L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

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ANNALES

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L’INSTITUT FOURIER

Liana DAVID

On cotangent manifolds, complex structures and generalized geometry Tome 66, n^o1 (2016), p. 1-28.

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ON COTANGENT MANIFOLDS, COMPLEX STRUCTURES AND GENERALIZED GEOMETRY

by Liana DAVID (*)

Abstract. — We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field trans- formations), which are analogous to the well-known results of Gualtieri on skew- symmetric generalized complex structures. Given an adapted (symmetric or skew- symmetric) generalized complex structureJand a linear connectionDon a mani- foldM, we construct an almost complex structureJ^J^,Don the cotangent manifold T^∗Mand we study its integrability. ForJ skew-symmetric, we relate the Courant integrability ofJ with the integrability of J^J,D. We consider in detail the case whenM =Gis a Lie group andJ,D are left-invariant. We also show that our approach unifies and generalizes various known results from special complex geometry.

Résumé. — Nous développons plusieurs propriétés des structures complexes généralisées symétriques (en relation avec leurs espaces holomorphes et les “B-field”

transformations), qui sont similaires aux résultats bien connus de Gualtieri sur les structures complexes généralisées anti-symétriques. Étant données une structure complexe généralisée adaptée (symétrique ou anti-symétrique)J et une connexion D sur une variété M, nous construisons une structure presque-complexe J^J,D sur la variété cotangente T^∗M et nous étudions son intégrabilité. Pour J anti- symétrique, nous relions l’intégrabilité deJau sens de Courant avec l’intégrabilité de J^J,D. Nous considérons en détail le cas où M =Gest un groupe de Lie et J, Dsont invariants à gauche. Nous montrons aussi que notre approche unifie et généralise plusieurs résultats connus en géométrie spéciale complexe.

Keywords:complex and generalized complex structures, holomorphic bundles, integrability, Lie groups, special complex geometry.

Math. classification:53C15, 53D18, 53C55.

(*) Part of this work was carried out during a Humboldt Research Fellowship at the Uni- versity of Mannheim (Germany). Hospitality at the University of Mannheim and finan- cial support from the Alexander von Humboldt-Foundation are greatly acknowledged.

Partial support from a CNCS-UEFISCDI grant, project no. PN-II-ID-PCE-2011-3-0362, is also acknowledged.

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1. Introduction 1.1. Motivation

The starting point of this note is a result proved in [1], which states that the cotangent manifold of a special symplectic manifold (M, J,∇, ω) in- herits, under some additional conditions, a hyper-Kähler structure. Recall that a manifold M with a complex structure J, a flat, torsion-free connection ∇ and a symplectic form ω is special symplectic if d^∇J = 0 (i.e.

∇_X(J)(Y) =∇_Y(J)(X), for anyX, Y ∈T M) and∇ω= 0.The connection

∇, acting on the cotangent bundleπ:T^∗M →M, induces a decomposition (1.1) T(T^∗M) =H^∇⊕π^∗(T^∗M) =π^∗(T M⊕T^∗M)

into horizontal and vertical subbundles. Assume now that the (1,1)-part ofω (with respect toJ) is non-degenerate and satisfies∇ω^1,1= 0. Under these additional conditions, the hyper-Kähler structure onT^∗M mentioned above is given, by means of (1.1), by (the pull-back of)

J₁:=

J 0 0 J^∗

, J₂:=

0 −(ω^1,1)⁻¹ ω^1,1 0

, g:=

g^1,1 0 0 (g^1,1)⁻¹

whereg^1,1 :=ω^1,1(J·,·). A key fact in the proof that (J₁, J₂, g) is hyper- Kähler is the integrability of J1 and J2. The integrability of J2 follows from a local argument, which uses∇-flat coordinates and ∇ω^1,1= 0. For the integrability of J1, one notices, using the special complex condition d^∇J = 0, that H^∇ ⊂ T(T^∗M) is invariant with respect to the canonical complex structure Jcan of T^∗M induced by J. Hence, J1 coincides with Jcanand is integrable. These arguments were developed in [1].

With special geometry as a motivation, in this note we consider the following setting: a manifoldM with a linear connectionDand a smooth field of endomorphismsJ of the generalized tangent bundleTM :=T M⊕T^∗M, such that J² = −Id. Following [10] (rather than the usual terminology from generalized geometry), we call J a generalized complex structure.

Motivated by J1 and J2 above, we assume that J is adapted, i.e. either symmetric or skew-symmetric with respect to the canonical metric of neutral signature of TM. From D and J we construct an almost complex structureJ^J^,D on the cotangent manifold T^∗M and we study its integrability. This provides a new insight, from the generalized complex geometry point of view, on the above arguments from [1]. Along the way, we prove various properties we need on symmetric generalized complex structures.

The relation with the Courant integrability is also discussed. As a main

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application, we construct a large class of complex structures on cotangent manifolds of real semisimple Lie groups.

In the remaining part of the introduction we describe in detail the results and the structure of the paper.

1.2. Structure of the paper

In Section 2 we prove basic facts we need from generalized geometry.

While skew-symmetric generalized complex structures are well-known (see e.g. [6] for basic facts), the symmetric ones do not seem to appear in the lit- erature. We begin by studying symmetric generalized complex structures on (real) vector spaces (see Definition 2.1). We find the general form of their holomorphic space (see Proposition 2.4) and we show that any symmetric generalized complex structure on a vector space is, modulo aB-field transformation, the direct sum of one determined by a complex structure and another determined by a pseudo-Euclidean metric (see Example 2.6 and Theorem 2.7). Therefore, there is an obvious analogy with the theory of skew-symmetric generalized complex structures developed by Gualtieri in [6]. We discuss this analogy in Subsection 2.2. For our purposes it is particularly relevant the common description of the holomorphic space L^τ(E, α) of a symmetric or, respectively, skew-symmetric generalized complex structure on a vector spaceV, in terms of a complex subspaceE⊂V^C and a skew-Hermitian, respectively skew-symmetric 2-formαonE, satisfying some additional conditions (see Corollary 2.8). These results extend pointwise to manifolds (see Subsection 2.3). Despite the above analogies, there is an important difference between symmetric and skew-symmetric generalized complex structures on manifolds: unlike the skew-symmetric ones, the symmetric generalized complex structures are never Courant integrable (see Lemma 2.13).

In Section 3 we consider a manifoldM together with a connectionDand an adapted generalized complex structureJ (see Definition 2.9). UsingD andJ we define an almost complex structure J^J^,D on T^∗M (see Defini- tion 3.1) and we discuss its integrability. It turns out that the integrability ofJ^J^,D imposes obstructions on the curvature of D and the data (E, α) which defines the holomorphic bundleL = L^τ(E, α) of J. In particular, the complex subbundleE⊂T^CM must be involutive andαmust satisfy a differential equation involvingD (see Theorem 3.3). As a straightforward

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application of Theorem 3.3, we relate the Courant integrability of a skew- symmetric generalized complex structureJ with the integrability ofJ^J^,D (see Corollary 3.4). In particular, we deduce that a left-invariant, skew- symmetric, generalized complex structureJ on a Lie groupGis Courant integrable, if and only if the almost complex structure J^J^,D^c on T^∗G is integrable, whereD^c is the left-invariant connection on G which on left- invariant vector fields is the Lie bracket (see Example 3.5). A systematic description of Courant integrable, left-invariant, skew-symmetric generalized complex structures on real semisimple Lie groups was developed in [2].

This is the motivation for our treatment from the next section.

Section 4 is devoted to applications of Theorem 3.3 to Lie groups. Our main goal here is to describe a large class of left-invariant symmetric (rather than skew-symmetric) generalized complex structuresJ on a semisimple Lie group, which, together with a suitably chosen left-invariant connection D⁰, determine an integrable complex structure J^J^,D⁰ on the cotangent group. (The connectionD⁰ plays the role ofD^c above). In the first part of Section 4, intended to fix notation, we briefly recall the basic facts we need on the structure theory of semisimple Lie algebras. We follow closely [9], Chapter VI. In Subsection 4.2 we develop an infinitesimal description, in terms of the so-called admissible triples (k,D, ), of pairs (J, D) formed by a left-invariant adapted generalized complex structureJ and a left-invariant connectionDon a (not necessarily semisimple) Lie groupG, such that the associated almost complex structureJ^J^,D onT^∗Gis integrable (see Def- inition 4.2 and Proposition 4.3). In this description, the pair (k, ) defines the fiber L^τ(k, ) at e∈ G of the holomorphic bundle of J and D is the restriction ofD to the space of left-invariant vector fields. The notion of admissible triple generalizes the notion of admissible pair, defined in [2] to encode the Courant integrability of left-invariant skew-symmetric generalized complex structures on Lie groups. WhenGis semisimple, we define the notion of regularity for the structures involved (see Definition 4.2); in the above notation, this means thatkis a regular subalgebra ofg^C, normalized by a maximally compact Cartan subalgebra ofg. The preferred connection D⁰ is introduced in Definition 4.8 and our motivation for its choice is explained before Lemma 4.7. Our main result in this section is Theorem 4.9, which provides a description (in terms of admissible triples) of regular symmetric generalized complex structuresJ on a semisimple Lie groupG, which, together with the connection D⁰, determine an (integrable) complex structure onT^∗G. The description from Theorem 4.9 requires further clarifications: one needs to construct the constants{να, α∈R^sym₀ }, which

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are subject to conditions (4.20), (4.21) and to study the non-degeneracy of the (symmetric) bilinear form g∆. A method to construct the να’s is provided by Lemma 4.10. When the root systemR₀ of the regular subal- gebrakis not onlyσ-parabolic, as required by Theorem 4.9, butσ-positive (see Definition 4.5), the non-degeneracy of g_∆ is straightforward (see Re- mark 4.11) and we obtain, on any semisimple Lie groupG, a large class of regular symmetric generalized complex structuresJ, such that J^J^,D⁰ is integrable. In the special case whenGis (semisimple) of inner type, the root systemR₀ ofk is always a positive root system and we obtain a full explicit description of all regular symmetric generalized complex structures J, such that J^J^,D⁰ is integrable (see Theorem 4.14).

In Section 5 we use Theorem 3.3 in order to reobtain and generalize various well-known results from special complex geometry, with emphasis on those from [1], already mentioned at the beginning of this introduction.

2. Symmetric generalized complex structures

In this section we study symmetric generalized complex structures. Sub- sections 2.1 and 2.2 are algebraic, while in Subsection 2.3 we discuss the Courant integrability.

2.1. Linear symmetric generalized complex structures

LetV be a real vector space. We denote by (2.1) gcan(X+ξ, Y +η) = 1

2(ξ(Y) +η(X)), X+ξ, Y +η∈V ⊕V^∗ the canonical pseudo-Euclidian metric of neutral signature onV ⊕V^∗.

Definition 2.1. — A generalized complex structure on V is an endo- morphismJ ∈End(V⊕V^∗), such thatJ²=−Id. The generalized complex structureJ is called symmetric (respectively, skew-symmetric) if it is symmetric (respectively, skew-symmetric) with respect tog_can. The generalized complex structureJ is called adapted if it is symmetric or skew-symmetric.

Remark 2.2. — In the classical terminology of generalized geometry (see e.g. [6, 8]), a generalized complex structure is, by definition, skew- symmetric. In this note we prefer the language of [10], where generalized complex structures are not assumed, a priori, to be compatible in any way withgcan.

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In the following proposition we describe the holomorphic space of symmetric generalized complex structures. Before we need to introduce a notation which will be used along the paper.

Notation 2.3. — For a complex subspace E⊂V^C, we denote by ¯E the image ofE through the antilinear conjugation V^C 3X → X¯ ∈V^C with respect to the real formV of V^C. In particular, ¯E is a complex subspace ofV^C(not to be confused with the conjugate vector space ofE).

Proposition 2.4. — A complex subspaceLof(V ⊕V^∗)^Cis the holomorphic space of a symmetric generalized complex structure onV if and only if it is of the form

(2.2) L=L⁻(E, α) :={X+ξ∈E⊕(V^C)^∗, ξ|E¯=i_Xα},

where E is any complex subspace of V^C, such that E+ ¯E = V^C, and α ∈ E^∗⊗E¯^∗ is any complex bilinear form satisfying the following two conditions:

i) it is skew-Hermitian, i.e.

(2.3) α(X,Y¯) +α(Y,X) = 0,¯ ∀X, Y ∈E;

ii) Im(α|∆)is non-degenerate. Here∆⊂V is the real part ofE∩E, i.e.¯

∆^C=E∩E.¯

Proof. — Let J be a symmetric generalized complex structure on V, with holomorphic spaceL. Thus L is a complex subspace of (V ⊕V^∗)^C, withL⊕L¯ = (V⊕V^∗)^C, andLisgcan-orthogonal to ¯L(from the symmetry ofJ). We denote by

π1: (V ⊕V^∗)^C→V^C, π2: (V ⊕V^∗)^C→(V^C)^∗ the natural projections. We defineE:=π1(L) and we let (2.4) α:E→E¯^∗, α(X) :=π₂◦(π₁|_L)⁻¹(X)|E¯.

We claim thatα∈E^∗⊗E¯^∗ is well defined. To prove this claim, we use (2.5) ξ( ¯Y) + ¯η(X) = 0, ∀X+ξ, Y +η∈L,

(which holds becauseLisgcan-orthogonal to ¯L). Thus, ifX+ξ1, X+ξ2∈ (π₁|L)⁻¹(X), i.e.X+ξ₁, X+ξ₂ ∈L, then, from (2.5), ξ₁ =ξ₂ on ¯E and we obtain thatαis well-defined, as required. From the very definition of α, L ⊂L⁻(E, α) and, being of the same dimension, we deduce that L= L⁻(E, α).SinceLisgcan-orthogonal to ¯L,αis skew-Hermitian. Moreover, L⊕L¯= (V⊕V^∗)^Cimplies thatE+ ¯E=V^C. We now claim thatL∩L¯ ={0}

implies that Im(α|∆) is non-degenerate. To prove this claim, we assume,

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by absurd, that there isX 6= 0 in the kernel of Im(α|∆). Defineξ∈(V^C)^∗ by

ξ(Z) =α(X,Z),¯ ξ( ¯Z) =α(X,Z),¯ ∀Z∈E.

Using that X ∈ Ker(Im(α|_∆)), one can check that ξ is well-defined and X+ξ∈L∩L, which is a contradiction. We proved that the holomorphic¯ spaceL ofJ is of the required form.

Conversely, it may be shown that any subspaceE ⊂V^C, which satisfies E+ ¯E = V^C, together with a skew-Hermitian form α∈ E^∗⊗E¯^∗, which satisfies the non-degeneracy property ii), define, by (2.2), the holomorphic space of a symmetric generalized complex structure onV. Corollary 2.5. — LetJ be a symmetric generalized complex structure onV, with holomorphic space L⁻(E, α). Then Re(α|∆) is a 2-form andIm(α|_∆)is a pseudo-Euclidean metric on∆ (the real part ofE∩E).¯

Proof. — Straightforward, from (2.3) and the non-degeneracy of Im(α|∆).

The second example below shows that symmetric generalized complex structures exist on vector spaces of arbitrary dimension.

Example 2.6.

i) A complex structureJ onV defines a symmetric generalized complex structure

J :=

J 0 0 J^∗

,

whereJ^∗ξ:=ξ◦J, for anyξ∈V^∗.Its holomorphic space isL⁻(V^1,0,0) = V^1,0⊕Ann(V^0,1), whereV^1,0 andV^0,1 are the holomorphic and anti- holomorphic spaces ofJ.

ii) A pseudo-Euclidean metric on V, seen as an isomorphismg:V →V^∗, defines a symmetric generalized complex structure

J :=

0 g⁻¹

−g 0

.

Its holomorphic space is L⁻(V^C, ig^C), where g^C ∈ (V^C⊗V^C)^∗ is the complex linear extension ofg.

iii) If J is a symmetric generalized complex structure on V, then so is itsB-field transformation exp(B)· J := exp(B)◦ J ◦exp(−B), where B∈Λ²(V^∗) and theB-field action is defined by

exp(B) :V ⊕V^∗→V ⊕V^∗, X+ξ→X+iXB+ξ.

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If L⁻(E, α) is the holomorphic space of J, then L⁻(E, α+B^C|_E⊗E¯) is the holomorphic space of exp(B)· J, where B^C ∈ Λ²(V^C)^∗ is the complex linear extension ofB.

In following theorem we show that any symmetric generalized complex structure can be (non-canonically) obtained from a complex structure, a pseudo-Euclidean metric and aB-field transformation.

Theorem 2.7. — Any symmetric generalized complex structure on a vector space V is a B-field transformation of the direct sum of one determined by a complex structure and another determined by a pseudo- Euclidean metric (as in Example 2.6).

Proof. — Let J ∈ End(V ⊕V^∗) be a symmetric generalized complex structure, with holomorphic spaceL =L⁻(E, α). Let ∆ be the real part ofE∩E¯ (i.e. ∆⊂V and ∆^C=E∩E) and¯ N a complement of ∆ inV. Thus

(2.6) V = ∆⊕N, E= ∆^C⊕(E∩N^C), E¯= ∆^C⊕( ¯E∩N^C).

We notice that ∆ comes with a pseudo-Euclidean metric, namely g_∆ :=

Im(α|∆), and N with a complex structure J^N, with holomorphic space E∩N^C(and anti-holomorphic space ¯E∩N^C). We claim that there isB∈ Λ²(V^∗) such that (as vector spaces with symmetric generalized complex structures)

(2.7) (V,exp(B)· J) = (∆, g∆)⊕(N, J^N), or, in terms of their holomorphic spaces,

(2.8) L⁻(E, α+B^C|_E⊗E¯) =L⁻(∆^C, i(g_∆)^C)⊕ E∩N^C⊕Ann( ¯E∩N^C) .

From the second and third relation (2.6), we obtain that (2.8) holds if and only if, for anyX ∈E, the covectoriX(α+B^C)∈E¯^∗ is given by

(2.9) iX(α+B^C)|_∆C =i(g∆)^C(pr_∆C(X),·), iX(α+B^C)|E∩N¯ ^C = 0, where pr_∆C :V^C →∆^C and pr_NC :V^C→N^C are the natural projections determined by the decompositionV^C= ∆^C⊕N^C.Moreover, it is easy to see that (2.9) is equivalent to

(Re(α) +B)|_∆⊗∆= 0, (α+B^C)|_(E∩NC)⊗∆^C = 0, (α+B^C)|_{E⊗( ¯}_E∩NC)= 0.

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Hence, we are looking for a (real) 2-form B ∈Λ²(V^∗) such that (2.10) is satisfied. In order to defineB, we useV = ∆⊕N and N^C= (E∩N^C)⊕

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( ¯E∩N^C).For anyX, Y ∈∆ andZ, W ∈N, let

B(X, Y) :=−Re(α)(X, Y), B(Z, W) :=−2Re(α)(z,w)¯ and

B(X, Z) =−B(Z, X) := 2Re(α)(z, X),

wherez, w∈E∩N^C(uniquely determined) are such that Z =z+ ¯z and W =w+ ¯w. Sinceα∈E^∗⊗E¯^∗ is skew-Hermitian,B is skew-symmetric and its complexification satisfies (2.10) (easy check). This concludes our

claim.

2.2. Analogy with skew-symmetric generalized complex structures

The theory of symmetric generalized complex structures from the previous section is similar to the theory of skew-symmetric generalized complex structures developed by Gualtieri in [6] and owing to this, one can treat these two types of structures in a unified way, using the notion of adapted generalized complex structure (see Definition 2.1). More precisely, it is well-known (see e.g. [6]) that complex and symplectic structures define skew-symmetric generalized complex structures and this corresponds to Example 2.6 i) and ii) from the previous section. In the same frame- work, Theorem 2.7 above is analogous to Theorem 4.13 from [6], which states that any skew-symmetric generalized complex structure, is, modulo a B-field transformation, the direct sum of a skew-symmetric generalized complex structure of symplectic type and of one of complex type.

The following unified description of the holomorphic space of symmetric and skew-symmetric generalized complex structures on vector spaces is a rewriting of Proposition 2.4 from the previous section and of Propositions 2.6 and 4.4 from [6]. We shall use it in the statement of Theorem 3.3.

Corollary 2.8. — A complex subspaceL ⊂(V ⊕V^∗)^C is the holomorphic space of an adapted generalized complex structureJ if and only if it is of the form

(2.11) L=L^τ(E, α) ={X+ξ∈E⊕(V^C)^∗, ξ|_τ(E)=iXα}

whereE⊂V^Cis a complex subspace withE+ ¯E=V^Candα∈E^∗⊗τ(E)^∗ is complex bilinear, such that

(2.12) α(X, τ(Y)) +τ(α(Y, τ(X))) = 0, ∀X, Y ∈E andIm(α|∆)is non-degenerate (where∆⊂V,∆^C=E∩E).¯

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In (2.11) and (2.12) the mapsτ :V^C→V^Candτ :C→Care both the standard conjugations (J symmetric), respectively both the identity maps (J skew-symmetric).

2.3. Remarks on integrability

The generalized tangent bundleTM =T M⊕T^∗M of a smooth manifold M has a canonical metric of neutral signature, defined like in (2.1), and the theory developed in the previous sections extends pointwise to manifolds, in an obvious way.

Definition 2.9. — A (symmetric, respectively skew-symmetric) generalized complex structure on a manifold M is a smooth field of endo- morphismsJ of TM, which, at anyp∈M, is a (symmetric, respectively skew-symmetric) generalized complex structure onTpM. An adapted generalized complex structure onM is a generalized complex structure which is either symmetric or skew symmetric.

Remark 2.10. — As opposed to the usual terminology, we do not assume that generalized complex structures on manifolds are Courant integrable (see Definition 2.11 below). In fact, the generalized complex structures we are mainly interested in, namely, the symmetric ones, turn out not to be Courant integrable (see Lemma 2.13).

Definition 2.11. — A generalized complex structureJ on a manifold M is called Courant integrable if the space of sections of its holomorphic bundle L ⊂ T^CM (the i-eigenbundle of J) is closed under the Courant bracket, defined by

[X+ξ, Y +η] = [X, Y] +LXη−LYξ+1

2(ξ(Y)−η(X))), for any vector fieldsX,Y and 1-formsξ, η.

The holomorphic bundleLof an adapted generalized complex structure onM may be described in terms of a complex subbundleE ⊂T^CM (the image ofL through the natural projection T^CM →T^CM) and a smooth sectionα∈Γ(E^∗⊗τ(E)^∗), with the algebraic properties from Corollary 2.8 (we assume that all points are regular, i.e.E is a genuine complex vector bundle). There is a basic result of Gualtieri (see [6, Propostition 4.19]) which expresses the Courant integrability of a skew-symmetric generalized complex structure in terms of its holomorphic bundle. Since we shall use it repeatedly, we state it here:

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Proposition 2.12. — [6]A skew-symmetric generalized complex structure on a manifoldM, with holomorphic bundleL=L(E, α), is Courant in- tegrable, if and only if the subbundleE⊂T^CM is involutive andd_Eα= 0, wheredEα∈Γ(Λ³E^∗)is the exterior differential of α∈Γ(Λ²E^∗), defined by

(dEα)(X, Y, Z) :=X(α(Y, Z)) +Z(α(X, Y)) +Y(α(Z, X))

+α(X,[Y, Z]) +α(Z,[X, Y]) +α(Y,[Z, X]), for anyX, Y, Z∈Γ(E).

The following simple lemma holds.

Lemma 2.13. — A symmetric generalized complex structure is never Courant integrable.

Proof. — As proved in Proposition 3.26 of [6], a Courant integrable subbundle ofT^CM is eitherg_can-isotropic or of the form (∆⊕T^∗M)^C, where

∆⊂T M is involutive (and non-trivial). Hence, it cannot be the holomorphic bundleLof a symmetric generalized complex structure (recall that L isgcan-orthogonal to ¯L, L⊕L¯=T^CM andgcanis non-degenerate).

3. Integrable complex structures on cotangent manifolds Let (M,J, D) be a manifold with a generalized complex structure J and a linear connectionD. The connectionDacts on the cotangent bundle π:T^∗M →M and induces a decomposition

(3.1) T(T^∗M) =H^D⊕T^vert(T^∗M) =π^∗(TM)

into horizontal and vertical subbundles. Above, we identified the horizontal bundleH^D withπ^∗(T M) and the vertical bundleT^vert(T^∗M) of the pro- jectionπwithπ^∗(T^∗M).From now on, we shall use systematically, without mentioning explicitly, the identification (3.1) between T(T^∗M) and π^∗(TM).

Definition 3.1. — The almost complex structure J^J^,D := π^∗(J) on the cotangent manifoldT^∗M is called the almost complex structure defined byJ andD.

In this section we study the integrability ofJ^J^,D, under the assumption thatJ is adapted. We begin by fixing notation.

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Notation 3.2. — In computations, we shall use the notation ˜X∈ X(T^∗M) for theD-horizontal lift of a vector fieldX ∈ X(M). Forms of degree one onM will be considered as constant vertical vector fields on the cotangent manifoldT^∗M. With these conventions, the Lie brackets [·,·]L of various vector fields onT^∗M are given by: at any γ∈T^∗M,

(3.2) [ ˜X,Y˜]L(γ) =[X, Y^](γ) +R^D_X,Y(γ), [ ˜X, ξ]L=DXξ, [ξ, η]L= 0 whereX, Y ∈ X(M),ξ, η∈Ω¹(M) and

R^D_X,Y :=−DXD_Y +D_YD_X+D_[X,Y_] is the curvature ofD.

The main result from this section is the following.

Theorem 3.3. — Let (M,J, D)be a manifold with a generalized complex structure J and a linear connection D. Assume that J is adapted and let L^τ(E, α) be its holomorphic bundle, where E ⊂ T^CM and α ∈ Γ(E^∗⊗τ(E)^∗)satisfy the algebraic properties from Corollary 2.8. The almost complex structureJ^J^,D from Definition 3.1 is integrable, if and only if the following conditions hold:

i) E is an involutive subbundle ofT^CM;

ii) the complex linear extensions ofD andR^Dsatisfy (3.3) D_Γ(E)Γ(τ(E))⊂Γ(τ(E)), R^D|_E×E(τ(E)) = 0.

iii) for anyX, Y, Z∈Γ(E),

(3.4) (DXα)(Y, τ(Z))−(DYα)(X, τ(Z)) +α(T_X^DY, τ(Z)) = 0, whereT^D is the complex linear extension of the torsion of the connec- tionD.

Proof. — We need to prove that the holomorphic bundleπ^∗L^τ(E, α)⊂ T^C(T^∗M) of J^J^,D is involutive if and only if the conditions i), ii) and iii) from the statement of the theorem hold. For this, we will compute the Lie brackets of basic sections of π^∗L^τ(E, α). (By a basic section of π^∗L^τ(E, α) we mean a vector field on the cotangent manifold T^∗M, of the form ˜X +ξ, where X +ξ is a section of L^τ(E, α)). Therefore, let X+ξ, Y +η ∈Γ(L^τ(E, α)). Then

(3.5) X, Y ∈Γ(E), ξ, η∈Γ(T^CM)^∗, ξ|τ(E)=i_Xα, η|τ(E)=i_Yα.

From (3.2), at anyγ∈T^∗M,

(3.6) [ ˜X+ξ,Y˜ +η]L(γ) =[X, Y^](γ) +R^D_X,Y(γ) +DXη−DYξ.

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We obtain that [ ˜X+ξ,Y˜ +η]_L is a section of π^∗L^τ(E, α) if and only if [X, Y] +R^D_X,Y(γ) +D_Xη−D_Yξ

belongs to the fiber ofL^τ(E, α) atπ(γ), for anyγ∈T^∗M, i.e.

(3.7) [X, Y]∈Γ(E), R^D_X,Y(γ)|_τ(E)= 0 and

(3.8) (D_Xη−D_Yξ)(τ(Z)) =α([X, Y], τ(Z)), ∀Z∈Γ(E).

We now rewrite (3.8). From (3.5), the left hand side of (3.8) is equal to Xα(Y, τ(Z))−Y α(X, τ(Z))−η(DX(τ(Z))) +ξ(DY(τ(Z))) and (3.8) becomes

Xα(Y, τ(Z))−Y α(X, τ(Z))−η(D_X(τ(Z)))+ξ(D_Y(τ(Z)))

=α([X, Y], τ(Z)), (3.9)

for any Z ∈ Γ(E). From (3.5) again, ξ|_τ(E) = i_Xα, but ξ can take any values on a complement ofτ(E) in T^CM. Similarly, the only obstruction onη isη|_τ(E)=iYα. Thus, if (3.9) holds for any sectionsX+ξandY+η ofL^τ(E, α), then

DX(τ(Z))∈Γ(τ(E)), ∀X, Z∈Γ(E)

and relation (3.9) becomes (3.4). We proved thatπ^∗L^τ(E, α) is involutive if and only if

[Γ(E),Γ(E)]⊂Γ(E), R^D|_E×Eτ(E) = 0, D_Γ(E)Γ(τ(E))⊂Γ(τ(E))

(3.10)

and relation (3.4) holds. Our claim follows.

We end this section by relating the Courant integrability of a skew- symmetric generalized complex structureJ with the integrability of the almost complex structure J^J^,D. This is a straightforward application of Theorem 3.3.

Corollary 3.4. — Let J be a skew-symmetric generalized complex structure, with holomorphic bundleL(E, α), andDa linear connection on M. Suppose thatE is involutive,D_Γ(E)Γ(E)⊂Γ(E),R^D_E,EE= 0 and the relation

(3.11) (DZα)(X, Y) +α(T_Z^DX, Y) +α(X, T_Z^DY) = 0, ∀X, Y, Z∈Γ(E) holds. ThenJ^J^,D is integrable if and only ifJ is Courant integrable.

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Proof. — From Proposition 2.12 and Theorem 3.3, we need to prove that dEα= 0 is equivalent to (3.4) (withτ :T^CM →T^CM the identity map).

This is a consequence of (3.11) and the following general identity: for any 2-formβ and vector fieldsX, Y, Z,

(D_Xβ)(Y, Z)−(D_Yβ)(X, Z) +β(T_X^DY, Z)

= (dβ)(X, Y, Z)− (D_Zβ)(X, Y) +β(T_Z^DX, Y) +β(X, T_Z^DY) (3.12) .

Example 3.5. — Let J be a left-invariant skew-symmetric generalized complex structure on a Lie groupG and D^c the (flat) left-invariant connection onG given by D^c_XY = [X, Y], for any left-invariant vector fields X, Y.Then D^c satisfies (3.11), for any left-invariant 2-form α. We obtain thatJ is Courant integrable if and only ifJ^J^,D^c is integrable.

4. Complex structures on cotangent manifolds of Lie groups

We begin by recalling basic facts we need about semisimple Lie algebras.

4.1. Semisimple Lie algebras Letg^Cbe a complex semisimple Lie algebra and

(4.1) g^C=h+g(R) =h+X

α∈R

gα

a Cartan decomposition. We identifyhwithh^∗, using the restriction of the Killing form B of g^C to h. By means of this identification, we denote by h_R ⊂ h the real span of the set of roots R ⊂ h^∗ of g^C relative to h and byH_α ∈h_R the vector which corresponds to the root α∈R. Recall that a Weyl basis of the root part g(R) := P

α∈Rgα consists of root vectors {Eα, α∈R}, satisfying the following conditions:

[E_α, E_−α] =H_α, B(E_α, E_−α) = 1, N_−α,−β =−N_αβ, N_αβ∈R, where the structure constantsN_αβ are defined by

[Eα, Eβ] =NαβEα+β, ∀α, β, α+β∈R.

A simple argument which uses the Jacobi identity forEα, Eβ, Eγ shows that for anyα, β, γ∈R, such that α+β+γ= 0,

(4.2) Nαβ=Nβγ=Nγα

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(see e.g. [7], page 146).

Recall now that a real formgofg^Cis the fixed point set of an antilinear involution

σ:g^C→g^C, x→σ(x) = ¯x,

i.e. an automorphism of real Lie algebras, which is complex antilinear and satisfiesσ²= Id. We review, following Theorem 6.88 of [9], the structure of such real forms. The idea is thatgis determined (up to isomorphism) by its Vogan diagram, which is the Dynkin diagram ofg^C(representing a set of simple roots Π relative to a chosen Cartan subalgebrah) together with two pieces of data: an involutive automorphismθ: Π→Π of the Dynkin diagram and some painted nodes, in the fixed point set ofθ.Choose a Weyl basis{Eα}ofg(R), whereR= [Π] is the set of roots ofg^Crelative toh. The action ofθon Π extends by linearity toh^∗_R∼=h_Rand this action preserves R. The antiinvolutionσpreserves hand it acts onRby

σ:R→R, σ(α) :=α◦σ.

This action coincides, up to a minus sign, with the action ofθ:σ|R=−θ|R. On root vectors from the chosen Weyl basis,σacts as

(4.3) σ(E_α) =−aαE_σ(α), ∀α∈R,

where{aα, α∈R}(determined by the painted nodes in the Vogan diagram) is a set of constants, satisfying

(4.4) a_α=a_−α=a_σ(α)∈ {±1}, ∀α∈R and

(4.5) aα+β=−aαaβN_αβ⁻¹N_σ(α)σ(β), ∀α, β, α+β∈R.

The real formh_g =h^σ=h⁺+h⁻, where

(4.6) h⁺:=hi(Hα+H_−σ(α)), α∈Ri, h⁻:=hHα+Hσ(α), α∈Ri (the signh· · · imeans the real span of the respective vectors) is a Cartan subalgebra of g. Up to isomorphism, g can be recovered from its Vogan diagram as

(4.7) g= (g^C)^σ=hg+X

α∈R

R(Eα−aαEσ(α)) +X

α∈R

Ri(Eα+aαEσ(α)).

Remark 4.1. — Sinceθ permutes Π, there is no root α∈R such that σ(α) =α. This means thath_g is a maximally compact Cartan subalgebra ofg(see [9, Proposition 6.70]). The real formg(and any Lie groupGwith Lie algebra g) is called of inner type if σ(α) = −α for any α ∈ R (the automorphismθ of the Vogan diagram is the identity). Any compact real

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form is of inner type, witha_α= 1, for anyα∈R.A real formg(and any Lie groupGwith Lie algebrag) which is not of inner type is called of outer type. For more details on real semisimple Lie algebras, Vogan diagrams, maximally compact Cartan subalgebras, see e.g. [9], Chapter VI.

4.2. Admissible triples on Lie groups

Let Gbe a Lie group. We identify TeGwith the space of left-invariant vector fields onGand with the Lie algebragof G, in the usual way. The following definition encodes the conditions from Theorem 3.3, whenM =G andJ,D are left-invariant. Recall that a connectionD is left-invariant if DXY is left-invariant, whenX andY are so.

Definition 4.2. — A (symmetric or skew-symmetric) g-admissible triple is a triple(k,D, ), with the following properties:

i) kis a complex subalgebra ofg^C, such thatk+ ¯k=g^C;

ii) D : g×g → g, (X, Y) → DX(Y), is a bilinear map whose complex linear extension satisfies

(4.8) D_kτ(k)⊂τ(k)

and

R^D_X,YZ:=−DXDY(Z) +DYDX(Z)+D[X,Y](Z) = 0,

∀X, Y ∈k, ∀Z∈τ(k).

(4.9)

iii) ∈k^∗⊗τ(k)^∗ satisfies

(X, τ(Y)) +τ((Y, τ(X))) = 0, ∀X, Y ∈k and

(X,D_Y(τ(Z)))−(Y,D_X(τ(Z))) =([X, Y], τ(Z)),

∀X, Y, Z∈k (4.10)

and, moreover,g∆:= Im(|∆)is non-degenerate on∆ = (k∩¯k)^σ. Above, the mapsτ :g^C→g^Candτ:C→Care both standard conjugations (symmetric case) or both the identity maps (skew-symmetric case).

The following correspondence holds and will play a key role in our treatment from the next subsection.

Proposition 4.3. — There is a one to one correspondence between:

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i) pairs (J, D) formed by a left-invariant adapted generalized complex structureJ and a left-invariant connection D onG, such that the associated almost complex structureJ^J^,D onT^∗Gis integrable;

ii) g-admissible triples (k,D, ).

In this correspondence D is the restriction of D to the space of left- invariant vector fields, k := Ee and := α|_k×τ(k), where L^τ(E, α) is the holomorphic bundle ofJ.

Proof. — Using the left-invariance ofE and α, one may check that the conditions from Theorem 3.3, on the integrability of J^J^,D, become the conditions for (k,D, ) to be ag-admissible triple. For example, to prove the equivalence between (3.4) and (4.10), we notice that (3.4) holds if and only if it holds for any X, Y, Z ∈ Γ(E) left-invariant, and for such arguments, α(Y, τ(Z)) andα(X, τ(Z)) are constant (becauseαis left-invariant).

4.3. Regular admissible triples and regular generalized complex structures

Here and until the end of Section 4 we fix a complex semisimple Lie algebrag^C, a real form g= (g^C)^σ given by (4.7), and a Lie groupG with Lie algebra g. A (complex) subalgebra k ⊂ g^C is called regular, if it is normalized by the (maximally compact) Cartan subalgebra h_g of g. It is known (see e.g. [11], Proposition 1.1, page 183) that such a subalgebra is of the form

(4.11) k=h_k+g(R₀) =h_k+ X

α∈R0

g_α

wherehk =k∩h andR0 ⊂R is a closed subset of roots (i.e. ifα, β ∈R0

andα+β ∈R, then α+β∈R₀). Remark that (4.12) ¯k=σ(k) = ¯hk+ X

α∈R₀

g_σ(α), k∩¯k=hk∩¯hk+ X

α∈R0∩σ(R0)

gα.

Definition 4.4. — LetJ be a left-invariant adapted generalized complex structure on G and L^τ(k, ) the fiber at e ∈ G of its holomorphic bundle. ThenJ is called regular if the subalgebrak⊂g^C is regular. Simi- larly, ag-admissible triple(k,D, )is called regular if the subalgebrak⊂g^C is regular.

We need to recall the notions of σ-parabolic and σ-positive systems [2].

They reduce, whengis of inner type, to the usual notions of parabolic and positive root systems, respectively (see e.g. [3]).

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Definition 4.5. — A closed set of rootsR₀⊂Ris called aσ-parabolic system, ifR0∪σ(R0) =R.If, moreover,R0∩σ(R0) =∅, thenR0 is called aσ-positive system.

The following simple lemma holds.

Lemma 4.6. — If a regular subalgebra k as in (4.11) belongs to a g- admissible triple, then its root part R₀ is a σ-parabolic system and its Cartan parthksatisfieshk+ ¯hk=h.If, moreover,R0is aσ-positive system, thenk∩¯k=h_k∩¯h_k.

Proof. — From the definition of g-admissible triples, k+ ¯k = h. This relation, together with (4.12), implies the statement of the lemma.

4.4. Complex structures on T^∗G

Our aim in this section is to define a natural left-invariant connectionD⁰ onGand to determine all regular symmetric generalized complex structures J, with the property that the almost complex structure J^J^,D⁰ onT^∗Gis integrable, or, equivalently (from Proposition 4.3), the associated triple (k,D⁰, ) is g-admissible (and regular, symmetric). From Definition 4.2, a bilinear mapD:g×g→gcan belong to a symmetricg-admissible triple (k,D, ) only if its complex linear extensionD:g^C×g^C→g^Csatisfies (4.8) and (4.9) (with τ =σ, hence τ(k) = ¯k). Recall now that k is of the form (4.11) and ¯kof the form (4.12). From these relations, it is immediate that if

(4.13) Dg_α(gβ)⊂g_σ(α)+β, Dh(gβ)⊂gβ, D_g_β(h)⊂g_σ(β), D_h(h) = 0,

for anyα, β∈R(withgσ(α)+β:= 0 ifα+σ(β)∈/ R), then (4.8) is satisfied.

A map whose complex linear extension satisfies (4.13) and (4.9) is provided by the following lemma.

Lemma 4.7. — LetD⁰:g^C×g^C→g^Cbe a complex bilinear map given by

D_E⁰

α(Eβ) =−aα[E_σ(α), Eβ], D⁰_H(Eβ) =σ(β)(H)Eβ, D_E⁰

β(H) =σ(β)(H)a_βE_σ(β), D⁰_H( ˜H) = 0 for anyα, β∈R andH,H˜ ∈h. ThenD⁰ satisfies (4.14) D_k⁰(¯k)⊂¯k, D⁰_g(g)⊂g, R^D⁰ = 0.

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Proof. — We already explained thatD_k⁰(¯k)⊂¯k. We now proveD⁰_g(g)⊂g.

For anyα∈R, letAα:=Eα−aαE_σ(α) andBα:=i(Eα+aαE_σ(α)). By a straightforward computation, which uses (4.4), we obtain:

D⁰_A

α(A_β) =−a_α [E_σ(α), E_β] +a_αa_β[E_α, E_σ(β)] + [Eα, Eβ] +aαaβ[E_σ(α), E_σ(β)]

, D_B⁰_α(Bβ) = [Eα, Eβ] +aαaβ[Eσ(α), Eσ(β)]

+a_α [E_σ(α), E_β] +a_αa_β[E_α, E_σ(β)] , D_A⁰_α(Bβ) = i [Eα, Eβ]−aαaβ[E_σ(α), E_σ(β)]

+aβi [Eα, E_σ(β)]−aαaβ[E_σ(α), Eβ] , D⁰_B_α(Aβ) =−i [Eα, Eβ]−aαaβ[Eσ(α), Eσ(β)]

−ia_α [E_σ(α), E_β]−a_αa_β[E_α, E_σ(β)] .

Moreover, for anyα, β∈R andH ∈h⁺,

D⁰_A_α(H) =iα(H)aαBσ(α), D⁰_B_α(H) =iα(H)Aα

D⁰_H(A_α) =iα(H)B_α, D⁰_H(B_α) =−iα(H)A_α, while for anyα, β∈R andH ∈h⁻,

D⁰_A_α(H) =α(H)aαAσ(α), D_B⁰_α(H) =α(H)Bα

D⁰_H(Aα) =α(H)Aα, D_H⁰(Bα) =α(H)Bα.

We now remark that for anyα, β∈R, the expressions

[Eα, Eβ] +aαaβ[Eσ(α), Eσ(β)], i [Eα, Eβ]−aαaβ[Eσ(α), Eσ(β)] belong to g. We also know that a_α =a_σ(α) ∈ {±1}. Moreover, any root takes real values onh⁻ and purely imaginary values onh⁺. Therefore, the above computations show thatD⁰_g(g)⊂g, as required.

It remains to prove that R^D_X,Y⁰ Z = 0, for any X, Y, Z ∈ g^C. Without loss of generality, we assume thatX, Y, Z are either root vectors from the Weyl basis or vectors from the Cartan subalgebrah.With this assumption, the above relation follows from a long but straightforward computation, which uses the definition ofD⁰. Consider for example the case whenX :=

E_α, Y :=E_β and Z :=E_γ are all root vectors from the Weyl basis (the remaining situations, with at least one argument from h, can be treated similarly). We distinguish three subcases: I)σ(α) +γ6= 0 andσ(β) +γ6= 0;

II)σ(α) +γ= 0 andσ(β) +γ6= 0; III)σ(α) +γ=σ(β) +γ= 0. In subcase III), R^D_E⁰

α,EβE_γ = 0 (trivial, α = β). Let us study I) in more detail. We

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need to distinguish two further subcases of I), namely α+β 6= 0 and, respectively,α+β = 0. Suppose first thatα+β 6= 0.Since σ(β) +γ6= 0, [E_σ(β), E_γ] is either a root vector (whenσ(β) +γ∈R) or [E_σ(β), E_γ] = 0 (whenσ(β) +γ /∈R). We obtain, from the definition ofD⁰,

(4.15) D_E⁰_αD⁰_E_βEγ=−aβD_E⁰_α([E_σ(β), Eγ]) =aαaβ[E_σ(α),[E_σ(β), Eγ]].

Similarly, sinceα+β6= 0, [E_α, E_β] is either a root vector or [E_α, E_β] = 0.

We obtain (4.16) D⁰_[E

α,E_β]Eγ =NαβD⁰_E_α+βEγ =−aα+βNαβ[E_σ(α+β), Eγ] (withE_α+β=E_σ(α+β)= 0 anda_α+β=N_αβ= 0 whenα+β /∈R). From (4.15), (4.16) and the Jacobi identity, we deduce that

(4.17) R^D_E⁰

α,E_βE_γ =a_αa_β[E_γ,[E_σ(α), E_σ(β)]] +a_α+βN_αβ[E_γ, E_σ(α+β)].

On the other hand, it is easy to check that

aα+βNαβE_σ(α+β)=−aαaβ[E_σ(α), E_σ(β)].

It follows thatR^D_E⁰

α,E_βEγ = 0, as required. Suppose now that I) holds and α+β = 0.Using again the definition ofD⁰, we obtain

R^D_E⁰

α,E−αE_γ = σ(γ)(H_α)−γ(H_σ(α)) E_γ.

But this last expression is zero: using σ(Hα) = Hσ(α) (easy check) and γ(H_σ(α))∈R, we obtainσ(γ)(Hα) =γ(H_σ(α)). The remaining case II) can

be treated similarly. Our claim follows.

The preferred connection we are looking for is defined as follows.

Definition 4.8. — The connectionD⁰is the unique (flat) left-invariant connection onGwhich on left-invariant vector fields coincides with the map D⁰ from Lemma 4.7.

With the above preliminary considerations, we can now state our main result from this section. Below we denote by {ωα ∈ (g^C)^∗, α ∈ R} the covectors defined byωα(Eβ) =δαβ for anyα, β∈R andωα|h= 0.We use the notationR₀^sym:=R₀∩(−R0) for the symmetric part ofR₀.

Theorem 4.9. — Consider a triple(k,D⁰, ), with kthe regular subalgebra (4.11),D⁰as in Lemma 4.7 and∈k^∗⊗¯k^∗ skew-Hermitian. Assume that

(4.18) (α+β)|h_k6= 0, ∀α, β∈R0∪ {0}, α+β6= 0.

Then (k,D⁰, ) is a (symmetric) g-admissible triple (and the associated pair(J, D⁰)defines a complex structureJ^J^,D⁰ onT^∗G) if and only if the following conditions hold: