also denoted by H, by the following formula : for all y in M s . The following decomposition of the tangent space T y M s holds : By Proposition 2.7, it follows that M s /G C has a unique (real) Banach manifold structure such that the quotient map is a submersion. Since for every element y in M s , H q(y) is invariant under the complex structure I of M, and since the G C -action on M is holomorphic, for every y in M s the subspace H y of T y M s is I-invariant and, further, M s /G C inherits a natural complex structure. Moreover, since the complex structure of the quotient M s /G C comes from the complex structure of M, the natural injection µ −1 (ξ) ֒→ M s induces a complex isomorphism between µ −1 (ξ)/G and M s /G C . Finally, M s being an open subset of M because of (H), the existence of holomorphic charts on M allow one to apply Proposition 2.7 to the holomorphic quotient M s /G C , thus implies the existence of holomorphic charts on M s /G C .

Hyperkähler quotient
Let M be a Banach manifold endowed with a weak hyperkähler metric g, with Kähler forms ω 1 , ω 2 and ω 3 , and corresponding complex structures I 1 , I 2 , and I 3 . Let G be a connected Banach Lie group with Lie algebra g, acting freely on M by hyperkähler diffeomorphisms. We assume in this Subsection that there exists a G-equivariant moment map µ i for each symplectic structure. Define µ : then the quotient space µ −1 (ξ)/G carries a structure of smooth Banach hyperkähler manifold.
Proof of Theorem 2.22: This is a direct application of Theorem 2.13 with respect to each complex structure I 1 , I 2 , and I 3 of M. By the very definition of the complex structures and symplectic forms on the reduced space, µ −1 (ξ)/G inherits a hyperkähler structure from the hyperkähler structure of M.
Let us now assume that the orthogonal (T as a topological direct sum. For each complex structure I k = k 1 I 1 + k 2 I 2 + k 3 I 3 in the 2-sphere of complex structures on M, indexed by k = (k 1 , k 2 , k 3 ) in S 2 , let us define an action · k of ig on M by : for all a in g and for all x in M. Assume that for a given k, the action · k integrates into an I kholomorphic action of G C on M. Let us choose an orthogonal complex structure to I k denoted by I l , and define a third complex structure by I m := I k .I l , so that I k , I l , I m satisfies the quaternionic identities. Denote by µ k the combination µ k := k 1 µ 1 + k 2 µ 2 + k 3 µ 3 and similarly µ l and µ m . Consider ξ l := l 1 ξ 1 + l 2 ξ 2 + l 3 ξ 3 and ξ m := m 1 ξ 1 + m 2 ξ 2 + m 3 ξ 3 , and similarly ω l := l 1 ω 1 + l 2 ω 2 + l 3 ω 3 and ω m := m 1 ω 1 + m 2 ω 2 + m 3 ω 3 .

Proposition 2.23
The map µ C := µ l + iµ m is an G C -equivariant holomorphic moment map for the I k -complex symplectic structure ω l + iω m .
✷ Proof of Proposition 2.23: By the same algebraic arguments than the ones given in [14], section 3 (D), the map µ C is holomorphic and a moment map for ω l + iω m . Since G is connected, the G C -equivariance of µ C is will follow from for all x in M, and all a, b ∈ g, where the bracket denotes the duality pairing. But this is an easy consequence of the G-equivariance of µ l and µ m and the following identities where X a denotes the vector field generated by a ∈ g. ✷

Lemma 2.24
The stable manifold M s k with respect to the complex structure I k , associated with the level set µ −1 (ξ), is a submanifold of the Banach manifold M, contained in the preimage by . Let x be an element in the level set µ −1 (ξ). Since ξ is a regular value of µ, the level set µ −1 (ξ) is a Banach submanifold of M. Consider an adapted chart (V, E × F, ϕ) of the submanifold µ −1 (ξ) at x, where V is a neighborhood of x in M, E and F are two Banach spaces, and ϕ is an homeomorphism from V onto a neighborhood of 0 in E × F such that U 1 := ϕ µ −1 (ξ) ∩ V ⊂ E. By assumption (S), one has : For a in the Lie algebra g of G, denote by X a the vector field on M generated by a. Since G acts freely on M, the map g → I k (T x G·x) which assigns to a ∈ g the vector I k X a is a continuous bijection of Banach spaces, hence an isomorphism, and similarly for the indexes l and m. Since by hypothesis the G-action on M extend to a I k -holomorphic action of G C on M, the vector fields I k X a , for a ∈ g, are complete, whereas the flows t → f t I l X b +I m X c of the vector fields I l X b + I m X c , with b and c in g, may not be globally defined. Nevertheless, by the smooth dependance of the solutions of a differential equation with respect to a parameter, there exists a small neighborhood 1]. Now consider the following map : Then Ψ provides an adapted chart in the neighborhood of any y in the fiber q −1

Kähler potential on a Kähler quotient
This Subsection is a generalization to the Banach setting of results obtained in the finite-dimensional case by O. Biquard and P. Gauduchon in [4] (see Theorem 3.1 there), and based on an idea in [14]. In this Subsection, we will again make use of the setting of Subsection 2.2 and we will suppose that the complex structure I of M is formally integrable.
and where x belongs to M, and X 1 , . . . , X n are elements in T x M. This action allows one to define the corresponding differential operator d c by d c := IdI −1 . Note that dd c = 2i∂∂ where ∂ and∂ are the Dolbeault operators. Recall also the following definitions : In the following we will make the assumption that M admits a G-invariant globally defined Kähler potential, which will be the case in the next Section. Under this assumption, the action of G is Hamiltonian with respect to the moment map µ defined by: for all x in M, and for all a in g.
for all σ in Γ(L) and a in g. This action corresponds to the action of g on the total space of L which assigns to an element a ∈ g the vector fieldX a over L whose value at a point ζ ∈ L over x ∈ M is : whereX a (ζ) denotes the horizontal lift at ζ of X a for the trivial connection, and where T denotes the vertical vector field given by T (ζ) = ζ. This action integrates into an action of the group G if and only if the following integrability condition is satisfied : (I) a → iξ(a) is the differential of a group homomorphism from G to S 1 at the unit element e of G.
Suppose it is the case and denote by χ the homomorphism such that (dχ) e = −iξ. The homomorphism χ extends to a homomorphism of G C to C * , which will be also denoted by χ. The corresponding action of G C on L is given by : where g is an element in G C . This induces an action of G C on Γ(L) by : where σ belongs to Γ(L), g is an element in G C and x is in M. In the remainder of this Subsection, we will assume that the integrability condition (I) is satisfied.
Definition 2.28 LetL be the complex line bundle over µ −1 (ξ)/G obtained as the G-orbit space of the restriction of the trivial line bundle L to the level set µ −1 (ξ). The fiber ofL over an element [x] in Definition 2.29 Define an Hermitian scalar productĥ onL by : where, for i ∈ {1, 2}, σ i is the G-invariant section of L |µ −1 (ξ) whose projection to Γ(L) isσ i .

Proposition 2.30
Let σ be a G-invariant section of L |µ −1 (ξ) whose projection to Γ(L) will be denoted byσ, and let X be a vector field on µ −1 (ξ)/G whose horizontal lift with respect to an arbitrary Ginvariant connection∇ on the bundle µ −1 (ξ) → µ −1 (ξ)/G will be denoted byX. Then the identitŷ where ∇ denotes the Chern connection on L, defines a connection∇ onL, which is independent of∇ and for whichĥ is parallel.
Consequently ∇X σ = ∇X 1 σ, hence∇ is independent of the connection∇. To check that ∇X σ is G-invariant, note that for all a ∈ g, one has : The Hermitian line bundle (L,ĥ,∇) prequantifies the quotient µ −1 (ξ)/G in the sense that R∇ = iω red .
✷ Proof of Proposition 2.31 : For any vector fields X and Y on µ −1 (ξ)/G, and for every sectionσ ofL given by a G-invariant section σ of L |µ −1 (ξ) , one has : is tangent to the G-orbit and since, for every a ∈ g, the identity a.σ = −∇ X a σ − iµ a σ + iξ(a).σ = 0 implies that the covariant derivative of σ with respect to a vertical vector vanishes. ✷

Corollary 2.32
The complex structure of the Hermitian line bundle (L,ĥ) is formally integrable and the connection∇ is the Chern connection.
✷ Proof of Corollary 2.32 : The curvature ofL being of type (1, 1), the operator∂ :=∇ 0.1 defines a formally integrable complex structure onL. Moreover,∇ is C-linear and preservesĥ, thus it is the associated Chern connection. ✷ To proceed, suppose that we are under the assumptions of Theorem 2.21, so that we have a submersion: given by the identification of µ −1 (ξ)/G with M s /G C . Note that the differential of p satisfies : dp • I = I red • dp, where I and I red are the complex structures on M s and µ −1 (ξ)/G respectively. From ✷ Proof of Proposition 2.33 : Since dp satisfies dp • I = I red • dp, the pull-back by p of the Chern connection associated withĥ is the Chern connection of p * L with respect to p * ĥ . Thus R p * ∇ = ip * ω red . In addition, the fiber over x ∈ M s of the bundle p * L is : Since the element q(x) in the class [q(x)] is distinguished, the fiber (x, [L q(x) ]) can be identified with the fiber (x, L q(x) ). One can therefore define an isomorphism Φ of complex bundles by : Clearly Φ * ĥ =h. Now let∇ be the Chern connection of the trivial Hermitian bundle (L |M s ,h). Its curvature is : Since the bundle is a complex line bundle, one has : Proof of Theorem 2.34 : The curvature of the Chern connection of the Hermitian line bundle (L |M s ,h) is given by: where σ is the canonical section. Moreover: Thus: 3 An Example of hyperkähler quotient of a Banach manifold by a Banach Lie group

Notation
Let us summarize the notation used in the remainder of this paper. H will stand for a separable Hilbert space endowed with an orthogonal decomposition H = H + ⊕H − into two closed infinite-dimensional subspaces. The orthogonal projection from H onto H + (resp. H − ) will be denoted by p + (resp. p − ). The restriction of p ± to a subspace P will generally be denoted by pr ± and the space P be specified.
Given two Banach spaces E and F , the set of Fredholm operators from E to F will be denoted by Fred(E, F ), the Hilbert space of Hilbert-Schmidt operators from E to F will be denoted by L 2 (E, F ), the Banach space of trace class operators from E to F by L 1 (E, F ), and the Banach space of bounded operators from E to F by B(E, F ). The argument F in the previous operator spaces will be omitted when F = E.
The set of self-adjoint trace class operators on a complex Hilbert space E will be denoted by S 1 (E), and the set of skew-Hermitian trace class operators on E by A 1 (E). Similarly, S 2 (E) will stand for self-adjoint Hilbert-Schmidt operators on E, and A 2 (E) for skew-Hermitian Hilbert-Schmidt operators on E.
The unitary group of H will be denoted by U(H) and the identity map of a Hilbert space E by Id E . Ran(A) will stand for the range of an operator A and Ker(A) for its kernel.

Introduction
The restricted Grassmannian Gr res of H, studied for instance in [38] and [50], is defined as follows: Gr res (H) = { P closed subspace of H such that pr + : P → H + ∈ Fred(P, H + ) where pr ± denotes the orthogonal projection from P to H ± . The space Gr res is a Hilbert manifold and a homogeneous space under the restricted unitary group: Note that the stabilizer of H + is U(H + ) × U(H − ). The space Gr res is a strong Kähler manifold whose Kähler structure is invariant under U res . The expressions of the metric g Gr , the complex structure I Gr and the symplectic form ω Gr at the tangent space of Gr res at H + are : where X and Y belong to the tangent space T H+ Gr res which can be identified with L 2 (H + , H − ). Two elements P 1 and P 2 of Gr res are in the same connected component Gr j res , (j ∈ Z), if and only if the projections pr 1 + : P 1 → H + and pr 2 + : P 2 → H + have the same index j. In particular Gr 0 res denotes the connected component of Gr res containing H + .
The aim of this Section is to construct a hyperkähler quotient whose quotient space will be identified with the cotangent space of Gr 0 res in Section 4. This will make T ′ Gr 0 res into a strong hyperkähler manifold. In Section 5, the same quotient space will be identified with a complexified orbit of Gr 0 res , which will therefore carry a hyperkähler structure in its own right. Since all the constructions that follow can be carried out substituting an arbitrary element of another connected component Gr j res for H + and its orthogonal for H − , it will follow in particular that the cotangent space of the whole restricted Grassmannian is strongly hyperkähler and that it can be identified with the union of the complexifications of all connected components Gr j res . The latter union is nothing but the orbit of p + under the action by conjugation of the (non-connected) group GL res which is the complexification of U res . In other words, T ′ Gr res ≃ GL res ·p + .

A weak hyperkähler affine space T M k
For k ∈ R * , let M k be the following affine Banach space : Then M k is modelled over the Banach space injects into the continuous cotangent space T ′ M k of M k via the application : Thus T M k inherits a structure of weak complex symplectic manifold with symplectic form Ω given by Ω ((Z 1 , T 1 ) ; (Z 2 , T 2 )) = Tr (T * 1 Z 2 ) − Tr (T * 2 Z 1 ), where (Z 1 , T 1 ) and (Z 2 , T 2 ) belong to T (x,X) (T M k ). We will denote by ω 2 and ω 3 the real symplectic forms given respectively by the real and imaginary parts of Ω. Besides, from the natural inclusion of L 1 (H + ) into L 2 (H + ), it follows that T M k admits a natural weak Riemannian metric whose expression is g (x,X) ((Z 1 , T 1 ) ; (Z 2 , T 2 )) = ℜ Tr Z * 1 Z 2 + ℜ Tr T * 1 T 2 where (Z 1 , T 1 ) and (Z 2 , T 2 ) belong to T (x,X) (T M k ). The complex symplectic form Ω and the Riemannian metric g give rise to a hyperkähler structure on T M k with complex structures : The real symplectic form associated with I 1 is given by

Tri-Hamiltonian action of a unitary group G
Let G be the following Banach Lie group of unitary operators : The Lie algebra g of G is the Lie algebra of skew-Hermitian operators of trace class. We will denote by G C the complexification of G : and by g C its complex Lie algebra g ⊕ ig. The group G acts on T M k by for all u in G and for all (x, X) in T M k . This action extends to an I 1 -holomorphic action of G C by for all g in G C and for all (x, X) in T M k .
Proposition 3.1 The action of G C on T M k is Hamiltonian with respect to the complex symplectic form Ω, with moment map µ C : ✷ It follows from Proposition 3.1 that the real symplectic forms ω 2 and ω 3 are Hamiltonian with respect to the real moment maps : The action of G on T M k is Hamiltonian with respect to the real symplectic form ω 1 with moment map : ✷ Proof of Proposition 3.2 : One has to check that µ 1 satisfies : for all (Z, T ) in T (x,X) M k and all a in g. One has : ω 1 ((Z, T )) .

✷
In the following, we will denote by µ the g ′ ⊗ R 3 -valued moment map defined by: X) ).
In the next Subsection we will consider the Ad * (G)-invariant value ξ k := − i 2 k 2 Tr , 0, 0 of the moment map µ and the level set:

Smooth Banach manifold structure on the level set W k
This Subsection is devoted to the proof of the following Theorem : Tr , 0, 0 is a smooth Riemannian submanifold of T M k . We will prove that ξ k = − i 2 k 2 Tr , 0, 0 is a regular value of the moment map µ. For this purpose we will need the following fact, which will be useful in other parts of the paper, so that we single it out here : We have: The level set W k is the preimage of (0, k 2 ) under F . To prove that W k is a smooth Banach submanifold of T M k it is sufficient to prove that the differential of F at a point (x, X) of W k is surjective and that its kernel splits. For this purpose consider the following decomposition of H + : The operator x is a Fredholm operator hence it has closed range, and X is a compact operator. The equality X * x = 0 implies that the range Ran x of x is orthogonal to the range Ran X of X. From the continuity of the orthogonal projection of H onto Ran x it follows that: Ran x ⊥ Ran X. Let us introduce the following decomposition of H: With respect to the decompositions (4) and (5) of H + and H into closed subspaces, x and X have the following expressions : where x 31 and x 42 are continuous bijections, thus isomorphisms, and where X 12 is 1-1 but not onto. Let (Z, T ) be a tangent vector to T M k at (x, X), and denote by (Z ij ) 1≤i≤4, 1≤j≤2 and (T ij ) 1≤i≤4, 1≤j≤2 the block decompositions of Z and T with respect to the direct sums (4) and (5). Note that : and that x * Z + Z * x − X * T − T * X equals : To show that the differential of F is onto, consider an element (U, V ) of L 1 (H + ) × S 1 (H + ) and denote be (U ij ) 1≤i,j≤2 and (V ij ) 1≤i,j≤2 the block decompositions of U and V with respect to the direct sum H + = Ker X ⊕ ( Ker X) ⊥ . A preimage of (U, V ) by d (x,X) F is given by the following ordered pair (Z, T ): Let us now show that for every (x, X) in W k , the kernel of the differential d (x,X) F splits. It is given by the following subspace of T (x,X) M k : where a 1 (resp. a 2 ) is an element of the space A 2 ( Ker X) (resp. A 2 ( Ker X) ⊥ of skew-Hermitian Hilbert-Schmidt operators on Ker X (resp. on ( Ker X) ⊥ ). An ordered pair (Z, T ) ∈ L 2 (H + , H) × L 2 (H + , H) is in E ⊥ḡ if and only if Z and T have the following form: where s 1 (resp. s 2 ) is an element of the space S 2 ( Ker X) (resp. S 2 ( Ker X) ⊥ of self-adjoint Hilbert-Schmidt operators on Ker X (resp. ( Ker X) ⊥ ). We will denote by (p E (Z), p E (T )) the orthogonal projection of an element (Z, T ) ∈ L 2 (H + , H) × L 2 (H + , H) to E, and (p E (Z ij )) 1≤i≤4, 1≤j≤2 (resp. (p E (T ij )) 1≤i≤4, 1≤j≤2 ) the block decomposition of p E (Z) (resp. p E (T )) with respect to the direct sums (4) and (5). Then p E (Z) and p E (T ) are given by : with: Denoting by p + the orthogonal projection of H onto H + , let us prove that p + • p E (Z) and p + • p E (T ) are of trace class whenever p + (Z) and p + (T ) are of trace class. Let is of trace class. Thus we obtain the following direct sum of closed subspaces : and the Theorem follows.

Hyperkähler quotient of T M k by G
In this Subsection we will prove that the assumptions (D) and (S) of Subsection 2.3 are satisfied, as well as the following Theorem.
Theorem 3.5 The hyperkähler quotient of the weak hyperkähler space T M k with respect to the trisymplectic action of G provides W k /G with a structure of strong hyperkähler smooth Hilbert manifold.
For the remainder of the paper, we will denote by (g red , I red 1 , I red 2 , I red 3 , ω red 1 , ω red 2 , ω red 3 ) the hyperkähler structure induced by T M k on the reduced space W k /G.

Proof of Theorem 3.5:
Let us first show that the quotient space W k /G endowed with the quotient topology is Hausdorff. Since G acts freely on W k , by Propositions 2.5 and 2.6, it it sufficient to prove that the graph C of the equivalence relation defined by G is closed in W k × W k and that the canonical application from C to G is continuous.
Note that, for all (y, Y ) in W k , the operator y is an injective Fredholm operator from H + to H, thus establishes an isomorphism between H + and its (closed) range. We will denote by y −1 the inverse operator of y, and extend it to an operator from H to H by demanding that the restriction of y −1 to the orthogonal complement of Ran y vanishes.
Let us remark that the canonical map from the graph C to G assigns to an ordered pair ((x, X), (y, Y )) the element g = y −1 x. It is continuous with respect to the topology of C inherited from the topology of W k × W k , and the topology of G induced by the L 1 -norm on Id + L 1 (H + ).
Let { (x n , X n ) ; (x n • g −1 n , X n • g −1 n ) } be a sequence in C which converges to an element ((x, X); (y, Y )) in W k × W k . The sequence { g −1 n = x −1 n • (x n • g −1 n ) } is a sequence of elements of G which converges to the element x −1 y in Id + L 1 (H + ). Since G is closed in Id + L 1 (H + ), it follows that x −1 y is in G and that C is closed in W k × W k . By Theorem 2.22, for W k /G to be hyperkähler, it is sufficient to have the following topological direct sum, referred to as (D) : For this purpose, we will use Lemma 3.4. Let us again consider the space L 2 (H + , H) × L 2 (H + , H) endowed with the complex structure I(Z, T ) = (iZ, −iT ) and the strong Riemannian metricḡ given by the real part of the natural Hermitian product. Let E (resp. F ) be the closure of T (x,X) W k (resp. T (x,X) G·(x, X)) in L 2 (H + , H) × L 2 (H + , H). We will show that theḡ-orthogonal projection of E onto F maps T (x,X) W k onto T (x,X) G·(x, X). The orthogonal of F in E is the set of ordered pairs (Z, T ) ∈ L 2 (H + , H) × L 2 (H + , H) of the form : The orthogonal projection of E onto F maps an ordered pair (Z, T ) ∈ E having the following block decompositions with respect to the direct sums (4) and (5) to an ordered pair (p G (Z), p G (T )) with decompositions where a satisfies : Let us show that the projection p G from E to F maps an element of T (x,X) W k into T (x,X) G·(x, X). Let p + = p 11 p 12 p 13 p 14 p 21 p 22 p 23 p 24 be the expression of the orthogonal projection onto H + with respect to the direct sums (5) and (4).
For (x, X) ∈ W k , the operators p 11 , p 12 , p 14 , p 21 , p 22 , p 23 are of trace class as well as p 13 − x −1 31 and p 24 − x −1 42 . The condition p + (Z) ∈ L 1 (H + ) implies in particular that a 1 , Z 32 and a 2 are of trace class. To conclude that p G maps T (x,X) W k into T (x,X) G·(x, X), it remains to show that a defined by Equation (6) is of trace class. This will follow by lemma 3.6 below. As a consequence, p + • p G (Z) and p + • p G (T ) are of trace class. Hence, F ⊥ḡ ∩ T (x,X) W k is a closed complement to T (x,X) G·(x, X). Since g is the restriction ofḡ to T (x,X) W k , it follows that : The map E x42 defined by is an isomorphism which restricts to an isomorphism of L 1 ( Ker X) ⊥ .
△ Proof of Lemma 3.6 : Indeed, x * 42 x 42 is a self-adjoint positive definite operator on the space ( Ker X) ⊥ satisfying x * 42 x 42 = k 2 Id + X * 12 X 12 , where X * 12 X 12 is a compact operator. It follows that there exists a diagonal operator D with respect to an orthogonal basis {f i } i∈J of ( Ker X) ⊥ , and a unitary operator u ∈ U ( Ker X) ⊥ such that x * 42 x 42 = u D u * . Denote by D i the eigenvalues of D. Remark that D i > k 2 . It follows that the equation and implies in particular that, for every i and j in J, one has : To see that the map E x42 is surjective, consider an element V in L 2 (( Ker X) ⊥ ), and define an operator a ∈ L 2 (( Ker X) ⊥ ) by : . Since E x42 is clearly continuous, it follows that E x42 is an isomorphism of L 2 (( Ker X) ⊥ ) that restricts to an isomorphism of L 1 (( Ker X) ⊥ ). △ Proposition 3.7 For every (x, X) in W k one has : where H (x, X) is the orthogonal of T (x,X) G·(x, X) in T (x, X) W k .
✷ Proof of Proposition 3.7: With the previous notation, it follows from the proof of Theorem 3.3 that the orthogonal projection from L 2 (H + , H)×L 2 (H + , H) to T (x,X) W k with respect to the strong Riemannian metricḡ takes T (x,X) M k to T (x,X) W k . It also follows from the proof of Theorem 3.5 that the orthogonal projection from T (x,X) W k to T (x,X) G·(x, X) with respect toḡ takes T (x,X) W k onto T (x,X) G·(x, X). Now let us remark that the complex structures I j , j = 1, 2, 3, extend to complex structures of L 2 (H + , H) × L 2 (H + , H) by the same formulas, making L 2 (H + , H) × L 2 (H + , H) into a hyperkähler space. Since, for j = 1, 2, 3, the complex structure I j fixes T (x,X) M k and is orthogonal with respect toḡ, it follows that for j = 1, 2, 3, the orthogonal projection from L 2 (H + , H) × L 2 (H + , H) onto I j T (x,X) G·(x, X) takes T (x,X) M k onto I j T (x,X) G·(x, X) . Hence, from the orthogonal sum it follows that Recall that the complex Banach Lie group G C : for all g in G C , and for all (x, X) in T M k . Let W s1 k be the stable manifold associated with W k with respect to the complex structure I 1 , i.e. the union of G C -orbits (for the above action) intersecting W k . Since by the proof of Theorem 3.5 and Proposition 3.7, assumptions (D) and (S) are satisfied, one has the following Corollary of Lemma 2.24 and Theorem 2.25: k is a I 1 -complex submanifold of T M k and the quotient space W s1 k /G C is a smooth complex manifold. The map from W k /G to W s1 k /G C induced by the natural injection of W k into W s1 k is an I 1 -holomorphic diffeomorphism. ✷ In the following Proposition we give an explicit characterization of the stable manifold W s1 k and we compute the projection : defined by Proposition 2.19.
and, for all (x, X) in W s1 k , the unique element g (x,X) of exp ig such that g (x,X) · (x, X) belongs to W k is defined by ✷ Proof of Proposition 4.2 : Let A be the set of elements (x, X) in T M k such that X * x = 0 and x is one-to-one. Let us show that W s1 k ⊂ A. Consider (x, X) ∈ W s1 k and g ∈ G C such that g · (x, X) ∈ W k . We have: If g −1 = |g| −1 .u −1 denotes the polar decomposition of g −1 , the equality: Thus |g| −1 x * x|g| −1 = k 2 Id H+ + |g|X * X|g| is a positive definite self-adjoint operator. Since |g| −1 is an isomorphism, the same holds true for x * x. It follows that x is one-to-one. To see that A ⊂ W s1 k , consider an element (x, X) of A. We are looking for a positive definite self-adjoint operator g (x,X) such that : The operator x being one-to-one, x * x is positive definite and its square root (x * x) 1 2 is an invertible operator on H + . Hence it is sufficient to find a positive definite operator γ := (x * x) 1 2 g −1 (x,X) such that: The unique positive definite solution of the latter equation is : is positive definite and self-adjoint, and its square root satisfies the required condition. ✷ 4.2 Identification of W s 1 k /G C with the cotangent space T ′ Gr 0 res of the restricted Grassmannian In this Subsection , we will use the following Theorem to identify the complex quotient space W s1 k /G C with the cotangent bundle T ′ Gr 0 res of the connected component Gr 0 res of the restricted Grassmannian. Recall that Ran x denotes the range of an operator x.
is an I 1 -holomorphic submersion whose fibers are the orbits under the I 1 -holomorphic action of the complex group G C on W s1 k ⊂ T M k .
Proof of Theorem 4.3: For (x, X) in W s1 k , the range P of x is an element of Gr 0 res since p + •x belongs to {Id H+ +L 1 (H + )}, thus is a Fredholm operator with index 0, and p − • x is a Hilbert-Schmidt operator. Furthermore the condition X * x = 0 implies that the restriction of η := 1 k 2 x • X * to P vanishes. Thus η can be identified with an element of L 2 (P ⊥ , P ) which is the cotangent space of Gr 0 res at P . Let us check that Ψ is onto. For P in Gr 0 res , denote by x P the operator from H + to H whose columns are the vectors of the canonical basis of P as defined in [38]. Then k x P is in M k (see [47] for the details of this affirmation). On the other hand, for every V ∈ L 2 (P ⊥ , P ), the operator X defined by X := k 2 V * • x * −1 P (where x P is viewed as an isomorphism between H + and P ) satisfies 1 k 2 x P • X * = V and is an element of L 2 (H + , P ⊥ ). Moreover, since p − : P → H − is Hilbert-Schmidt, p + : P ⊥ → H + is also Hilbert-Schmidt and it follows that p + • X ∈ L 1 (H + ). Thus Ψ (kx P , k 2 V * • x * −1 P ) = (P, V ). Let us show that two elements (x 1 , X 1 ) and (x 2 , X 2 ) in W s1 k have the same image by Ψ if and only if they are in the same orbit under G C . We have: thus : Let us explicit the differential of Ψ at (x, X). Denote by P the range of x, by U P ⊂ Gr 0 res the open subset of elements P ′ ∈ Gr 0 res such that the orthogonal projection of P ′ onto P is an isomorphism and by ϕ P the chart from U P onto L 2 (P, P ⊥ ) which maps P ′ to the unique element U in L 2 (P, P ⊥ ) whose graph is P ′ . Let (Z, T ) ∈ T (x,X) W s1 k and (x(t), X(t)) ∈ C 1 ((−ǫ, ǫ), W s1 k ) be such that :ẋ (0) = Z andẊ(0) = T.

Corollary 4.4
The quotient space W s1 k /G C is isomorphic as a smooth complex manifold to the cotangent space T ′ Gr 0 res endowed with its natural complex structure via the following isomorphism Hence T ′ Gr 0 res carries a 1-parameter family of hyperkähler structures indexed by k ∈ R * . By restriction to the zero section of the tangent space T M k one deduces from the previous Theorem the following result, which has been, as already mentioned in the Introduction, partially obtained by T. Wurzbacher (cf [51]) : The hyperkähler manifold T M k admits a globally defined hyperkähler potential, i.e. a Kähler potential with respect to all complex structures, which has the following expression : The theory of Subsection 2.4 applied to the particular case of T ′ Gr 0 res yields the following Theorem : with γγ * := k 2 2 Id H+ + Id H+ + 4 k 4 (x * x) 1 2 X * X(x * x)

4.4
The Kähler potential K 1 as a function of the curvature of Gr 0 res Theorem 4.9 The potential K 1 has the following expression in terms of the curvature : K 1 ((x, X)) = k 2 4 log det x * x k 2 + k 2 g Gr (f (I 1 R I1V,V )V, V ) , Proof of Theorem 4.9: The Grassmannian Gr 0 res is a Hermitian-symmetric orbit of the connected component U 0 res of the restricted unitary group. Its curvature is therefore given by (see [46]) : for all X, Y, Z ∈ T P Gr 0 res . The operator R I1V,V acts on T P Gr 0 res by : It follows that : , and : g Gr (I 1 R I1V,V ) j V, V = ℜ Tr 4 j (V * V ) j+1 = 1 4 ℜ Tr (4V * V ) j+1 . Therefore one has : Id + ( Id + 4V * V ) Remark 4.10 The first summand in the expression of K 1 is the pull-back to W s1 k of the Kähler potential of the restricted Grassmannian ( defined on the stable manifold M s k of Gr 0 res ) via the canonical injection M s k ֒→ W s1 k . Note that the Kähler potential of Gr 0 res is the pull-back of the Kähler potential of the complex projective space of a separable Hilbert space by Plücker's embedding. The second summand is expressed as a function of the curvature of the restricted Grassmannian applied to the image of an element (x, X) in W s1 k by the identification W s1 k /G C = T ′ Gr 0 res given by Corollary 4.4. }. An element P of Gr 0 res can be identified with ik 2 pr P , k = 0, where pr P denotes the orthogonal projection of H onto P . Via this identification, the natural action of U 2 (H) on Gr 0 res is given by the conjugation. The complexified orbit O C of Gr 0 res is the orbit of an element P ∈ Gr 0 res under the action of the complex Lie group GL 2 (H) := GL(H) ∩ {Id H + L 2 (H)}. It is the set of operators z ∈ B(H) whose spectrum is the pair {ik 2 , 0} with k = 0, and such that the eigenspace associated with ik 2 (resp. 0) is an element of Gr 0 res (resp. of the Grassmannian Gr 0 * res obtained from the definition of Gr 0 res by exchanging the roles of H + and H − ). This complexified orbit has been introduced in particular by J. Mickelsson in [32]. and is continuous. Since Id H − p 1 takes its values in Ker (x * − X * ), one has Ran(x + X) ⊕ Ker (x * − X * ) = H as a direct topological sum. Moreover, for (x, X) ∈ T M k , Ran(x + X) and Ran(x − X) are elements of Gr 0 res , thus Ker (x * − X * ) = ( Ran(x − X)) ⊥ is an element of Gr 0 * res . It follows that ψ takes values in O C .
the block decomposition of g with respect to the directs sums H = H + ⊕ H − and H = P ⊕ P ⊥ . The block decomposition of g with respect to H = H + ⊕ H − is : g = au 1 bu 2 cu 1 du 2 .
One has dψ (x,X) (I 3 ((Z , T )) = idψ (x,X) ((Z , T )), thus ψ is holomorphic. Let z be in O C , and let P (resp. Q) be the eigenspace of z with respect to the eigenvalue ik 2 (resp. 0). Let (U , V ) be an element of L 2 (P , P ⊥ ) × L 2 (Q , Q ⊥ ). A preimage of (U , V ) by dψ (x,X) is given by the ordered pair (Z , T ) ∈ L 2 (H + , H) × L 2 (H + , H) defined by Using equations (8) page 27, one gets : ) . Moreover, for (x , X) ∈ W s3 k , p + • Z and p + • T are trace class operators. It follows that the differential dψ (x,X) is onto. The orbit of an ordered pair (P, Q) in Gr 0 res ×Gr 0 * res under the natural action of GL 2 (H) is characterized by the dimension of P ∩ Q. The orbit of (P, Q) under the action of U 2 (H) on Gr 0 res ×Gr 0 * res is characterized by the set of characteristic angles θ i .
Proposition 5.5 allows to express the Kähler potential K 3 on the complexified orbit either in terms of the eigenvalues a 2 i of A * A or in terms of the characteristic angles θ i : Theorem 5.8 The form ω red