Symplectic Convexity Theorems and Applications to the Structure Theory of Semisimple Lie Groups

Home , Real form (Lie theory)

SYMPLECTIC CONVEXITY THEOREMS AND APPLICATIONS TO THE STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirement for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

Michael Otto

*****

The Ohio State University 2004

Dissertation Committee: Approved by

Professor T. Kerler

Professor B. Kr¨otz,Co-Adviser Adviser

Professor R. Stanton, Co-Adviser Department of Mathematics

ABSTRACT

Atiyah’s well known convexity theorem states that for a Hamiltonian torus action T × M → M on a compact connected symplectic manifold M the image Φ(M) under the associated moment map Φ : M → t∗ is convex. Duistermaat in addition considered antisymplectic involutions τ on M satisfying Φ ◦ τ = Φ. He showed that

Φ(M) = Φ(Q) for Lagrangian submanifolds Q that arise as ﬁxed point sets of such involutions.

We prove a generalization of Duistermaat’s symplectic convexity theorem for involutions τ which satisfy several compatibility properties with the torus action, but which are not necessarily antisymplectic. By the same method we can also extend a

Duistermaat-type theorem for non-compact M with proper Φ.

All the symplectic convexity theorems mentioned have applications to the structure theory of semisimple Lie groups. With the generalization of Duistermaat’s theorem we are able to extend existing symplectic proofs for Kostant’s and Neeb’s convexity theorems to all semisimple groups. In addition we develop the symplectic framework to reﬁne a recently discovered complex convexity theorem.

ii ACKNOWLEDGMENTS

First and foremost, I wish to thank Bernhard Kr¨otz for his role as adviser throughout my graduate studies. With his energy and enthusiasm he provided a great working environment.

I would also like to thank Bob Stanton for his support, especially during the process of writing this dissertation, and for his frequent valuable advice.

iii VITA

April 19, 1974 ...... Born — Braunschweig, Germany

2001 ...... Diplom, TU Clausthal/Germany 2000–present ...... Graduate Teaching Associate, The Ohio State University

PUBLICATION

B. Kr¨otzand M. Otto, Vanishing properties of analytically extended matrix coeﬃ- cients, Journal of Lie Theory 12, 2002, 409-421.

FIELDS OF STUDY

Major Field: Mathematics

iv TABLE OF CONTENTS

Page Abstract ...... ii Acknowledgments ...... iii Vita ...... iv

Chapters:

1. Introduction ...... 1

2. Symplectic Geometry ...... 7

2.1 Background ...... 8 2.2 Local normal forms ...... 18 2.3 Symplectic convexity theorems ...... 28

3. Lie theory ...... 41

3.1 Kostant’s convexity theorems ...... 42 3.1.1 Kostant’s theorems ...... 42 3.1.2 The linear version ...... 44 3.1.3 The nonlinear version ...... 47 3.2 Neeb’s convexity theorem for semisimple symmetric spaces . . 56 3.2.1 Statement of the theorem and reduction of the problem 57 3.2.2 A symplectic proof ...... 60 3.3 A reﬁnement of the complex convexity theorem of Gindikin- Kr¨otz ...... 65 3.3.1 Notation and basic facts ...... 66 3.3.2 Complex groups ...... 69 3.3.3 The complex convexity theorem for non-complex groups 76 Appendix: Poisson Lie groups and Manin triples ...... 81 Bibliography ...... 86

v CHAPTER 1

INTRODUCTION

A complex (n × n)-matrix A is called Hermitian if A = A¯T . Here the bar sym- bolizes the matrix operation where each matrix entry is replaced with its complex conjugate. The exponent T denotes the transposed matrix. It is a well known fact that a given Hermitian matrix A is diagonalizable. This means one can ﬁnd (complex) numbers λ1, . . . , λn, not necessarily distinct, and a complex matrix V with the property V −1 = V¯ T (such a V is called unitary), such that   λ  1    −1  ..  V AV =  .  .     λn

The λi’s are the eigenvalues of A and can easily shown to be real numbers.

How do the entries of A depend on the λi’s, or in other words, how much information on the matrix A is encoded in its eigenvalues ?

A partial answer to this question was provided by Schur in 1923 [21]. He showed that the range of the diagonal elements of A was restricted by the eigenvalues. Let

1 us be more speciﬁc. We denote the n-dimensional vector of diagonal elements of A by diag(A), and we write the eigenvalues in vector form (λ1, . . . , λn). Note that both

n diag(A) and (λ1, . . . , λn) are points in R . For the ﬁnite set of points which are obtained from (λ1, . . . , λn) by permuting the entries we write Sn.(λ1, . . . , λn) (the orbit of (λ1, . . . , λn) under the symmetric group Sn). The convex hull conv(Sn.(λ1, . . . , λn))

n of the ﬁnite set Sn.(λ1, . . . , λn) is a compact region in R which is determined uniquely by the eigenvalues λ1, . . . , λn.

Schur’s result asserts that the diagonal vector must lie in this region, i.e.

diag(A) ∈ conv(Sn.(λ1, . . . , λn)).

It turns out that diagonal vectors of Hermitian matrices with eigenvalues λ1, . . . , λn not only lie in conv(Sn.(λ1, . . . , λn)) but actually make up all that region, i.e.

{diag(A): A Hermitian with eigenvalues λ1, . . . , λn} = conv(Sn.(λ1, . . . , λn)).

This was proven by Horn in 1954 [12].

Statements on square matrices can often be extended to semisimple Lie groups.

But it was not until 1973 that Kostant [14] proved such a generalization for the

Schur-Horn result.

Let G be a connected semisimple Lie group with Lie algebra g. The Iwasawa decomposition of G generalizes the Gram-Schmidt decomposition of a complex matrix g with determinant one into a unique product g = nak of an upper triangular matrix n with ones on the diagonal, a real diagonal matrix a with determinant one and a unitary matrix k with determinant one. The usual notation is G = NAK for

2 certain subgroups N, A, K of G that are unipotent, abelian, compact, respectively.

On the Lie algebra level the Iwasawa decomposition states g = n ⊕ a ⊕ k, where n = Lie(N), a = Lie(A), k = Lie(K). It is a generalization of the fact that any complex matrix with trace zero can be uniquely written as the sum of a strictly upper triangular matrix, a real diagonal matrix with trace zero and a skew-Hermitian matrix.

The group Sn from above becomes the Weyl group W = NK (a)/ZK (a). We denote bya ˜ : G = NAK → A, nak 7→ a, and pra : g = n + a + k → a,Xn + Xa + Xk 7→ Xa the middle projections in the Iwasawa decompositions. Then Kostant’s result is the following.

Kostant’s convexity theorems. Let Y ∈ a. Then

1. pra(Ad(K).Y ) = conv(W.Y ) linear version

2. loga ˜(K exp(Y )) = conv(W.Y ) nonlinear version where conv(·) denotes the convex hull of (·).

The Schur-Horn theorem is a special case of the linear version in case G =

SL(n, C).

Atiyah [2] took this subject into a new and unexpected direction. He related the

Lie theoretic results of Kostant with symplectic geometry. He, and independently,

Guillemin and Sternberg [6], discovered a symplectic convexity theorem from which

Kostant’s linear result could be derived.

3 AGS-Theorem. Let (M, ω) be a compact connected symplectic manifold. Sup- pose a torus T acts on M in a Hamiltonian way with moment map Φ: M → t∗.

Then Φ(M) is convex. More precisely, Φ(M) is the convex polytope spanned by the

ﬁnite set Φ(Fix(M)), where Fix(M) denotes the T -ﬁxed points of M.

Subsequently, the AGS-theorem has been generalized in diﬀerent directions by numerous authors. Some consider not only torus actions but more general Hamil- tonian actions. Others study non-compact symplectic manifolds. In many instances additional conditions on the moment map are required.

One of the early extensions of the AGS-theorems will be of particular interest to us.

It was given by Duistermaat [4] who considered certain involutions on the underlying symplectic manifold.

Duistermaat’s Theorem. Let (M, ω) be a compact connected symplectic manifold. Suppose a torus T acts on M in a Hamiltonian fashion with moment map

Φ: M → t∗. In addition, let τ : M → M be an antisymplectic involution which satisﬁes Φ ◦ τ = Φ.

If Q denotes the set of τ-ﬁxed points of M, then Φ(Q) = Φ(M). In particular, Φ(Q) is convex. The statement remains true if one replaces Q with any of its connected components.

It turns out that antisymplecticity is a rather strong condition on the involution

τ that is not always satisﬁed in applications. One of the main results of this thesis is

4 the following theorem which states how the conditions on τ in Duistermaat’s theorem can be weakened.

Theorem A. [15] Let M be a compact connected symplectic manifold with Hamil- tonian torus action T × M → M and moment map Φ: M → t∗. Furthermore, let

τ : M → M be an involutive diﬀeomorphism with ﬁxed point set Q such that

1. t ◦ τ = τ ◦ t−1 for all t ∈ T .

2. Φ ◦ τ = Φ.

3. Q is a Lagrangian submanifold of M.

Then Φ(Q) = Φ(M) = conv Φ(Fix(M)) = conv Φ(Fix(Q)). In particular Φ(Q) is a convex subset of t∗. Moreover, the same assertions hold if Q is replaced with any of its connected components.

Note that this theorem extends the class of Lagrangian submanifolds Q in Duis- termaat’s theorem for which the image Φ(Q) is still the full convex set Φ(M).

We will also consider the convexity theorem by Hilgert, Neeb and Plank [11] which deals with the situation that M is non-compact and the moment map Φ is proper.

We will show that, again, the condition of Φ being antisymplectic can be weakened as in Theorem A.

All the symplectic convexity theorems listed have interesting applications to the structure theory of semisimple Lie groups. The AGS- and Duistermaat’s theorem

5 can be used to give a symplectic proof of Kostant’s linear convexity theorem.

Lu and Ratiu found a way to put even Kostant’ s nonlinear theorem into a symplectic framework [18]. The use of Duistermaat’s theorem, however, restricts their method to special cases of groups G. We will show that by using Theorem A these restrictions can be overcome.

Neeb’s convexity theorem for semisimple symmetric spaces is a generalization of

Kostant’s (nonlinear) theorem. In Kostant’s theorem one considers orbits of the compact group K. This K comes about as ﬁxed point group of the Cartan involution

θ on G. For arbitrary involutions σ on G the ﬁxed point group H = Gσ is not necessarily compact anymore. Neeb’s theorem describes the image of certain H- orbits under a projection map which is the middle projection of an Iwasawa-type decomposition of G adapted to the involution σ. A symplectic proof has been given in [10], but it has the same constraints as the method of Lu and Ratiu for Kostant’s theorem. The proof can be ﬁnished by applying our version of Theorem A for non- compact symplectic manifolds.

Finally, we will use symplectic techniques to reﬁne a complex convexity theorem recently discovered by Gindikin and Kr¨otz[5].

6 CHAPTER 2

SYMPLECTIC GEOMETRY

In this chapter we develop the symplectic convexity results mentioned in the introduction. These will serve as the main ingredients for the proofs in our Lie theoretic applications in the following chapter. However, this chapter is completely independent of the following one, being formulated entirely in symplectic geometric terms.

The ﬁrst section introduces the fundamental concepts, in particular the notion of a Hamiltonian torus action with its corresponding moment map on a symplectic manifold. We are especially interested in involutions on this manifold which satisfy certain compatibility properties with the torus action. The second section is of a rather technical nature. Here, we consider local properties of the moment map and

ﬁnd local normal forms. This local information will be used in the third section to derive the convexity theorems which state global properties of the moment map

(provided this map is proper). Major parts of this chapter appear in [15].

7 2.1 Background

We introduce notation and basic concepts in symplectic geometry. Most of the material can be found in standard text books such as [7].

Deﬁnition 2.1. A real vector space V equipped with a bilinear form Ω is called symplectic if Ω is antisymmetric and non-degenerate.

One can show by induction that for a symplectic vector space (V, Ω) one can always ﬁnd a basis for V with respect to which Ω takes the matrix form   0 1        −1 0     .  Ω =  ..  .        0 1      −1 0

In particular, a symplectic vector space V has to be of even dimension.

For a subspace W ⊂ V denote by W ⊥ its orthogonal complement with respect to

Ω. One calls W

isotropic, if W ⊂ W ⊥,

coisotropic, if W ⊥ ⊂ W,

Lagrangian, if W = W ⊥.

A smooth map φ :(V, Ω) → (V, Ω) is called symplectic if it preserves the symplectic form Ω, i.e.

Ω(x, y) = Ω(Dφ(v).x, Dφ(v).y) ∀ v, x, y ∈ V,

8 where Dφ(v) denotes the diﬀerential of φ at v ∈ V .

This notion for vector spaces can be generalized to manifolds in the following way.

Let M be a connected smooth manifold of dimension 2n. Let us denote the tangent

∗ space at a point m ∈ M by TmM, the cotangent space by TmM . A 2-form ω on M deﬁnes a smooth family of bilinear forms ωm on TmM × TmM.

Deﬁnition 2.2. A 2-form ω on M is called symplectic if it satisﬁes

(1) ωm(X,Y ) = −ωm(Y,X) ∀ m ∈ M,X,Y ∈ TmM.

∗ ∗ (2) ωm ∈ TmM ⊗ TmM is non-degenerate for all m ∈ M.

(3) dω = 0, i.e. ω is a closed form.

The pair (M, ω) is called symplectic manifold.

Note that the first two conditions define a family {(TmM, ωm): m ∈ M} of symplectic vector spaces, for which the symplectic forms ωm are unrelated. The closedness condition on ω forces some relation on the ωm’s. In particular, it implies that a symplectic manifold has local charts ϕi : Ui → V into a fixed symplectic vector

−1 space (V, Ω) such that the coordinate changes ϕi ◦ ϕj preserve Ω. A submanifold P ⊂ M is called

isotropic, if TpP ⊂ TpM is isotropic for all p ∈ P,

coisotropic, if TpP ⊂ TpM is coisotropic for all p ∈ P,

Lagrangian, if TpP ⊂ TpM is Lagrangian for all p ∈ P.

The symplectic form ω, as every 2-form on M, induces a mapping from the vector

fields on M to the set of differential forms: A vector field X gives a differential form

9 ι(X )ω = ω(X , ·).

Since ω is non-degenerate this mapping is invertible. In particular, for every smooth

∞ function f ∈ C (M) there is a unique vector ﬁeld Xf such that df = ι(Xf )ω. This vector ﬁeld Xf is called Hamiltonian, and f is its Hamiltonian function.

Deﬁnition 2.3. A smooth manifold M is called a Poisson manifold if there is a bracket operation {, } : C∞(M) × C∞(M) → C∞(M) such that

1. {, } is bilinear and antisymmetric,

2. For f, g, h ∈ C∞(M),

{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 (Jacobi identity),

3. For f, g, h ∈ C∞(M),

{f, gh} = {f, g}h + g{f, h} (Leibniz rule).

A symplectic manifold (M, ω) is automatically a Poisson manifold. Its Poisson bracket is deﬁned by

∞ ∞ ∞ {, } : C (M) × C (M) → C (M), (f, g) 7→ {f, g} = ω(Xf , Xg).

For basic properties of general and some special Poisson manifolds, see the Appendix.

A map φ : M1 → M2 between two symplectic manifolds (M1, ω1) and (M2, ω2) is

∗ ∗ called symplectic if it preserves the symplectic forms, i.e. φ ω2 = ω1. Here, φ stands for the pull back. If Dφ denotes the diﬀerential of φ (at some point), then φ∗ is the extension of the dual map (Dφ)∗ to the exterior algebra of diﬀerential forms.

10 Let G be a Lie group that acts on the manifold M via diﬀeomorphisms,

G × M → M, (g, m) 7→ g.m =: φg(m).

The G-action is called symplectic if each g ∈ G acts as a symplectic map on (M, ω),

∗ i.e. φgω = ω ∀ g ∈ G. Let g denote the Lie algebra of G. Each X ∈ g deﬁnes a vector ﬁeld X˜ on M via

˜ d Xm = exp(tX).m ∀ m ∈ M. dt t=0

Moreover, since ω is symplectic, X ∈ g gives rise to a diﬀerential form ι(X˜)ω. The form ι(X˜)ω is closed as the following argument shows. Recall ”Cartan’s magic formula”

LX = ι(X )d + dι(X )

which, for a vector ﬁeld X , relates the Lie derivative LX , the inner product ι and the exterior diﬀerential d.

In our case ι(X˜)dω = 0, because ω is closed by assumption. Since X ∈ g and G was assumed to act symplectically,

d d ∗ LX˜ ω = φexp(tX)ω = ω = 0. dt t=0 dt t=0

It follows that dι(X˜)ω = 0.

If ι(X˜) turns out to be even exact with

˜ i(X)ω = dΦX , (2.1)

∞ we call ΦX ∈ C (M) a Hamiltonian function (for X ∈ g).

11 Deﬁnition 2.4. Suppose G acts on M in a way such that there exists a Hamiltonian function ΦX for every X ∈ g and such that the mapping

∞ g → (C (M), {, }),X 7→ ΦX is a G-equivariant Lie algebra homomorphism. Then we call the action of G on M

Hamiltonian. In this case the mapping Φ : M → g∗ of M into the dual g∗ of g deﬁned by

hΦ(m),Xi = ΦX (m) ∀ m ∈ M,X ∈ g, is called the moment map for the Hamiltonian action of G on M.

We will be interested in Hamiltonian torus actions, i.e. in the case when G = T is a torus. Then we have a moment map Φ : M → t∗ where t denotes the abelian Lie algebra of T and t∗ its dual. Write Fix(M) for the set of T -ﬁxed points in M. Note that

˜ d t.m = m ∀ t ∈ T ⇔ Xm = exp(tX).m = 0 ∀ X ∈ t dt t=0 ˜ ⇔ dΦX (m) = ι(Xm)ωm = 0 ∀ X ∈ t

⇔ dΦ(m) = 0.

Hence, Fix(M) is the critical set of the moment map Φ : M → t∗.

Let τ : M → M be an involutive diﬀeomorphism. We will denote by Q its ﬁxed point set, i.e. Q = {m ∈ M : τ(m) = m}, and assume Q to be non-empty. Notice that Q is a closed submanifold of M.

We will be interested in involutions τ satisfying special conditions.

12 T -τ-compatibility: t ◦ τ = τ ◦ t−1 for all t ∈ T . (2.2)

Φ ◦ τ = Φ. (2.3)

Q is a Lagrangian submanifold of M. (2.4)

Remark 2.5. 1. Taking derivatives one sees that condition (2.2) is equivalent to

˜ ˜ dτ(m)Xm = −Xτ(m) ∀ m ∈ M,X ∈ t. (2.5)

In particular we have

˜ ˜ dτ(m)Xm = −Xm ∀ m ∈ Q, X ∈ t. (2.6)

2. Suppose τ is antisymplectic, i.e. τ ∗ω = −ω. Then (2.2) and (2.3) are equivalent,

and (2.4) is automatically satisﬁed.

3. Considering involutions τ with properties (2.2), (2.3) and (2.4) seems to be

a rather unmotivated generalization of the more natural setup of an antisym-

plectic involution satisfying (2.2). However, in the Lie theoretic applications of

the following chapters we will encounter quite natural involutions which satisfy

(2.2), (2.3) and (2.4) but which are not antisymplectic.

To illustrate the concepts introduced we want to give an example. The setup is that of a symplectic vector space. However, it is central to our treatment of more general symplectic manifolds in the next section where we linearize torus action and involution.

13 Example 2.6. Consider the symplectic vector space (R2n, Ω) where Ω has the matrix form   0 1        −1 0     .  Ω =  ..  .        0 1      −1 0

The symplectic group Sp(2n) (with respect to Ω) is the group of invertible real

(2n) × (2n)-matrices A that preserve Ω, i.e. AΩAT = Ω. A linear symplectic action by the torus T is a representation ρ : T → Sp(2n). Since T is compact and abelian the elements of ρ(T ), viewed as operators on C2n, are simultaneously diagonalizable.

Therefore, on R2n one can choose a basis such that ρ : T → Sp(2n) diﬀerentiates to the Lie algebra homomorphism   0 λ (X)  1       −λ1(X) 0     .  ρ : t → sp(2n),X 7→  ..  , (2.7)        0 λn(X)      −λn(X) 0

∗ for certain weights λ1, . . . , λn ∈ t .

The action of T is Hamiltonian with moment map

n X 1 Φ: 2n → t∗, (q , p , . . . , q , p ) 7→ λ (q2 + p2) (2.8) R 1 1 n n 2 j j j j=1

14 The linear involution τ given by   1 0        0 −1     .  τ =  ..         1 0      0 −1 is antisymplectic. It is immediate from (2.8) that Φ ◦ τ = Φ.

We shall investigate the condition of T -τ-compatibility in the linear case, i.e. when (M, ω) is a symplectic vector space (V, Ω) with linear torus action and linear involution τ.

Let (V, Ω) denote a ﬁnite dimensional symplectic vector space and τ : V → V a linear involution. As τ ◦ τ = idV , the linear operator τ is semisimple with eigenvalues +1 and −1. Accordingly we have an eigenspace decomposition V = V1 ⊕ V−1.

Next we endow (V, Ω) with a linear symplectic torus action T × V → V . Then V decomposes into ﬁxed and eﬀective part

V = Vﬁx ⊕ Veﬀ , (2.9) where

Vﬁx = {v ∈ V : X.v = 0 ∀ X ∈ t}

= {v ∈ V : t.v = v ∀ t ∈ T }, and

Veﬀ = t.V.

15 Notice that Ω remains non-degenerate when restricted to Vfix or Veff . Hence both Vfix and Veff become symplectic subspaces of V .

The following lemma gives equivalent characterizations of τ being antisymplectic.

Note that the ﬁrst three statements do not rely on the existence of the torus action at all.

Lemma 2.7. Let (V, Ω) be a symplectic vector space and τ : V → V be a linear involution with eigenspace decomposition V = V1⊕V−1. Then the following statements are equivalent:

1. τ is antisymplectic.

2. V1 and V−1 are Lagrangian.

3. V1 is Lagrangian and there exists a symplectic isomorphism ϕ : V → V with

ϕ(V1) = V−1 and ϕ(V−1) = V1.

4. V1 is Lagrangian and there exists a linear symplectic torus action T × V → V

with the following properties:

(a) t ◦ τ = τ ◦ t−1 ∀ t ∈ T.

(b) Vﬁx = {0}.

Proof. For v ∈ V let v1 ∈ V1, v−1 ∈ V−1 be such that v = v1 + v−1.

(1) =⇒ (2): Assume that τ is antisymplectic. Then, for v, w ∈ V ,

Ω(v1, w1) = Ω (τ(v1), τ(w1)) = −Ω(v1, w1), and similarly,

Ω(v−1, w−1) = Ω (−τ(v−1), −τ(w−1)) = −Ω(v−1, w−1).

16 This implies that V1 and V−1 are isotropic. As V = V1 ⊕ V−1, it follows that both V1 and V−1 are maximally isotropic, i.e. Lagrangian subspaces of V .

(2) =⇒ (1): Assume that V1 and V−1 are Lagrangian. We compute

Ω(v1 + v−1, w1 + w−1) = Ω(v1, w1) + Ω(v−1, w1) + Ω(v1, w−1) + Ω(v−1, w−1)

= Ω(v−1, w1) + Ω(v1, w−1), and

Ω(τ(v1 + v−1), τ(w1 + w−1)) = Ω(v1 − v−1, w1 − w−1)

= Ω(v1, w1) − Ω(v−1, w1) − Ω(v1, w−1) + Ω(v−1, w−1)

= −Ω(v1 + v−1, w1 + w−1).

Hence, τ is antisymplectic.

(2) =⇒ (3): As Lagrangian subspaces both V1 and V−1 have dimension n =

1 2 dimV . Let {e1, . . . en} be a basis of V1. As Ω is non-degenerate, assumption (2) implies that there is a basis {f1, . . . fn} of V−1 such that Ω(ei, fj) = δij for all 1 ≤ i, j ≤ n.

Deﬁne ϕ by

ϕ(ei) = fi, ϕ(fi) = −ei ∀ i.

Then for all 1 ≤ i, j ≤ n,

Ω(ei, fj) = δij = Ω(ej, fi) = −Ω(fi, ej) = Ω(ϕ(ei), ϕ(fj)), completing the proof of (2) =⇒ (3).

(3) =⇒ (2): It suﬃces to show that V−1 is isotropic. But this follows from the fact that ϕ is surjective and that for v, w ∈ V1,

Ω(ϕ(v), ϕ(w)) = Ω(v, w) = 0.

17 (3) =⇒ (4): For t = Rϕ, the corresponding torus action by T = exp t clearly has the desired properties.

(4) =⇒ (3): Let T × V → V be a torus action which satisﬁes (4a) and (4b). We notice that (4a) is equivalent to its inﬁnitesimal version

τ ◦ X = −X ◦ τ ∀ X ∈ t.

For any X ∈ t this implies

X(V1) ⊆ V−1 and X(V−1) ⊆ V1 .

If we set Y = exp X ∈ T , then

Y (V1) = V−1 and Y (V−1) = V1 .

Since Y ∈ T acts as a symplectic isomorphism on V we have found the isomorphism

ϕ in (3).

2.2 Local normal forms

This section is essentially Section 2.2 in [15]. We will assume that (M, ω) is a connected symplectic manifold endowed with a Hamiltonian torus action with momentum map Φ. Also, we have an involution τ such that the T -τ-compatibility is satisfied. We denote the set of τ-fixed points of M by Q. Our objective is to provide a local normal form for Φ Q near a point m ∈ Q. To that end we first recall a method of finding suitable local descriptions for ω and Φ in the neighborhood of a generic point m ∈ M. We then consider points m ∈ Q and obtain a refined form of

Φ Q which is adapted to the involution τ. We start with a simple observation (cf.[4], Lemma 2.1):

Lemma 2.8. Let m ∈ Q and X ∈ t. Then d(ΦX Q)(m) = 0 implies dΦX (m) = 0.

In particular, m is ﬁxed under the action of the one parameter subgroup exp(RX).

Proof. Write E = TmM for the tangent space at m. Let E = E1 ⊕ E−1 be the decomposition of E into ±1-eigenspaces of the involution dτ(m). Notice that TmQ =

E1. In order to show that dΦX (m) = 0 it hence suﬃces to prove dΦX (m)(v) = 0 for all v ∈ E−1.

Let v ∈ E−1. Then it follows from (2.3) that

dΦX (m)(v) = −dΦX (m)(dτ(m)v) = −d(ΦX ◦ τ)(m)(v) = −dΦX (m)(v),

and so dΦX (m)(v) = 0. The last assertion in the Lemma follows from (2.1). This concludes the proof of the lemma.

For m ∈ M we write Tm for the stabilizer of T in m, i.e.

Tm = {t ∈ T : t.m = m}.

The Lie algebra tm of Tm is then given by

˜ tm = {X ∈ t : Xm = 0}.

If in addition m ∈ Q, then it follows from Lemma 2.8 that we can equally characterize tm by

tm = {X ∈ t : d(ΦX Q)(m) = 0}. (2.10)

Fix now m ∈ M. Next we provide local normal forms for ω and Φ near m. We will recall the procedure of momentum reconstruction (cf.[6], Chapter 41): The moment map near m is uniquely characterized by Φ(m), the stabilizer Tm and the linear representation of Tm on the tangent space TmM. This is even true for a general

19 compact Lie group T and in case of a torus was further exploited in [11], Sect. 2.

Notice that Tm(T.m) is an isotropic subspace of TmM. Thus ωm induces on the quotient

⊥ V = Tm(T.m) /Tm(T.m) a symplectic form Ω. We write Sp(V, Ω) for the corresponding symplectic group.

Notice that the isotropy subgroup Tm acts on TmM symplectically. Clearly this action

⊥ leaves Tm(T.m) and hence Tm(T.m) invariant, thus giving rise to a representation on V , say

π : Tm → Sp(V, Ω).

Write (Tm)0 for the connected component of Tm containing 1. Notice that (Tm)0 < T is a subtorus and so we can ﬁnd a torus complement Sm to (Tm)0 in T , i.e.

T = Sm × (Tm)0.

We denote the Lie algebra of Sm by sm. Then t = tm ⊕ sm and we have a canonical

∗ ∗ ∗ ∗ identiﬁcation t = sm × tm. Denote by T Sm the cotangent bundle of Sm with its

∗ ∗ canonical symplectic structure. We will use the identiﬁcation T Sm = sm × Sm.

∗ Hence T Sm × V carries a natural symplectic structure. Further T = Sm × (Tm)0

∗ acts symplectically on T Sm × V via

(s, t).(β, s0, v) = (β, ss0, π(t)v), (2.11)

0 ∗ for s, s ∈ Sm, t ∈ (Tm)0, β ∈ sm and v ∈ V . It follows from [11], Lemma 2.1 that there is a symplectic diﬀeomorphism

∗ φ : T Sm × V ⊃ U → U ⊂ M

20 ∗ from an open neighborhood U of (0, 1, 0) ∈ T Sm × V to an open neighborhood U of m such that φ is locally T -equivariant and satisﬁes φ(0, 1, 0) = m.

∗ In the following we will identify U with U ⊂ T Sm × V via our symplectic, locally

T -equivariant chart φ : U → U. Write V = Vfix ⊕ Veff for the decomposition of V into effective and fixed part for the linear action of Tm on V (cf. (2.9)). Furthermore we have the tm-weight space decomposition V = ⊕λ∈ΛVλ. We decompose elements v ∈ V P as v = λ∈Λ vλ with vλ ∈ Vλ. Recall that there is a Tm-invariant complex structure

J on Veff such that hv, wi = Ω(v, Jw) defines a positive definite scalar product on

Veﬀ . ( Choose a basis for Veﬀ with respect to which an element X ∈ tm acts as in

(2.7) (the λi are not necessarily distinct). For some Z ∈ tm with λi(Z) 6= 0 ∀i, let J be the endomorphism of Veﬀ with matrix representation ρ(Z). )

Then it follows from [11], Lemma 2.2 that the local normal form of Φ near a generic point m ∈ M is given on U by

1 X Φ: U → t∗ = s∗ × t∗ , (β, s, v) 7→ Φ(0, 1, 0) + β, kv k2λ . (2.12) m m 2 λ λ∈Λ λ6=0

Assume now that m ∈ Q. So far we have not adressed the question of the nature

∗ of Q and τ within our new coordinates in T Sm × V . And we would like to ﬁnd a normal form for Φ Q. In case τ is antisymplectic on M, there is a beautiful answer (cf. Remark 2.10 below).

However, with our restricted assumption of T -τ-compatibility we cannot hope for such a nice form.

Near (0, 1, 0) the shape of Q is essentially determined by the linear involution σ = dτ(0, 1, 0) on E = T(0,1,0)U'TmM.

∗ ∗ We will use the natural identification E = sm × sm × V . Define W = sm × sm × Vfix.

21 Then it follows from (2.11) that E = W ⊕ Veff is the decomposition of E into fixed and effective part of the isotropy action of (Tm)0 on E 'TmM. Then (2.2) implies that the involution σ leaves the decomposition E = W ⊕ Veff invariant. Hence  

σ 0 σ =  W  (2.13)   0 σ Veﬀ

Accordingly we have a splitting

T(0,1,0)Q = T(0,1,0)Q ∩ W ⊕ T(0,1,0)Q ∩ Veﬀ .

Next we will analyze the pieces σ and σ . We start with σ . Notice that it W Veﬀ Veﬀ follows from (2.2) that

−1 π(t) ◦ σ = σ ◦ π(t )(t ∈ Tm). (2.14) Veﬀ Veﬀ

Write Veff = Veff,1 ⊕ Veff,−1 for the decomposition of V into eigenspaces of σ . Thus Veff   id 0 Veff,1 σ =   . (2.15) Veff   0 −idVeff,−1

Notice that Veff,1 = T(0,1,0)Q ∩ Veff . Since W and Veff are orthogonal with respect to

ωm condition (2.4) of the T -τ-compatibility implies that dim(Veﬀ,1) = dim(Veﬀ,−1).

In particular, dim(Veﬀ,1) is a Lagrangian subspace of Veﬀ .

Thus we can apply Lemma 2.7 (to V = Veff , τ = σ and V1 = T(0,1,0)Q ∩ Veff ) and Veff conclude:

σ is antisymplectic. (2.16) Veﬀ

For the scalar product h·, ·i on Veﬀ this means that we may assume in addition that it is invariant under σ . Veﬀ

Next we turn our attention to σ W . From (2.6) and the concrete formula (2.11) for the Sm-action it follows that   ∗ 0 ∗     σ =   (2.17) W  ∗ −idsm ∗      ∗ 0 ∗

∗ with respect to a basis of W compatible with W = sm × sm × Vﬁx. A consequence of

(2.3) is dΦ(0, 1, 0) ◦ σ = dΦ(0, 1, 0). From (2.13) it hence follows that dΦ(0, 1, 0) W ◦

∗ σW = dΦ(0, 1, 0) W . As dΦ(0, 1, 0) W is the projection W → sm along sm × Vﬁx, the pattern (2.17) further simpliﬁes to   id ∗ 0 0  sm    σ =   (2.18) W  ∗ −idsm ∗      ∗ 0 D

for some linear operator D : Vﬁx → Vﬁx. As σ W ◦ σ W = idW we derive from

2 (2.18) that D = idVﬁx . Accordingly we obtain an eigenspace decomposition Vﬁx =

Vfix,1 ⊕ Vfix,−1 for D. Again, (2.4) implies dim(Vfix,1) = dim(Vfix,−1). With respect to

∗ the refined decomposition W = sm × sm × Vfix,1 × Vfix,−1 we then have   ids∗ 0 0 0  m     A −id B B   sm 1 2  σ =   . W    C 0 id 0   1 Vfix,1   

C2 0 0 −idVﬁx,−1

Employing again σ W ◦ σ W = idW we obtain that B2 = 0 and C1 = 0. Thus   ids∗ 0 0 0  m     A −id B 0   sm  σ =   (2.19) W    0 0 id 0   Vﬁx,1   

C 0 0 −idVﬁx,−1

∗ ∗ for linear operators A : sm → sm, B : Vfix,1 → sm and C : sm → Vfix,−1. Combining (2.13) and (2.19) we then obtain   id ∗ 0 0 0  sm       A −idsm B 0       0 0 id 0   Vfix,1  σ =   (2.20)    C 0 0 −id   Vfix,−1       idVeff,1 0      0 −idVeff,−1

∗ with respect to a basis of E compatible with E = sm×sm×Vfix,1×Vfix,−1×Veff,1×Veff,−1.

With the help of (2.20) we can now determine the tangent space T(0,1,0)Q ⊂ E. Notice that

T(0,1,0)Q = {v ∈ E : σ(v) = v},

24 so that (2.20) implies that     x         1    (Ax + By)    2       y     ∗  T(0,1,0)Q =   : x ∈ sm, y ∈ Vﬁx,1, z ∈ Veﬀ,1 . (2.21)  1    2 Cx          z             0 

∗ Write pr : E → sm × Vfix,1 × Veff,1 for the projection along sm × Vfix,−1 × Veff,−1. Then

∗ (2.21) implies that pr : T(0,1,0)Q → sm × Vﬁx,1 × Veﬀ,1 is a linear isomorphism. T(0,1,0)Q This in turn allows us to apply the implicit function theorem: there exist an open

∗ neigborhood U1 of 0 in sm × Vﬁx,1 × Veﬀ,1, an open neigborhood U2 of (1, 0, 0) in

Sm × Vfix,−1 × Veff,−1 and a differentiable map

ψ = (ψS, ψfix, ψeff ): U1 → U2 such that ψ(0, 0, 0) = (1, 0, 0) and     x            ψS(x, y, z)          y     ∗  Q ∩ (U1 × U2) =   ∈ T Sm × V :(x, y, z) ∈ U1 . (2.22)     ψfix(x, y, z)          z             ψeff (x, y, z)) 

One can say a little bit more about the map ψ when we notice that T(0,1,0)Q can

25 equally be expressed as     u            dψS(0)(u, v, w)          v     ∗  T(0,1,0)Q =   ∈ E : u ∈ sm, v ∈ Vfix,1, w ∈ Veff,1 . (2.23)     dψfix(0)(u, v, w)          w             dψeff (0)(u, v, w) 

Comparing (2.21) with (2.23) yields

dψeﬀ (0) = 0. (2.24)

We are now ready to summarize the discussion of this subsection. In more compact notation we have proved the following:

Theorem 2.9. (Local normal forms for Φ and Φ Q) Let (M, ω) be a connected symplectic manifold of dimension 2n. Suppose there exist a Hamiltonian torus action and an involution

τ : M → M which satisfy the T -τ-compatibility. Let m ∈ Q. Write k = dim sm =

∗ k ∗ dim t/tm and identify t ' R × tm. Then there exist an open neighborhood U of m, symplectic coordinates x, y, q, p = x1, . . . , xk, y1, . . . , yk, q1, . . . , qn−k, p1, . . . , pn−k with x(m) = y(m) = q(m) = p(m) = 0 and an integer 0 ≤ l ≤ n − k such that:

∗ k ∗ 1. On U the moment map Φ: M → t ' R × tm is given by

l 1 X Φ = Φ(m) + x, λ (q2 + p2) , 2 j j j j=1

∗ with weights λ1, . . . , λl ∈ tm \{0}.

26 2. The restriction (x, q): Q∩U → Rn of (x, q) to Q∩U is a diﬀeomorphism onto

n n an open ball Br (0) of radius r > 0 in R . Furthermore

l 1 X 2 2 Φ = Φ(m) + x, λj(q + ψj(x, q) ) , Q 2 j j=1

n l where ψ = (ψ1, . . . , ψl): Br (0) → R is a diﬀerentiable map with ψ(0) = 0 and dψ(0) = 0.

Proof. To explain the notation in the theorem: (x, y) are symplectic coordinates for

∗ ∗ ∗ T Sm = sm × Sm with x corresponding to sm and y to Sm; next (q, p) are symplec- L tic coordinates for V compatible with the weight space decomposition V = λ Vλ, where q1, . . . , ql correspond to Veﬀ,1 and ql+1, . . . , qN correspond to Vﬁx,1 (and similar for p).

From (2.12) we obtain (1) by choosing any orthonormal basis of Veff and let q1, . . . , ql, p1, . . . , pl represent coefficients with respect to this basis. The antisymplecticity of σ in (2.16) implied that the inner product on Veff could be assumed Veff

σ -invariant. Then the orthonormal basis of Veﬀ can be chosen such that the coef- Veﬀ

ficients qi belong to basis vectors in Veff,1 and the pi belong to vectors in Veff,−1.

Finally, the assertion in (2) follows from (1) and (2.22) combined with (2.24). Here the map ψ corresponds to what was called ψeﬀ above.

Remark 2.10. Under the condition that τ is antisymplectic on M and satisﬁes (2.2)

(or, equivalently, (2.3)) the local description of Q, τ and Φ becomes much simpler.

The reason is that for antisymplectic τ an equivariant version of Darboux’s theorem can be applied. This reduces the situation to a linear Hamiltonian action of T and a linear antisymplectic involution τ on the symplectic vector space (TmM, ωm). Then,

27 local coordinates (x, y, q, p) can be found such that τ(x, y, q, p) = (x, −y, q, −p).

Hence Q takes the form Q = {(x, 0, q, 0)}, and

l 1 X 2 Φ = Φ(m) + x, λjq . Q 2 j j=1 See [4], Prop. 2.2, for details.

2.3 Symplectic convexity theorems

In this section we want to discuss a series of convexity theorems. ”Convexity”, in this context, is a feature of the image of the moment map.

In the following (M, ω) always denotes a connected symplectic manifold of dimension

2n. We assume that a torus T acts on M in a Hamiltonian way with moment map

Φ: M → t∗. By Fix(M) we denote the T -ﬁxed points of M.

The starting point is the well known symplectic convexity theorem of Atiyah and

Guillemin-Sternberg which reads as follows ([2],[6]).

Theorem 2.11. If M is compact, then Φ(M) is convex. More precisely, Φ(M) is the convex polyhedron spanned by the ﬁnite set Φ(Fix(M)).

This theorem has been generalized by Duistermaat [4]. Assume that M carries an antisymplectic involution τ : M → M and write Q for the ﬁxed point set of τ.

We require that Q is non-empty. Then Q is a Lagrangian submanifold of M, and

Duistermaat’s convexity theorem [4], Th. 2.5, says:

Theorem 2.12. If M is compact, and τ : M → M is an antisymplectic involution which satisﬁes Φ ◦ τ = Φ, then Φ(Q) = Φ(M). In particular Φ(Q) is convex.

The statement remains true if one replaces Q with any of its connected components.

28 We want to show that the conditions on τ in Duistermaat’s theorem can be weakened. The statement of the theorem remains true if the T -τ-compatibility condition from Section 2.1 holds.

Later in this section we will consider the case where M is not necessarily compact anymore.

Our setup is as follows. Let (M, ω) be a (not necessarily compact) connected symplectic manifold with Hamiltonian torus action T × M → M. The corresponding moment map Φ : M → t∗ is assumed to be proper. Note that for compact M this is satisfied automatically. Without loss of generality we can assume that T acts effec- T T tively, i.e. m∈M Tm = {1} (otherwise we replace T with T/ m∈M Tm). Then the set of points m ∈ M with trivial stabilizer is open and dense. In addition, we have an involutive diffeomorphism τ : M → M which satisfies the T -τ-compatibility and for which the fixed point set Q in M is not empty.

We introduce some notation. Fix m ∈ Q with stabilizer Tm and tm = Lie(Tm).

+ + ∗ Let R =]0, ∞[ and R0 = [0, ∞[ and deﬁne a closed convex cone in tm by

l X + l Γm = { sjλj : s = (s1, . . . , sl) ∈ (R0 ) }, j=1 where the λj are the weights from Theorem 2.9. Note that since T acts eﬀectively

∗ the λj linearly span tm. Theorem 2.9(1) implies that an open neighborhood U of m in M is mapped by Φ

k onto an open neighborhood of Φ(m) in the cone Cm = Φ(m) + R + Γm. We want to see what the image of a neighborhood of m in Q looks like.

29 Lemma 2.13. Let Q0 be an arbitrary connected component of Q.

∗ 0 1. Suppose Γm = tm for m ∈ Q . For any neighborhood U of m in M the image Φ(U ∩ Q0) contains an open neighborhood of Φ(m) in t∗.

2. The set int(Φ(Q0)) of inner points of Φ(Q0) ⊂ t∗ is dense. More speciﬁ-

cally, for any m ∈ Q0 there is an m0 ∈ Q0 arbitrarily close to m such that

0 0 Φ(m ) ∈ int(Φ(Q )) ∩ Cm.

∗ Proof. 1. Since Γm = tm, there exists a vector

l k + l X ∗ v = (0; v1, . . . , vl) ∈ R × (R ) such that vjλj = 0 ∈ tm. (2.25) j=1

For any neighborhood U of v ∈ Rk × (R+)l the set

{(u1, . . . uk, uk+1λ1 + ··· + uk+lλl):(u1, . . . , uk+l) ∈ U}

contains an open neighborhood of 0 in t∗.

n l Recall the map ψ : Br (0) → R from Theorem 2.9(2). We deﬁne

1 1 Ψ: Bn(0) → k ×( +)l, (x, q) 7→ (x, (q2 +ψ (x, q)2),..., (q2 +ψ (x, q)2)). r R R0 2 1 1 2 l l (2.26)

According to Theorem 2.9(2) it is suﬃcient to show that imΨ contains a point

v as in (2.25) as inner point. Fix v satisfying (2.25). The claim will be proved

if we can show that imΨ contains sv as an inner point for some s > 0. In the

following we will verify this. We may assume that kvk = 1.

The properties ψ(0) = 0, dψ(0) = 0 from Theorem 2.9(2) are crucial. Together

30 with Taylor’s formula they imply that we can ﬁnd a constant K > 0 such that for all 1 ≤ j ≤ l,

2 |ψj(x, q)| ≤ Kk(x, q)k , (2.27) for every (x, q) ∈ Rk × (R+)n−k suﬃciently close to (0, 0).

k+l There is another constant C > 0 such that the ball BsC (sv) lies entirely in

k + l k+l R × (R ) for all s > 0. We notice that every point in BsC (sv) can be written

1 2 1 2 + as (x1, . . . , xk, 2 q1,..., 2 ql ) with unique x1, . . . , xk ∈ R, q1, . . . , ql ∈ R . From now on let 0 < s ≤ 1. Observe that the condition

1 1 (x , . . . , x , q2,..., q2) ∈ Bk+l(sv) 1 k 2 1 2 l sC

˜ puts restrictions on the vector (x1, . . . , xk, q1, . . . , ql): there is a constant C > 0

(independent of the vector) such that

√ ˜ k(x1, . . . , xk, q1, . . . , ql)k ≤ sC.

In particular, (2.27) implies that for some K˜ > 0,

1 1 k(0,..., 0, ψ (x, ˜q)2,..., ψ (x, ˜q)2k ≤ Ks˜ 2. (2.28) 2 1 2 l

k + l n−k−l for all (x, ˜q) = (x1, . . . , xk, q1, . . . , ql, 0,... 0) ∈ R × (R ) × R with

1 2 1 2 k+l (x1, . . . , xk, 2 q1,..., 2 ql ) ∈ BsC (sv). ˜ 2 Choose 0 < s0 ≤ 1 small enough such that Ks0 < s0C holds. Set = s0C. We are now in a position to apply Brouwer’s ﬁxed point theorem: consider the mapping

k+l k+l ξ : B (0) → B (0), 1 1 1 1 (x , . . . , x , q2,..., q2) − s v 7→ −(0,..., 0, ψ (x, ˜q)2,..., ψ (x, ˜q)2). 1 k 2 1 2 l 0 2 1 2 l

31 1 2 1 2 If (x, 2 q1,..., 2 ql ) − s0v is a ﬁxed point of ξ, then 1 1 Ψ(x, ˜q) = (x, (q2 + ψ (x, ˜q)2),..., (q2 + ψ (x, ˜q)2)) = s v. 2 1 1 2 l l 0

We want to show that s0v is an inner point of imΨ. Notice that by choosing

s0 small enough we can assume that the point (x, ˜q) with Ψ(x, ˜q) = s0v is

arbitrarily close to (0, 0). The mapping Ψ is submersive at most points close

enough to the origin as a look at its derivative shows:   idk 0 ··· 0        q1 ··  dΨ(x, q) =    .   .. ··      ql 0 ··· 0   0 0   +   ∂ψi ∂ψi (ψi(x, q) (x, q))i,j (ψi(x, q) (x, q))i,j ∂xj ∂qj Relation (2.27) implies that for (x, q) approaching (0, 0) the entries in the sec-

ond summand become arbitrarily small compared to those in the ﬁrst summand.

Since the ˜q under consideration satisfy q1, . . . , ql > 0, we conclude that

rank(dΨ(x, q)) = k + l for (x, q) close enough to (0, 0).

n Since the ball Br (0) can be chosen arbitrarily small the claim follows.

2. This follows from the normal form of Φ Q in Theorem 2.9(2). In local charts the points in Q (and Q0) have the form (x, q). For an m0 = (x, q) close to the

∗ origin for which all entries in (x, q) are not zero, Γm0 = tm0 = {0}. Therefore

0 1 Pl 2 2 0 Φ(m ) = Φ(m) + (x, 2 j=1 λj(qj + ψj(x, q) )) ∈ Cm is in int(Φ(Q )) by (1).

32 We can now state and prove our generalization of Theorem 2.12. The goal is to obtain global information on the image of Q under the moment map Φ. Theorem

2.9 and Lemma 2.13 provide the necessary local information. For the transition from local to global statement we will make use of a Morse theoretic argument which was used also in the proofs of Theorems 2.11 and 2.12 [6, 4]. At this point it is necessary to assume that the underlying symplectic manifold M is compact.

Theorem 2.14. Let M be a compact connected symplectic manifold with Hamiltonian torus action T × M → M and momentum map Φ: M → t∗. Further let τ : M → M be an involutive diﬀeomorphism with ﬁxed point set Q such that

1. t ◦ τ = τ ◦ t−1 for all t ∈ T .

2. Φ ◦ τ = Φ.

3. Q is a Lagrangian submanifold of M.

Then Φ(Q) = Φ(M) = conv Φ(Fix(M)) = conv Φ(Fix(Q)). In particular Φ(Q) is a convex subset of t∗. Moreover, the same assertions hold if Q is replaced with any of its connected components.

Proof. We begin by showing that for a ﬁxed connected component Q0 of Q the image

Φ(Q0) is convex. According to [17], Satz 3.3, it is enough to show that Φ(Q0) has dense interior in t∗ and that for any boundary point Φ(m) of Φ(Q0) there is a hyperplane passing through Φ(m) such that Φ(Q0) lies entirely in one of the halfspaces deﬁned by the hyperplane.

We know from Lemma 2.13(2) that int(Φ(Q0)) is dense in Φ(Q0). Fix m ∈ Q0 such

0 ∗ ∗ that Φ(m) is a boundary point of Φ(Q ). Then Cm 6= t otherwise Γm = tm and Φ(m)

33 is an inner point of Φ(Q0) by Lemma 2.13(1). Therefore, locally, Φ(M) lies entirely

∗ in a halfspace which contains the cone Cm 6= tm. This means there exists some X ∈ t such that in a small neighborhood U of m

0 0 0 ΦX (m ) = hΦ(m ),Xi ≤ hΦ(m),Xi = ΦX (m) ∀ m ∈ U,

i.e. the function ΦX has a local maximal value at m.

Since M is compact the same Morse-theoretic argument as in [4] and [6] implies that in fact ΦX has a global maximal value at m, i.e.

0 0 0 ΦX (m ) = hΦ(m ),Xi ≤ ΦX (m) ∀ m ∈ M,

∗ i.e. Φ(M) lies in the halfspace {λ ∈ t : hλ, Xi ≤ ΦX (m)}.

This argument shows that Φ(m) is also a boundary point of Φ(M). And it ﬁnishes the proof that Φ(Q0) is convex.

We use the convexity of Φ(M) established in [2] and [6] to show that Φ(Q0) = Φ(M).

As closed convex sets Φ(Q0) and Φ(M) are the convex hulls of their respective boundary points. Assume that some boundary point η of Φ(M) does not lie in Φ(Q0). From

Lemma 2.13 we know there is a point ζ in int(Φ(Q0)) ⊂ int(Φ(M)). On the line segment connecting η and ζ there must be a boundary point β of Φ(Q0). Since Φ(M) is convex (and since η 6∈ Φ(Q0)), we see that β must be contained in int(Φ(M)). But then β cannot be a boundary point of Φ(Q0) as was shown in the ﬁrst part of the proof. This contradiction proves Φ(Q0) = Φ(M).

If, for m ∈ Q0, the image Φ(m) is an extremal point of the convex polytope Φ(Q0) =

Φ(M), the normal form in Theorem 2.9(1) implies k = 0, i.e. m must be a T -ﬁxed point.

34 Remark 2.15. The proof of Theorem 2.14 given here diﬀers from the proof of theorem

2.12 in [4]. However, one can repeat Duistermaat’s arguments (adjusting notation) for our setup if one establishes the following facts.

0 0 1. Let m, m ∈ M. If Φ(m) = Φ(m ), then Cm = Cm0 .

2. For any neighborhood U of m ∈ Q0 in M the image Φ(U ∩Q0) contains an open

neighborhood of Φ(m) in Cm (generalization of Lemma 2.13(1)).

Given (1) and (2) one proceeds as follows. In the proof of Theorem 2.14 it was shown that Φ(Q0) is convex. Convexity of Φ(M) follows with the same arguments. Both

Φ(Q0) and Φ(M) contain inner points. The boundary points of Φ(Q0) and Φ(M) are

0 ∗ exactly those Φ(m) (with m ∈ Q or m ∈ M) for which Cm 6= tm. Therefore,

\ \ 0 Φ(M) = Cm ⊂ Cm = Φ(Q ). ∗ 0 ∗ {m∈M:Cm6=tm} {m∈Q :Cm6=tm}

Below we will prove (1) in the more general case of proper Φ. For compact M it follows from the Morse theoretic argument mentioned above.

Part (2) follows from the fact that a boundary point of Φ(Q0) is also a boundary point of Φ(M). Note that for antisymplectic involution τ it is immediate from the normal form of Φ Q in Remark 2.10.

In Theorem 2.14 M was required to be compact. We give a simple example of a non-compact M where the image Φ(M) fails to be convex.

Example 2.16. Recall the Hamiltonian torus action on (R2n, Ω) with moment map

n X 1 Φ: 2n → t∗, (q , p , . . . , q , p ) 7→ λ (q2 + p2) R 1 1 n n 2 j j j j=1

35 from Example 2.6.

For an open set U ⊂ t∗ its preimage P = Φ−1(U) is an open T -invariant submanifold of M. As such, P is a symplectic manifold on which T acts in a Hamiltonian way with moment map Φ P . Then Φ P (P ) = U.   0 r 0 0        −r 0 0 0  E.g. for n = 2, dim t = 2, ρ :(r, s) →  ,    0 0 0 s      0 0 −s 0 4 2 2 2 2 2 2 choose P = {(q1, p1, q2, p2) ∈ R : 4 < (q1 + p1) + (q2 + p2) < 8}. Then P is connected and Φ(P ), in R2 the region in the ﬁrst quadrant between the circles of radius 1 and 2 centered at the origin, is not convex.

However, the compactness assumption can be relaxed such that the image of Φ is still convex. Hilgert, Neeb and Plank showed [10] that for proper moment maps the

AGS-theorem and Duistermaat’s theorem can be generalized in the following way.

Theorem 2.17. Consider a Hamiltonian torus action of T on the connected symplectic manifold (M, ω). Suppose the corresponding moment map Φ: M → t∗ is proper, i.e. Φ is closed and Φ−1(Z) is compact for every Z ∈ t∗.

Then Φ(M) is a closed locally polyhedral convex set.

Suppose that, in addition, there is an antisymplectic involution τ : M → M which satisﬁes Φ ◦ τ = Φ. If Q = Fix(M) is non-empty, then Φ(Q0) = Φ(Q) = Φ(M) for every connected component Q0 of Q.

Sketch of proof. The details of the proof can be found in [11], Ch.1-3. We mention only the main steps.

36 The essential properties of the moment map Φ : M → t∗ are the following:

1. Φ is proper (by assumption)

2. The map m 7→ Cm (with Cm the closed convex cone with vertex Φ(M) as deﬁned

just before Lemma 2.13) deﬁnes local convexity data, i.e. for each m ∈ M there

exists an arbitrarily small neighborhood Um of m such that

(a) Φ : Um → Cm is an open map.

−1 0 0 (b) Φ (Φ(m )) ∩ Um is connected for all m ∈ Um.

( Properties (a) and (b) follow from the normal form of Φ in Lemma 2.9(1).

For (a) confer also Lemma 1.17 in [11]. )

One can deﬁne an equivalence relation ∼ on M via

m ∼ m0 iﬀ Φ(m) = Φ(m0) and m and m0 belong to the same

connected component of Φ−1(Φ(m)).

Then one has the quotient space M˜ = M/ ∼ and the quotient map π : M → M˜ . For the induced map Φ:˜ M˜ → t∗ the preimage of a point is ﬁnite, due to the properness of Φ.

The main observation is that the quotient space M˜ = M/ ∼ carries a metric d which defines its topology. The metric d is defined as follows. Form, ˜ n˜ ∈ M˜ let d(m, ˜ n˜) be the infimum of the lengths of all the curves Φ˜ ◦ γ where γ connectsm ˜ andn ˜ in M˜ and Φ˜ ◦ γ is piecewise differentiable.

The so called ”Lokal-global-Prinzip” ([11], Theorem 3.10) now shows how the properness of Φ and the local convexity data imply the ﬁrst assertion (and more) of the

Theorem. In particular ([11], Theorem 4.1),

37 1. Φ(M) is a closed locally polyhedral convex set.

2. Φ : M → Φ(M) is an open mapping.

3. The inverse images of points in Φ(M) are connected.

The second statement of the theorem follows essentially from the fact that the map

0 Φ Q0 locally spans the same cone Cm as Φ at points m ∈ Q (cf. Remark 2.15). In Theorem 3.2 below we will give a diﬀerent proof for more general τ.

As in the case of compact M it is suﬃcient to assume T -τ-compatibility. Thus we obtain our convexity result for non-compact M.

Theorem 2.18. Let M be a connected symplectic manifold with Hamiltonian torus action T × M → M and proper moment map Φ: M → t∗. Furthermore, let τ : M →

M be an involutive diﬀeomorphism with ﬁxed point set Q such that

1. t ◦ τ = τ ◦ t−1 for all t ∈ T .

2. Φ ◦ τ = Φ.

3. Q is a Lagrangian submanifold of M.

Then Φ(Q) = Φ(M). In particular Φ(Q) is a convex subset of t∗. Moreover, the same assertions hold if Q is replaced with any of its connected components.

Proof. Let Q0 be a connected component of Q. We want to use the convexity of Φ(M) asserted by Theorem 2.17 to show Φ(Q0) = Φ(M). The following lemma shows that we only need to exploit local properties of Φ(Q0). The lemma generalizes the second part of the proof of Theorem 2.14.

38 Lemma 2.19. Let C be a convex set with non-empty interior int(C) in a ﬁnite dimensional vector space V . Consider a closed subset A ⊂ C with non-empty interior int(A). Then,

A = C iﬀ ” For a point a ∈ A : a ∈ int(C) ⇒ a ∈ int(A).”

Proof. We only have to show the if-part.

Suppose A 6= C. Fix two points c ∈ C \ A and a ∈ int(A). The line segment L connecting a and c lies entirely in the convex set C. The fact that c 6∈ A and the closedness of A imply that there is point b ∈ L \{c} which is a boundary point of

A. On the other hand, since a is an interior point of C every point in L \{c}, is also in int(C), therefore, by assumption, in int(A). In particular, b ∈ int(A), which is a contradiction. This proves the lemma.

We want to employ Lemma for C = Φ(M),A = Φ(Q0).

From Lemma 2.13(2) we know that int(Φ(Q0)) is not empty. Fix m ∈ Q0 such that

Φ(m) ∈ int(Φ(M)). We want to show that Φ(m) is also an inner point of Φ(Q0).

First we establish two facts.

0 0 1. Let m, m ∈ M. If Φ(m) = Φ(m ), then Cm = Cm0 .

As mentioned in the proof of Theorem 2.17, the inverse image Φ−1(Φ(m)) of a

point is connected for any m ∈ M. But the local normal form of Φ in Theorem

0 2.9(1) shows that Cm = Cm0 if m, m ∈ M lie in the same connected component

of Φ−1(Φ(m)).

∗ 2. For m ∈ M the image Φ(m) is an inner point of Φ(M) if and only if Cm = tm. From (1) we know that the particular choice of a preimage point of Φ(m)

39 does not matter. We need another result from [11] mentioned in the proof of

Theorem 2.17, namely that Φ : M → Φ(M) is an open mapping. Therefore a

neighborhood U of m is mapped onto a neighborhood Φ(U) of Φ(m) in Φ(M).

Then, Φ(m) ∈ int(Φ(M)) is equivalent to Φ(U) being a neighborhood of Φ(m)

in t∗. But the local normal form of Φ at m implies now that Φ(U) is open in t∗

∗ if and only if Cm = tm.

0 ∗ Now ﬁx m ∈ Q such that Φ(m) ∈ int(Φ(M)). By (2) this implies Γm = tm. But then Lemma 2.13(1) asserts that Φ(m) ∈ int(Φ(Q0)).

40 CHAPTER 3

LIE THEORY

We will discuss several applications of the symplectic convexity results from Chap- ter 2. The goal is to obtain information on the structure of semisimple Lie groups. In each case we consider a certain group orbit and recognize it as a symplectic manifold with a Hamiltonian torus action. Finding the appropriate symplectic structure is usually the hardest part. Most of the symplectic manifolds we consider here arise as symplectic leaves of certain Poisson Lie groups. The latter are obtained by the deﬁnition of appropriate Manin triples. The facts on Poisson Lie groups and Manin triples used are reviewed in the Appendix. The theorems from Chapter 2 then assert convexity of the image of the corresponding moment map. By interpreting this convexity in Lie theoretic terms we obtain the desired structural results.

Most of the results in this chapter have been established earlier using non-symplectic methods.

41 3.1 Kostant’s convexity theorems

In 1973, Kostant proved two convexity results for semisimple Lie groups [14]. The linear version involved adjoint orbits on the Lie algebra level, the nonlinear one group orbits. Atiyah showed that Kostant’s linear convexity theorem had an interpretation in symplectic geometric terms, at least if the underlying group was assumed to be complex [2]. Duistermaat was able to give a symplectic proof also for real groups by considering antisymplectic involutions [4].

In 1991, Lu and Ratiu found a way to put even Kostant’s nonlinear convexity theorem into a symplectic framework [18]. They covered completely the case where the underlying group was complex. However, as pointed out in [10], their method did not provide a way to deal with arbitrary real semisimple groups.

In this section we want to discuss all the results mentioned. In particular, repeating the arguments of Chapter 4 in [15], we show how Theorem 2.14 can be applied to complete the symplectic proof of Kostant’s nonlinear convexity theorem.

First we introduce some notation and state the linear and the nonlinear version of

Kostant’s theorem. Then we deal with both versions in separate sections.

3.1.1 Kostant’s theorems

Let G denote a connected semisimple Lie group. Universal complexiﬁcations of

Lie groups will be denoted by a subscript C, i.e. GC is the universal complexiﬁcation of G etc. For what follows it is no loss of generality when we assume that G ⊂ GC and that GC is simply connected. Write g for the Lie algebra of G. Complexiﬁcations of Lie algebras shall be denoted

42 by the subsript C, i.e. gC is the complexiﬁcation of g etc. Let g = k + p be a Cartan decomposition of g with k a maximal compact subalgebra.

Fix a maximal abelian subspace a ⊂ p and denote by Σ = Σ(g, a) the corresponding restricted root system in a∗, the dual of a. For each α ∈ Σ let gα = {Y ∈ g :[H,Y ] =

α(H) ∀ H ∈ a} be the associated root space. With m = zk(a) one then has the root space decomposition M g = a + m + gα. α∈Σ + L α Select a positive system Σ ⊂ Σ and deﬁne the nilpotent subalgebra n = α∈Σ+ g . The decomposition g = n ⊕ a ⊕ k is called Iwasawa decomposition. We deﬁne the middle projection pra : g → a, i.e. pra(X) is the a-component of X ∈ g in the

Iwasawa decomposition.

On the group level we denote by A, K and N the analytic subgroups of G with Lie algebras a, k and n. Then there is the Iwasawa decomposition of G which states that the multiplication mapping

N × A × K → G, (n, a, k) 7→ nak is an analytic diﬀeomorphism. For g ∈ G let us denote bya ˜(g) the A-component of g in the Iwasawa decomposition.

We deﬁne the Weyl group of Σ by W = NK (a)/ZK (a).

We are ready to state Kostant’s theorems [14].

Theorem 3.1. ( Kostant’s convexity theorems )

Let Y ∈ a. Then

1. pra(Ad(K).Y ) = conv(W.Y ) linear version

43 2. loga ˜(K exp(Y )) = conv(W.Y ) nonlinear version where conv(·) denotes the convex hull of (·).

3.1.2 The linear version

First we assume that G is complex, i.e. g is a Lie algebra with complex structure i. Then the Cartan decomposition of g is given by g = k+ik, i.e. p = ik. Furthermore a = it with t a maximal toral subalgebra in k. Set T = exp t.

For Y ∈ a the compact orbit M = Ad(K).Y can be identiﬁed with a coadjoint orbit of the compact group K. It therefore carries a natural symplectic structure ω. At

Z = Ad(k0).Y ∈ M,

˜ ˜ ωZ (UZ , VZ ) = κ(−iZ, [U, V ]) ∀ U, V ∈ k,

where κ denotes the Killing form on gC.

Proposition 3.2. The adjoint action of the torus T on M = Ad(K).Y is Hamilto-

∗ nian. The corresponding moment map Φ: M → t ' t satisﬁes iΦ(Z) = pra(Z), i.e. the moment map is essentially the middle projection of the Iwasawa decomposition.

Proof. Deﬁne Φ(Z) = −ipra(Z) for Z ∈ M. To see that Φ is indeed the moment map for the Hamiltonian action T × M → M we have to show three things.

1. K, therefore in particular T , acts symplectically on M.

Let k ∈ K,Z = Ad(k0).Y ∈ M and U, V ∈ k. Then

∗ ˜ ˜ (k ω)Z (UZ , VZ ) = ωAd(k)Z (Ad(^k)U Ad(k)Z , Ad(^k)V Ad(k)Z )

= κ(−iAd(k)Z, Ad(k)[U, V ])

˜ ˜ = κ(−iZ, [U, V ]) = ωZ (UZ , VZ )

44 ˜ 2. ι(X)ω = dΦX for all X ∈ t.

We compute both sides.

˜ ˜ ˜ ˜ (ι(X)ω)Z UZ = ωZ (XZ , UZ ) = κ(−iZ, [X,U])

˜ d dΦX (Z)UZ = ΦX (Ad(exp tU)Z) dt t=0 d = κ(Φ(Ad(exp tU)Z),X) dt t=0 d = κ(−ipra(Ad(exp tU)Z),X) dt t=0

= κ(−ipra([U, Z]),X) = κ(−i[U, Z],X) = κ(−iZ, [X,U])

The second last equality follows from the fact that a ⊥κ n and a ⊥κ k and

X ∈ t = ia.

3. {ΦX , ΦY } = Φ[X,Y ] = 0 ∀ X,Y ∈ t.

˜ ˜ {ΦX , ΦY }(Z) = ωZ (XZ , YZ ) = κ(−iZ, [X,Y ]) = 0.

According to the lemma Atiyah’s theorem (Thm. 2.11) can be applied. It means that

pra(Ad(K)Y ) = conv pra(Fix(Ad(K)Y )).

The T -ﬁxed points of M = Ad(K)Y are exactly the elements Z = Ad(k0)Y for k0 in the Weyl group W (cf. [9], Ch.V). This proves Theorem 3.1(1) for complex G.

If G is not complex the orbit M = Ad(K).Y is not necessarily a symplectic manifold anymore. It does not even have to have even dimension. However, one

45 can consider M as a subset of a symplectic manifold that comes about as a compact adjoint orbit of the complex group GC, the complexiﬁcation of G. We want to work out the details.

Deﬁne a maximal compact subalgebra u in gC by u = k + ip and let U be the corresponding maximal compact subgroup of GC. Notice that gC = u + iu is a

Cartan decomposition of gC. If t1 denotes a maximal torus in m = zk(a), then a1 = a + it1 deﬁnes a maximal abelian subspace of iu. The Iwasawa decomposition of gC is gC = n1 ⊕ a1 ⊕ u. Let us deﬁne the tori T1 = exp(ia1) and T = exp(ia) ⊂ T1. ˜ For Y ∈ a the orbit M = Ad(U).Y in gC carries a symplectic structure as shown ˜ above. The torus T1 acts on M in a Hamiltonian way. Let Φ1 and Φ denote the moment maps with respect to the actions of T1 and T , respectively. Then Φ = pria ◦ Φ1. In view of Proposition 3.2 this means that ˜pra(Z) := iΦ(Z) represents the a-part in the Iwasawa decomposition Z ∈ n1 + a + it1 + u for Z ∈ gC.

Let τ : gC → gC denote the complex conjugation of gC with respect to the real form g. It can be lifted to an involution on the group level which we also denote by τ.

Clearly, the τ-ﬁxed subset of M˜ is M. Since τ(t) = t−1 for all t ∈ T the involution

τ anticommutes with the action of T . In addition, the involution τ is antisymplectic on M˜ as the following calculation shows.

Let V,W ∈ u,Z ∈ Ad(U).Y . Then

∗ ˜ ˜ ¯ ¯ ¯ (τ ω)Z (VZ , WZ ) = ωτZ (τVf τZ , τWgτZ ) = κ(−iZ, [V, W ])

˜ ˜ = κ(iZ, [V,W ]) = κ(iZ, [V,W ]) = −ωZ (VZ , WZ )

In view of Remark 2.5(2) we can apply Duistermaat’s theorem (Thm. 2.12). There-

46 fore,

pra(Ad(K)Y ) = ˜pra(Ad(K)Y ) = conv ˜pra(W1.Y ),

where W1 = NU (a1)/ZU (a1) is the Weyl group for gC.

Since ˜pra(W1.Y ) = pra(W.Y ) = W.Y , we obtain the statement of Kostant’s linear convexity theorem.

3.1.3 The nonlinear version

As in Section 3.1.2 we will consider the case of complex G ﬁrst. In particular, the relevant symplectic structure is deﬁned for complex groups. In a second step we use the setup for complex G and apply Theorem 2.14 (instead of Duistermaat’s theorem) to prove Kostant’s nonlinear theorem for real G.

In case G is complex a symplectic proof of Theorem 3.1 was given by Lu and Ratiu

[18]. We now recall their method.

Let us assume that G is complex. As before, the Cartan decomposition of g is given by g = k + ik. Set a = it with t a maximal toral subalgebra in k and T = exp(t).

Deﬁne a solvable subalgebra of g by b = a + n. Write B = AN for the corresponding group and notice that B is invariant under conjugation by the torus T .

Let us denote by κ the Cartan-Killing form of the complex Lie algebra g and deﬁne a symmetric R-valued bilinear form on g by

B : g × g → R, B(X,Y ) = =κ(X,Y ).

Notice that B is invariant and non-degenerate. The important fact is that both b and k are isotropic for B; in other words (g, b, k) becomes a Manin-triple (see the

Appendix for deﬁnition). Likewise (G, B, K) is a Manin-triple. It is shown in the

47 Appendix that this implies that B ' G/K carries a natural structure of a Poisson

Lie group. In order to describe the symplectic leaves write ˜b(g) for the B-part of g ∈ G in the decomposition G = B · K. Then the symplectic leaves in B passing ˜ through points a ∈ A are given by Ma = b(Ka). As manifolds Ma ' K/Ka with

Ka = ZK (a). The symplectic form ω of Ma is given by

˜ ˜ −1 −1 ωb(Xb, Yb) = B(prk(Ad(b) X), Ad(b) Y ) for b ∈ Ma,X,Y ∈ k, (3.1)

where prk : g → k is the projection along b.

Notice that K does not act symplectically on Ma; however T does and the T -action is Hamiltonian. It was established in [18] that the corresponding moment map is the non-linear Iwasawa projection.

∗ ˜ Φ: Ma → a ' t , b(ka) 7→ loga ˜(ka). (3.2)

(See also Proposition 3.9 for a proof.)

Again, standard structure theory implies that Fix(Ma) = W.a. Thus (3.2) combined with the Atiyah-Guillemin-Sternberg convexity theorem gives a symplectic proof of

Theorem 3.1(2) in the case of G complex [18].

We now consider the real case.

Let g = k + p be the Cartan decomposition of the real semisimple Lie algebra g. Let u = k + ip a maximal compact subalgebra gC by and U the corresponding maximal compact subgroup of GC. Then gC = u + iu is a Cartan decomposition of gC. If t1 denotes a maximal torus in m = zk(a), then a1 = a + it1 deﬁnes a maximal abelian subspace of iu. Write Σ1 = Σ1(gC, a1) for the corresponding root system. Fix a

+ + positive system Σ1 of Σ1. Without loss of generality we may assume that Σ1 and

48 + + + L α Σ are compatible, i.e. Σ ⊂ Σ ∪ {0}. Write n1 = + g for the nilpotent 1 a α∈Σ1 C + subalgebra of gC associated to Σ1 . Likewise we denote by N1 the corresponding subgroup of GC. Notice that NC ⊆ N1, but generally NC 6= N1, unless m = t1 is abelian. Clearly we have A ⊂ A1 and so B ⊂ B1 where B = AN and B1 = A1N1.

Finally let us deﬁne the torus T = exp(ia). ˜ Fix a ∈ A ⊂ A1 and consider the symplectic manifold Ma = b(Ua) of the Poisson Lie group B1 (cf. the paragraph on the complex case). As explained above the action of

∗ T on Ma is Hamiltonian with moment map Φ : Ma → a ' t given by (3.1)

˜ Φ(b1(ua)) = log a˜(ug) for u ∈ U,

˜ with a˜(g) the A-part of g ∈ GC in the decomposition GC = N1A exp(it1)U. ˜ Set Qa = b(Ka). Then the restriction of Φ to Qa is the nonlinear Iwasawa projection, ˜ i.e Φ(b(ka)) = loga ˜(ka) for all k ∈ K. We are interested in the image Φ(Qa).

Denote by τ : GC → GC the complex conjugation with respect to the real form G.

Notice that τ is an involutive diﬀeomorphism of GC which induces an involution on

Ma, say τa, by the prescription

˜ ˜ ˜ τa(b(ua)) = b(τ(ua)) = b(τ(u)a) for u ∈ U.

Standard structure theory shows that the connected component containing a ∈ Ma ˜ of the ﬁxed-point set of the involution τa : Ma → Ma is given by Qa = b(Ka). We collect some important properties of τa and Qa.

Lemma 3.3. The following assertions hold.

1. Qa ⊂ Ma is Lagrangian submanifold of Ma.

−1 2. For all t ∈ T and m ∈ Ma one has τa(t.m) = t .τa(m).

49 3. Φ ◦ τa = Φ.

˜ ˜ Proof. (1) It is suﬃcient to show that ωb(Xb, Yb) = 0 for all b ∈ Qa and X,Y ∈ k.

Note that Ad(b−1)X, Ad(b−1)Y ∈ g. Since B is deﬁned as the imaginary part of the

Killing form, (3.1) implies

˜ ˜ −1 −1 ωb(Xb, Yb) = B pru(Ad(b) X), Ad(b) Y

−1 −1 = B prk(τ(Ad(b) X)), τ(Ad(b) Y )

= 0

˜ (2) Let b ∈ Ma. Then b = b(ua) for some u ∈ U. Hence for t ∈ T ,

˜ ˜ ˜ −1 −1 τa(t.b) = τa(b(tua)) = b(τ(tua)) = b(t τ(ua)) = t .τa(b), establishing (2).

(3) Write N+ for the complex subgroup of N1 corresponding to the Lie algebra n+ =

L α + g . Notice that N1 = N N+. Fix g ∈ G. Then g can be uniquely α∈Σ1 ,α|a=0 C C o expressed as

g = nn+atu (3.3)

with n ∈ NC, n+ ∈ N+, a ∈ A, t ∈ exp(it1) and u ∈ U. Replacing g with τ(g) one obtains a decomposition

0 0 0 0 0 τ(g) = n n+a t u (3.4)

0 0 0 0 0 0 for n ∈ NC, n+ ∈ N+, a ∈ A, t ∈ exp(it1) and u ∈ U. We claim that a = a . Clearly this will prove the assertion in (3).

We apply τ to (3.3)

−1 τ(g) = τ(n)τ(n+)at τ(u).

50 With (3.4) we now get the equality

0 0 0 0 0 −1 n n+a t u = τ(n)τ(n+)at τ(u). (3.5)

Observe that τ(NC) = NC and τ(U) = U, but τ(N+) = N−. Here, N− denotes the

L α analytic subgroup of G with Lie algebra n− = − g . Let θ denote the C α∈Σ1 ,α|a=0 C

Cartan-involution on GC with ﬁxed point group U. Symmetrizing (3.5) we obtain

−1 0 0 0 2 0 2 0 −1 0 −1 τ(g)θ(τ(g )) = n n+(a ) (t ) θ(n+) θ(n )

2 −2 −1 −1 = τ(n)τ(n+)a t θτ(n+) θτ(n) .

0 0 2 0 2 0 −1 2 −2 −1 Notice that n+(a ) (t ) θ(n+) and τ(n+)a t θτ(n+) belong to the reductive group Z (A). Thus the Bruhat-decomposition of G with respect to the parabolic GC C subgroup Z (A)N implies that n0 = τ(n). Hence we obtain the identity GC C

0 0 2 0 2 0 −1 2 −2 −1 n+(a ) (t ) θ(n+) = τ(n+)a t θτ(n+) (3.6) in Z (A). Notice that A lies in the center of Z (A). Thus (3.6) implies that GC GC

(a0)2x0 = a2x for some x0, x ∈ [Z (A),Z (A)] . GC GC 0

Hence (a0)2 = a2 and so a = a0 as was to be shown.

The lemma says that the torus action and the involution τa on Ma satisfy the

T -τ-compatibility from Section 2.1.

We can now ﬁnish the proof of Theorem 3.1(2). In view of Lemma 3.3 the as- sumptions of Theorem 2.14 are satisﬁed and we can conclude that Φ(Qa) = Φ(Ma) = conv Φ(Fix(Ma)). But, for a = exp Y , this translates to loga ˜(K exp Y ) = conv(log a˜(W1.a)).

The right hand side equals conv(W.Y ).

Hence, we have proven Kostant’s nonlinear convexity theorem for real G.

51 Remark 3.4. 1. If m = t1 is abelian, then the involution τa : Ma → Ma is anti-

symplectic (cf.[10]). In fact, if m is abelian, then τ(B1) = B1 and pru ◦ τ =

∗ τ ◦ pru holds. By the deﬁnition of B we have τ B = −B. Therefore, for

b ∈ Ma,X,Y ∈ u,

∗ ˜ ˜ −1 −1 (τa ω)b(Xb, Yb) = B pru(Ad(τb) τX), Ad(τb) τY

−1 −1 = B pru(τ(Ad(b) X)), τ(Ad(b) Y )

−1 −1 = B τ(pru(Ad(b) X)), τ(Ad(b) Y )

−1 −1 = −B pru(Ad(b) X), Ad(b) Y

˜ ˜ = −ωb(Xb, Yb)

2. If m is not abelian, then τ(B1) 6= B1 and pru ◦ τ 6= τ ◦ pru. One can verify from

(3.1) that τa is not antisymplectic in this case (see the following Example 3.5).

Example 3.5. We will show that τa is not antisymplectic in case m is not abelian.

In this situation g contains a subalgebra of type so(1, 4). Therefore it is enough to consider the case of G = SOe(1, 4).

We start with some comments of general nature. From the polar decomposition ˜ G = KAK it follows that every n ∈ N is contained in Ma = b(Ka) for some a ∈ A.

Now ﬁx n ∈ N and a ∈ A such that n ∈ Ma. Let X,Y ∈ ip. Then τa(n) = n and

˜ ˜ ˜ ˜ dτa(n)Xn = −Xn and dτa(n)Yn = −Yn.

Thus if τa were antisymplectic, then

˜ ˜ ˜ ˜ ˜ ˜ ωn(Xn, Yn) = −ωτa(n)(dτa(n)Xn, dτa(n)Yn) = −ωn(Xn, Yn),

52 ˜ ˜ i.e. ωn(Xn, Yn) = 0. ˜ ˜ Below we will show that ωn(Xn, Yn) 6= 0 for a speciﬁc choice of elements n, X, Y .

Let now G = SOe(1, 4). Then the Lie algebra of G is given by     0 ut  g   4 so =   : u ∈ R ,X ∈ (4) .  u X  The complexiﬁcation of g then is     0 wt  g =   : w ∈ 4,Z ∈ so(4, ) . C   C C  w Z 

Our choice of a and t1 will be     0 0 0 0 1 0 0 0 0 0              0 0 0 0 0   0 0 1 0 0          a = R  0 0 0 0 0  and t1 = R  0 −1 0 0 0  .              0 0 0 0 0   0 0 0 0 0          1 0 0 0 0 0 0 0 0 0

+ With appropriate Σ1 the nilpotent Lie algebras nC and n+ are given by     0 zt 0        n =   : z ∈ 3 , C  z 0 −z  C     t    0 z 0  and   0 0 0 0 0        0 0 0 i 0      n+ = C  0 0 0 1 0  .        0 −i −1 0 0      0 0 0 0 0

53 Deﬁne an element n ∈ N by   5 1 1 1 − 3  2 2       1 1 0 0 −1      n =  1 0 1 0 −1  ∈ N.        1 0 0 1 −1     3 1  2 1 1 1 − 2 Next we introduce elements X,Y ∈ ip by     0 0 0 i 0 0 i 0 0 0              0 0 0 0 0   i 0 0 0 0          X =  0 0 0 0 0  and Y =  0 0 0 0 0  .              i 0 0 0 0   0 0 0 0 0          0 0 0 0 0 0 0 0 0 0

A simple computation gives   0 −i −i 3 i −i  2       −i 0 0 −i i    −1   Ad(n) X =  −i 0 0 −i i       3 1   i i i 0 − i   2 2   1  −i −i −i 2 i 0 and   0 3 i −i −i −i  2     3 1   2 i 0 i i − 2 i    −1   Ad(n) Y =  −i −i 0 0 i  .        −i −i 0 0 i     1  −i 2 i −i −i 0

54 Then   0 0 0 i −i        0 0 0 −1 0    −1   pru(Ad(n) X) =  0 0 0 1 0  .        i 1 −1 0 0      −i 0 0 0 0 Up to scalar we may identify B with the imaginary part of the trace form. With this normalization we then have     0 0 0 i −i 0 3 i −i −i −i    2         3 1   0 0 0 −1 0   2 i 0 i i − 2 i      ˜ ˜     ωn(Xn, Yn) = =tr  0 0 0 1 0  ·  −i −i 0 0 i               i 1 −1 0 0   −i −i 0 0 i         1  −i 0 0 0 0 −i 2 i −i −i 0   0 ∗ ∗ ∗ ∗        ∗ i ∗ ∗ ∗      = =tr  ∗ ∗ 0 ∗ ∗  = 2 6= 0.        ∗ ∗ ∗ 1 + i ∗      ∗ ∗ ∗ ∗ −1

Remark 3.6. Recently, T. Ratiu pointed out to the author that an alternate symplectic proof of Theorem 3.1 was obtained earlier by P. Sleewaegen [23]. His proof diﬀers considerably from the one presented here.

55 3.2 Neeb’s convexity theorem for semisimple symmetric spaces

We now consider a generalization of Kostant’s nonlinear convexity theorem. We want to present a symplectic proof of Neeb’s convexity theorem for a semisimple Lie group G [20]. In Kostant’s theorem the manifolds to consider are K-orbits, where

K is the group of fixed points under the Cartan involution θ. Neeb’s theorem is concerned with H-orbits, where H is a certain subgroup of the set of fixed points under an arbitrary involution σ on G. The main difference, in view of our approach via symplectic convexity theorems, is that the H-orbits are not necessarily compact anymore.

We ﬁrst introduce the notation to state Neeb’s theorem precisely. An important step is the reduction of the problem, from a general semisimple group G to one special case. The material is presented as in Neeb’s paper [20].

In the second part of the chapter we use symplectic techniques to deal with the special case for G which remains. Most of the section is a review of the arguments in [10].

The approach is similar to the one explained in the section on Kostant’s nonlinear theorem. For complex G a symplectic structure on certain H-orbits is introduced via

Manin triples. Since the orbits are not necessarily compact one needs the convexity theorem of Hilgert-Neeb-Plank (Thm. 2.17) to obtain the desired result. For real G, the H-orbits are viewed as ﬁxed point sets of symplectic manifolds under an involution

τ. The methods from [10] are suﬃcient to deal with the case of an antisymplectic τ.

To prove Neeb’s theorem in full generality we use Theorem 2.18.

56 3.2.1 Statement of the theorem and reduction of the problem

Let G be a connected semisimple Lie group with Lie algebra g. We assume G is equipped with an involution σ, i.e. we have a connected semisimple symmetric Lie group (G, σ). We also write σ for the corresponding Lie algebra endomorphism on g.

With respect to σ : g → g we have a decomposition g = h + q of g into +1-eigenspace h and −1-eigenspace q.

We can choose a Cartan involution θ on G and g that commutes with σ. If g = k + p denotes the Cartan decomposition of g we get

g = hk + hp + qk + qp,

where hk = h ∩ k, hp = h ∩ p, qk = q ∩ k, qp = q ∩ p.

a a a a Note that the symmetric Lie algebra (h , σ ) with h = hk + qp and σ = σ ha = θ ha is reductive.

0 We choose a maximal abelian subspace a of qp and a maximal abelian a in p with a ⊂ a0. Then g decomposes into simultaneous eigenspaces with respect to a.

M α g = zg(a) ⊕ g . α∈∆

Here ∆ denotes the set of non-zero linear functionals on a for which the root space

α a g = {Y ∈ g :[X,Y ] = α(X)Y ∀ X ∈ a} is non-zero. For c = z(h ) ∩ qp ⊂ a the set of compact roots ∆k is deﬁned as

∆k = {α ∈ ∆ : α(X) = 0 ∀ X ∈ c}.

+ Then ∆p = ∆ \ ∆k is called the set of noncompact roots. A positive system ∆ =

{α ∈ ∆ : α(X0) > 0} can be deﬁned where X0 ∈ a satisﬁes α(X0) 6= 0 for all α ∈ ∆

57 + + + + and such that α(X0) < β(X0) whenever β ∈ ∆p = ∆ ∩ ∆p and α ∈ ∆k = ∆ ∩ ∆k. One considers the cones

+ Cmax = {X ∈ a : α(X) ≥ 0 ∀ α ∈ ∆p } and

∗ Cmin = Cmax = {X ∈ a : κ(X,Y ) ≥ 0 ∀ Y ∈ Cmax}, where κ is the Killing form on g.

One also deﬁnes the following subalgebras of g and subgroups of G.

M n = gα,N = exp n,A = exp a,K = Gθ = h exp ki, α∈∆+ 0 0 M = ZK (a ) = {k ∈ K : Ad(k) a0 = ida },

0 0 m = Lie(M) = zk(a ) = {X ∈ k :[X, a ] = 0}.

It can be shown that the projection mapping

L : NAMGσ → a, g = namh 7→ log a (3.7) from the open subset NAMGσ of G to a is well deﬁned and analytic (for a proof, see

[20], Cor. I.4). Note that we slightly changed the notation from [20] to emphasize the similarities with Kostant’s theorem as presented in Section 3.1.3.

σ Let H ⊂ G be an open subgroup and H0 the connected component of the identity in H. The subgroup H is called essentially connected if H = ZK∩H (a)H0. Fixing an essentially connected H, we call an element a ∈ A admissible if Ha ⊂ NAMGσ.

Clearly, admissible elements Aadm are exactly those a ∈ A for which L(Ha) is well deﬁned.

By W we denote the Weyl group of the pair (ha, a).

We can now state Neeb’s convexity theorem ([20], Thm. I.7).

58 Theorem 3.7. Let (G, σ) be a connected semisimple symmetric Lie group, H ⊂ Gσ essentially connected and a ∈ A an admissible element.

Then L(Ha) is a closed convex set. More precisely,

L(Ha) = conv(W. log a) + C(a), with the cone

+ C(a) = {Y ∈ Cmin : α(Y ) ≤ 0 for all α ∈ ∆ for which α(W. log a) = {0}}.

The theorem can be proven by considering only special cases of symmetric groups

(G, σ) and subgroups H ⊂ Gσ. Neeb reduces the problem in the following way.

First one can assume that H is connected and that G has trivial center (Lemmas I.10 and I.12 in [20]). Lemma I.14 reduces the problem to the case that (g, σ) is irreducible.

But then there are only a few possibilites left. They are listed in Theorem I.20: (g, σ) must satisfy one of the three cases (recall that a semisimple Lie algebra with Cartan decomposition g = k + p is called Hermitian if z(k) 6= {0}).

a 1. (g, σ) is orthogonal, i.e. σ = θ, h = g,Aadm = A, Cmin = {0}.

a 2. z(h ) ⊂ hk,Aadm = {1},Cmin = {0} .

a c 3. z(h ) ∩ qp 6= {0}, g := h + iq ⊂ gC is a semisimple Hermitian Lie algebra and g is simple.

The possible subcases are

(a) gc is simple.

(b) g ' hC, and h is simple Hermitian.

59 Note that in case (1) Neeb’s convexity theorem reduces to Kostant’s nonlinear convexity theorem:

Since H = K,Aadm = A, C(a) = {0}. The Weyl group W is then the Weyl group of the pair (g, a), and Neeb’s theorem says

L(Ka) = conv(W. log a) ∀ a ∈ A.

We saw a symplectic proof in Section 3.1.3.

In case (2) the statement of the theorem, L(H) = {0} is trivially satisﬁed.

Therefore, only case (3) is left to prove. This is the objective of the next Section.

3.2.2 A symplectic proof

Case (3b) is considered ﬁrst. Let G be a connected Lie group whose Lie algebra

σ g is as in case (3b). In particular, g = hC, where h = g is a simple Hermitian

Lie algebra. From the assumption it follows that the pair (h, θh) is an irreducible orthogonal symmetric Lie algebra (cf. [9] Ch.VIII, Thm. 5.4). Therefore, we can

ﬁnd a compactly embedded Cartan subalgebra t of h. One obtains a simultaneous eigenspace decomposition of g with respect to t,

M α g = tC ⊕ g . α∈∆ As usual, gα = {Y ∈ g :[X,Y ] = α(X)Y ∀ X ∈ t}, ∆ = {α ∈ t∗ \{0} : gα 6= {0}}. C Choose a positive system ∆+ ⊂ ∆ and set n = L gα. We set a = it and α∈∆+ C b = a ⊕ n. This notation is consistent with the one in Section 3.2.1 (here: a0 = a, since t is a Cartan subalgebra). Since σ(gα) = g−α for each α ∈ ∆ we ﬁnd that h ∩ b = {0}. This implies

g = b ⊕ h = n ⊕ a ⊕ h.

60 We write l : n ⊕ a ⊕ h → a for the linear middle projection.

Denote by A, N, B the analytic subgroups of G corresponding to the Lie algebras a, n, b, respectively. According to Prop. 4.1 in [10] the group B is simply connected and closed and the mapping N × A × H → G, (n, a, h) 7→ nah is a diﬀeomorphism onto its open image. In particular, there is a well deﬁned projection map ˜b : NAH →

B, nah 7→ na.

Let a ∈ A be admissible, i.e. Ha ⊂ NAH (note that M ⊂ H in our special case). ˜ We are interested in the H-orbit of a in B, i.e. in Ma = b(Ha). Note that an element

˜ 0 ˜ 0 ˜ 0 −1 h ∈ H acts on b = b(h a) ∈ Ma via h.b = b(hh a) = b(hh ah ). We are about to deﬁne a Poisson structure on B such that the symplectic leaf through a is just Ma.

It is enough to deﬁne an appropriate Manin triple on g.

Lemma 3.8. Let B = =κ be the imaginary part of the Killing form κ on g.

With B as bilinear form (g, b, h) deﬁnes a Manin triple.

Proof. We have already shown that g = b ⊕ h. Clearly, h is isotropic with respect to

B since κ takes real values on h × h. The same argument shows that a is isotropic.

From the invariance of κ it follows that κ(gα, gβ) = 0 for α + β 6= 0. But then

B(b, b) = B(a, a) = 0.

From the general theory laid out in the Appendix it follows now that B carries a

Poisson structure and that Ma is a (noncompact) symplectic manifold with symplectic form ω induced by the Manin form B. More precisely,

˜ ˜ −1 −1 ωb(Xb, Yb) = B(prh(Ad(b) X), Ad(b) Y ) for b ∈ Ma,X,Y ∈ h, (3.8)

where prh : g → h denotes the projection with kernel b.

We deﬁne the torus T = exp t ⊂ H.

61 Proposition 3.9. The action of T on Ma is Hamiltonian with moment map Φ:

∗ ˜ Ma → t , Φ(b(ha)) = −iL(ha) with the projection map L : NAH → a from (3.7).

˜ Proof. Throughout we ﬁx b = b(ha) ∈ Ma.

1. The action of T on Ma is symplectic.

Note that [t, h] ⊂ h and [t, b] ⊂ b. Thus, for t ∈ T , both h and b are left

invariant by Ad(t). On the group level this means that H and B are invariant

under conjugation by t ∈ T . In particular, t.b = ˜b(that−1) = t˜b(ha)t−1 = tbt−1.

From the invariance of B = =κ we obtain, for U, V ∈ h,

∗ ˜ ˜ (t ω)b(Ub, Vb) = ωt.b(Ad(^t)U t.b, Ad(^t)V t.b)

−1 −1 −1 −1 = B(prhAd(tb t )Ad(t)U, Ad(tb t )Ad(t)V )

−1 −1 = B(Ad(t)prhAd(b )U, Ad(t)Ad(b )V )

−1 −1 = B(prhAd(b )U, Ad(b )V )

˜ ˜ = ωb(Ub, Vb).

˜ 2. ι(X)ω = dΦX for all X ∈ t.

Fix U ∈ h,X ∈ t. Note that Ad(b−1)X ∈ X + n. Therefore,

˜ ˜ ˜ ˜ −1 −1 −1 (ι(X)ω)bUb = ωb(Xb, Ub) = =κ(prhAd(b )X, Ad(b )U) = =κ(X, Ad(b )U),

and

˜ d d dΦX (b)Ub = ΦX (exp(tU).b) = κ(−iL(exp(tU)ha),X) dt t=0 dt t=0 d = − κ(iL(bb−1 exp(tU)b),X) dt t=0 d = − κ(iL(b) + iL(exp(tAd(b−1)U)),X) dt t=0 = −iκ(l(Ad(b−1)U, X) = =κ(l(Ad(b−1)U, X) = =κ(Ad(b−1)U, X).

62 3. {ΦX , ΦY } = 0 ∀ X,Y ∈ t.

−1 −1 Since Ad(b )X ∈ X + n, Ad(b )Y ∈ Y + n and a ⊥κ n, we get

˜ ˜ −1 −1 {ΦX , ΦY } = ωb(Xb, Yb) = =κ(prhAd(b )X, Ad(b )Y ) = =κ(X,Y ) = 0.

The moment map Φ on Ma can be shown to be proper. This proof, which involves quite some work, is carried out in [10], Ch.5.

We are now in the situation that the action of the torus T on the connected symplectic manifold Ma is Hamiltonian and the corresponding moment map Φ is proper. We can therefore apply Theorem 2.17 and conclude that the image of Ma under the moment map, which is essentially L(Ha), is convex. This is just the main statement of Neeb’s theorem.

Neeb’s result as stated in Theorem 3.7 includes a more detailed description of the closed convex set L(Ha) as the sum of a convex polytope and a cone. We do not want to go into the details of the proof but refer to Chapter 6 in [10] for a detailed description of the image of Ma under the moment map.

We are left with the case (3a). Again, we just want to establish convexity of the image of the moment map. Now G is connected simple and gc is simple Hermitian. We will obtain Neeb’s result in the following way. First we go over to the complexiﬁcation

GC of G. We denote by τ the complex conjugation on GC and gC with respect to

G and g, respectively. The complexified Lie algebra gC is equipped with another involution σ]: For X,Y ∈ g define σ](X +iY ) = σ(X) − iσ(Y ), i.e. σ] is the complex antilinear extension of σ. The σ]-fixed point set of g is then gσ] = h+iq = gc which, C C

63 by assumption, is simple Hermitian. We can apply the arguments from case (3b) to

c ] gC = (g )C. We modify the notation for g from case (3b) above by adding a to

] c denote subalgebras of gC or subgroups of GC. We have a decomposition gC = b ⊕ g ,

] c ] a Manin triple (gC, b , g ) and a corresponding Poisson structure on B . The relevant

] ] ] c ] ] projections are L : N A G → a and pra : a → a.

Consider an element a ∈ Aadm. It is also admissible in the bigger group GC, i.e.

] c ] ] a ∈ Aadm. The G -orbit in B through a, denote it by Ma, is a symplectic manifold. Its symplectic form ω] is given by

] ˜ ˜ −1 −1 ] c ωb(Xb, Yb) = B(prgc (Ad(b) X), Ad(b) Y ) for b ∈ Ma,X,Y ∈ g . (3.9)

] ] The torus T = exp t, with t = ia ⊂ ia , acts on Ma in a Hamiltonian fashion. The

] ] corresponding moment map is Φ = −i(pra ◦ L ). Moreover, Ma is invariant under the

] involution τa := τ ] . The H-orbit Ma ⊂ M can be viewed as the ﬁxed point set of Ma a

] ] Ma under τa. The involution τa on Ma is not necessarily antisymplectic. Therefore the generalized version of Theorem 2.17 with involution can not always be used. As in Section 3.1 we can show that complex conjugation satisﬁes the T -τ-compatibility.

] ] ] Lemma 3.10. Consider the symplectic manifold (Ma, ω ) with ω as in (3.9). Let

] T denote the (small) torus exp t acting on Ma in a Hamiltonian way with moment

] ] ] ] map Φ = −i(pra ◦ L ). Let τa : Ma → Ma be the restriction to Ma of the complex conjugation with respect to g.

Then the following conditions are satisﬁed.

] 1. Ma is a Lagrangian submanifold of Ma.

] −1 2. For all t ∈ T and m ∈ Ma one has τa(t.m) = t .τa(m).

64 3. Φ ◦ τa = Φ.

Proof. The proof is essentially a redoing of the proof of Lemma 3.3.

Since T -τ-compatibility holds and the moment map Φ is proper we can apply

Theorem 2.18. We conclude that the image of Ma under the moment map is convex,

] i.e. (pra ◦ L )(Ha) = L(Ha) is convex. This ﬁnishes our symplectic proof of Neeb’s theorem.

3.3 A reﬁnement of the complex convexity theorem of Gindikin- Kr¨otz

We now consider a complex convexity theorem. In the applications of Sections

3.1 and 3.2 we were interested in certain orbits in a connected semisimple Lie group

G. Now, we go over to the complexiﬁcation GC of G and consider orbits in GC. Let G be a semisimple Lie group and G = NAK be an Iwasawa decomposition. The

Iwasawa decomposition on the Lie algebra level is g = n ⊕ a ⊕ k. Complexifying Lie algebras yields gC = nC ⊕ aC ⊕ kC. In the following GC denotes the complexiﬁcation of G, and NC,AC,KC denote analytic subgroups of GC with Lie algebras nC, aC, kC, respectively.

π π Let X ∈ a = Lie(A) be such that Spec(adX) ⊆] − 2 , 2 [ and set a = exp(iX). It is known that Ka ⊆ NCACKC. The main result of this section asserts that

= loga ˜(Ka) = conv(W.X). (3.10)

Herea ˜ : NCACKC → AC denotes a middle projection and conv(W.X) stands for the convex hull of the Weyl group orbit W.X. We note that the inclusion 00 ⊆00 in (3.10)

65 is the complex convexity theorem from [5].

We proceed as follows. First we state the problem precisely. In particular, we have to describe the projectiona ˜ in 3.10. Also, since we will deal with the complexiﬁcation

GC of a complex group G, we say a few words about double complexiﬁcation of a Lie algebra. To prove statement 3.10, we ﬁrst consider the case where G is a complex group. In this situation we show that BC = ACNC carries a natural structure of a

Poisson Lie group. Locally, we can identify BC inside of GC/KC and consequently we obtain a local action of GC on BC. Within this identiﬁcation the symplectic leaf

Pa through a = exp(iX) ∈ BC becomes a local KC-orbit. All of this follows from the deﬁnition of a Manin triple (gC, bC, k) as explained in the Appendix. However, unlike in the applications of Sections 3.1 and 3.2, the symplectic leaf Pa is not the group orbit we are interested in. The relevant orbit in 3.10 is the totally real K-orbit

Ma ⊂ Pa. Interestingly, the symplectic form on Pa remains non-degenerate on Ma.

We then exhibit a Hamiltonian torus action on the compact symplectic manifold

Ma and show that (3.10) becomes a consequence of the Atiyah-Guillemin-Sternberg convexity. Finally, the general case of arbitrary G can be handled by descent to certain Lagrangians Qa ⊆ Ma by means of Theorem 2.14.

The material is presented as in [16].

3.3.1 Notation and basic facts

In this section we recall some basic facts about semisimple Lie algebras and groups.

We make an emphasis on the complexiﬁed Iwasawa decomposition. Furthermore we review some standard facts on the double complexiﬁcation of a semisimple Lie algebra.

66 We let g denote a semisimple Lie algebra and let g = k + p be a Cartan decomposition of g. For a maximal abelian subspace a of p let Σ = Σ(g, a) ⊆ a∗ be the corresponding root system. Then g admits a root space decomposition

M g = a ⊕ m ⊕ gα, α∈Σ α where m = zk(a) and g = {X ∈ g :[H,X] = α(H)X ∀ H ∈ a}.

+ L α For a ﬁxed positive system Σ deﬁne n = α∈Σ+ g . Then we have the Iwasawa decomposition on the Lie algebra level:

g = n ⊕ a ⊕ k.

For any real Lie algebra l we write lC for its complexiﬁcation. In the following GC will

denote a simply connected Lie group with Lie algebra gC. We write G, K, KC, A, AC,N and NC for the analytic subgroups of GC corresponding to the subalgebras g, k, kC, a, aC, n and nC , respectively.

The Weyl group of Σ can be deﬁned by W = NK (a)/ZK (a).

Following [1] we deﬁne a bounded and convex subset of a that plays a central role:

π Ω = {X ∈ a : |α(X)| < ∀ α ∈ Σ}. (3.11) 2

With Ω one can deﬁne a G − KC-double coset domain in GC by

˜ Ξ = G exp(iΩ)KC.

Also we write ˜ Ξ = Ξ/KC

˜ for the union of right KC-cosets of Ξ in the complex symmetric space GC/KC. One refers to Ξ as the complex crown of the symmetric space G/K. Notice that Ξ is

67 independent of the choice of a, hence generically deﬁned through G/K. ˜ It is known that Ξ is an open and G-invariant subset of NCACKC (cf. [19] for a short proof).

Next we consider the open and dense cell NCACKC ⊆ GC in more detail. Deﬁne

F = KC ∩ AC and recall that F = K ∩ exp(ia) is a ﬁnite 2-group. The group F acts on AC × KC by

−1 f.(a, k) = (af , fk) ∀ f ∈ F, a ∈ AC, k ∈ KC.

In particular, the elements (af, k) and (a, fk) are in the same F -orbit. We write

AC ×F KC for the quotient of AC × KC under the F -action. Standard techniques imply that the mapping

NC × (AC ×F KC) → NCACKC, (n, [a, k]) 7→ nak (3.12)

is a biholomorphism. In particular (3.12) induces a holomorphic mapn ˜ : NCACKC →

NC and a multi-valued holomorphic mappinga ˜ : NCACKC → AC such that x ∈ n˜(x)˜a(x)KC for all x ∈ NCACKC.

Set BC = NCAC. Clearly, bC = aC + nC is the Lie algebra of BC. We deﬁne a multi- ˜ ˜ valued holomorphic map b : NCACKC → BC by b(x) =n ˜(x)˜a(x). Recall that Ξ is contractible, and hence simply connected. It follows that the re- ˜ strictiona ˜|Ξ˜ has a unique single-valued holomorphic lift loga ˜ : Ξ → aC such that ˜ ˜ loga ˜(1) = 0. Consequently, b|Ξ˜ lifts to a single-valued holomorphic map Ξ → BC which shall also be denoted by ˜b.

For the remainder of this section we will assume that g carries a complex structure, say j. Then g can be viewed as the complexiﬁcation of its compact real form k and

68 the Cartan decomposition becomes g = k+jk. The Cartan involution θ on g coincides with the complex conjugation¯: g → g with respect to k.

A second complexiﬁcation yields gC, which carries another complex structure i : gC → gC. The following map ϕ deﬁnes a real Lie algebra isomorphism.

¯ ¯ ϕ : gC → g × g,X + iY 7→ (X + jY, X + jY ).

Its inverse is given by

1 i ϕ−1 : g × g → g , (A, B) 7→ (A + B) + (−jA + jB). C 2 2

Under ϕ we have the following identiﬁcations:

kC = {(Z,Z): Z ∈ g}, (3.13)

aC = {(Z, −Z): Z ∈ a + ja}, (3.14)

¯ nC = {(X + jY, θ(X − jY )) : X,Y ∈ n} = n × n, (3.15)

L α where n¯ = −α∈Σ+ g .

3.3.2 Complex groups

If G is complex, then we can endow BC with a natural structure of a Poisson

Lie group. The symplectic leaves become local KC-orbits. We show that the totally real K-orbit in each leaf is again a symplectic manifold. From that we obtain compact symplectic manifolds with appropriate Hamiltonian torus actions. The complex convexity theorem then becomes a consequence of the Atiyah-Guillemin-Sternberg convexity (Theorem 2.11).

69 Throughout this section g denotes a complex semisimple Lie algebra. Our first task is to define a bilinear form h, i on gC = g×g which gives (gC, bC, kC) the structure of a Manin triple. To that end let κ be the Killing form of the complex Lie algebra g. Then define on gC = g × g a bilinear form by

h(X,Y ), (X0,Y 0)i = <κ(X,X0) − <κ(Y,Y 0) (3.16)

0 0 for (X,Y ), (X ,Y ) ∈ gC.

Proposition 3.11. The bilinear form h, i on gC = g×g from (3.16) has the following properties:

1. h, i is symmetric, non-degenerate and GC-invariant.

2. hbC, bCi = {0} and hkC, kCi = {0}.

Proof. (1) is immediate from the deﬁnition. For (2) we observe that the relations hkC, kCi = {0} and haC, aCi = {0} are straightforward from the identiﬁcations (3.13) and (3.14). Finally, the fact that root spaces gα and gβ are κ-orthogonal if α + β 6= 0 implies hnC, aC + nCi = {0}.

Proposition 3.11 says that h, i turns (gC, bC, kC) into a Manin triple. Accordingly

BC becomes a Poisson Lie group whose symplectic leaves are local KC-orbits (see Appendix).

Here we shall only be interested in the leaves through points a = exp(iX) for X ∈ Ω.

Denote the leaf containing a by Pa. Then

˜ Pa = {b(ka) ∈ BC : k ∈ KC, ka ∈ NCACKC}0,

70 where {·}0 refers to the connected component of {·} containing a. ˜ For an element Z ∈ kC we write Z for the corresponding vector ﬁeld on Pa, i.e. if ˜ b = b(ka) ∈ Pa, then ˜ d ˜ Zb = b(exp(tZ)ka). dt t=0

Write pr : g → k and pr : g → b for the projections along b , resp. k . kC C C bC C C C C ˜ We notice that TbPa = {Zb : Z ∈ kC}. The symplectic formω ˜ on Pa is then given by

˜ ˜ −1 −1 ω˜b(Yb, Zb) = hpr (Ad(b )Y ), Ad(b )Zi ∀ Y,Z ∈ k . (3.17) kC C

Our interest is not so much with Pa as it is with its totally real submanifold

˜ Ma = b(Ka).

Thenω ˜ induces a closed 2-form ω on Ma by ω =ω ˜|T Ma×T Ma . A priori it is not clear that ω is non-degenerate, i.e. that (Ma, ω) is a symplectic manifold. This will be shown now. We start with a simple algebraic fact.

⊥ Lemma 3.12. With respect to h, i one has k = kC + b.

⊥ Proof. Because of the non-degeneracy of h, i, it is suﬃcient to verify that kC +b ⊆ k .

⊥ Clearly, kC ⊆ k since kC is isotropic. As b = a + n, it thus remains to show that hk, ai = {0} and hk, ni = {0}. Now hk, ai = {0} follows from (??) and <κ(k, a) =

{0}. Finally we show that hk, ni = {0}. For that ﬁx an arbitrary element W =

P P ¯ α P ( α∈Σ+ Yα, α∈Σ+ Yα) of n ⊆ g×g; here Yα ∈ g . Likewise let U = (V + α∈Σ+ (Zα +

¯ P ¯ α Zα),V + α∈Σ+ (Zα + Zα)) be an element of k ⊆ g × g; here V ∈ m and Zα ∈ g .

71 α α β As m ⊥κ g and g ⊥κ g for α + β 6= 0, we obtain

X ¯ ¯ ¯ hU, W i = <κ(Zα + Zα,Yα) − <κ(Zα + Zα, Yα) α∈Σ+ X ¯ ¯ = <κ(Zα,Yα) − <κ(Zα, Yα) = 0. α∈Σ+ This concludes the proof of the lemma.

Lemma 3.13. At any point b ∈ Ma, the bilinear form ωb : TbMa × TbMa → R is non-degenerate.

˜ 0 0 Proof. Let b = b(ka) ∈ Ma. Then b = kak for some k ∈ K, k ∈ KC. Notice that ˜ TbMa = {Yb : Y ∈ k}. ˜ ˜ Assume that there is an U ∈ k such that ωb(Ub, Yb) = 0 for all Y ∈ k, i.e.

hpr (Ad(b−1)U), Ad(b−1)Y i = 0 ∀ Y ∈ k. kC

˜ −1 ⊥ We have to show that Ub = 0. Set Z = pr (Ad(b )U). Then Ad(b)Z ∈ k . Thus kC Lemma 3.12 implies that

−1 Z ∈ Ad(b )(kC + b). (3.18)

On the other hand, by deﬁnition,

−1 −1 Z ∈ Ad(b )k + bC = Ad(b )(k + bC). (3.19)

From (3.18) and (3.19) it follows that

−1 −1 −1 Z ∈ Ad(b )(kC + b) ∩ Ad(b )(k + bC) = Ad(b )g.

Moreover, since Z is an image point of pr , kC

−1 Z ∈ Ad(b )g ∩ kC,

72 i.e.

Ad(b)Z ∈ g ∩ Ad(b)kC.

As b = kak0 we now get

Ad(b)Z ∈ g ∩ Ad(k)Ad(a)kC = Ad(k)(g ∩ Ad(a)kC) .

Using standard techniques (see [1] or [19], Lemma 2), it follows from (3.11) that

g ∩ Ad(a)kC = zk(X).

Hence Ad(b)Z ∈ Ad(k)zk(X).

From

−1 Ad(b )U ∈ Z + bC we conclude

U ∈ Ad(b)Z + bC ⊆ Ad(k)zk(X) + bC.

Since U was assumed to lie in k, we ﬁnally get

U ∈ Ad(k)zk(X).

˜ But this just means that Ub = 0, concluding the proof that ωb is non-degenerate.

It follows from Lemma 3.13 that (Ma, ω) is a compact symplectic manifold.

Clearly, the torus T = exp(ja) acts on Ma as K does. We want to show that the action of T on Ma is Hamiltonian and identify the corresponding moment map

∗ Φ: Ma → t , where t = ja is the Lie algebra of T . In the following we will identify t∗ with a via the linear isomorphism

a → t∗,Y 7→ (Z 7→ hiY, Zi) . (3.20)

73 We shall write pr : g → a for the projection along k + n . aC C C C C

Proposition 3.14. The action of the torus T = exp(ja) on Ma is Hamiltonian with moment map

∗ ˜ Φ: Ma → a ' t , b(ka) 7→ = loga ˜(ka).

Proof. 1. We ﬁrst show that T = exp(ja) acts on Ma symplectically. For that we

ﬁrst notice that T normalizes BC. Hence the action of T on Ma ⊆ BC is given by conjugation, i.e.

−1 T × Ma → Ma, (t, b) 7→ t.b = tbt .

Moreover, for each t ∈ T the map Ad(t) commutes both with pr and pr . kC bC Combining these facts, it is then straightforward from (3.17) that T acts indeed

symplectically on Ma (cf. part (1) in the proof of Proposition 3.9).

2. Next we show that

˜ ˜ {ΦY , ΦZ } = ω(Y, Z) = 0 ∀ Y,Z ∈ t. (3.21)

Fix b ∈ Ma. From the deﬁnition (3.17) we obtain that

˜ ˜ −1 −1 ωb(Yb, Zb) = hpr (Ad(b )Y ), Ad(b Z)i. kC

−1 −1 Now for Y,Z ∈ t we have Ad(b )Y ∈ Y + nC and Ad(b )Z ∈ Z + nC. From

Proposition 3.11 we know hkC, kCi = {0}. Hence, to prove (3.21) it suﬃces to

show ht, nCi = {0}. P P ¯ ¯ Let U = (jZ, jZ) ∈ t and V = ( α∈Σ+ Xα +jYα, α∈Σ+ Xα +jYα) ∈ nC, where

α Z ∈ a and Xα,Yα ∈ g . Then,

X ¯ ¯ hU, V i =

74 α β since g ⊥κ g for α + β 6= 0.

˜ 3. It remains to show that ι(Z)ω = dΦZ for all Z ∈ t. Fix b ∈ Ma and Y ∈ k.

With the identiﬁcation (3.20) we then compute

˜ d ˜ dΦZ (b)(Yb) = ΦZ (b(exp(tY ).b) dt t=0 d = hiΦ(˜b(exp(tY )b),Zi dt t=0 d = hiΦ(˜b(b exp(tAd(b−1)Y )),Zi dt t=0 d = hi= loga ˜(b exp(tAd(b−1)Y )),Zi dt t=0 = hi=pr (Ad(b−1)Y ),Zi aC

= hpr (Ad(b−1)Y ),Zi. aC

For the last equality we have used the fact that a ⊥ k with respect to h, i (cf.

Lemma 3.12).

On the other hand,

˜ ˜ ˜ ˜ −1 −1 (ι(Z)ω)b(Yb) = ωb(Zb, Yb) = hpr (Ad(b )Z), Ad(b )Y i kC

= hAd(b−1)Z, pr (Ad(b−1)Y )i bC

= hZ, pr (Ad(b−1)Y )i bC

= hZ, pr (Ad(b−1)Y )i. aC

−1 The last two equations hold because Ad(b )Z ∈ Z + nC, and ht, nCi = {0}.

As Ma is compact and the action of T on (Ma, ω) is Hamiltonian, the Atiyah-

Guillemin-Sternberg convexity theorem (Thm. 2.11) asserts

Φ(Ma) = conv(Φ(Fix(Ma))).

75 In this formula conv(·) denotes the convex hull of (·) and Fix(Ma) stands for the

T -ﬁxed points in Ma. Standard structure theory implies that Fix(Ma) = W.a. We have thus proved:

Theorem 3.15. Let G be a complex semisimple Lie group and X ∈ Ω. Then

=(loga ˜(K exp(iX))) = conv(W.X) .

3.3.3 The complex convexity theorem for non-complex groups

For real groups the totally real K-orbits are no longer symplectic manifolds. How- ever, they can be viewed as fixed point sets of an involution τ on the compact symplectic manifold Ma as introduced in Section 3.3.2. We will define τ and show that it satisfies the T -τ-compatibility from Section 2.1. Then Theorem 2.14 can be applied.

Let g0 be a non-compact real form of the complex Lie algebra g. It is no loss of generality if we assume that g0 is θ-invariant. With k0 = g0 ∩ k and p0 = g0 ∩ p we then obtain a Cartan decomposition g0 = k0 +p0 of g0. We ﬁx a maximal abelian subalgebra a0 of p0 which is contained in a. Write Σ0 = Σ(g0, a0) for the corresponding restricted root system and set

π Ω = {X ∈ a : |α(X)| < ∀α ∈ Σ } . 0 0 2 0

As Σ0 = Σ|a0 \{0} we record that

Ω0 ⊆ Ω and Ω0 = Ω ∩ a0 . (3.22)

+ + It is no loss of generality to assume that Σ0 = Σ |a0 \{0} deﬁnes a positive system of

+ L α Σ . We form the nilpotent Lie algebra n0 = + g and observe that n0 = g0 ∩ n. 0 α∈Σ0 0

The analytic subgroups of G with Lie algebras g0, k0, a0 and n0 will be denoted by

76 G0,K0,A0 and N0. If Ξ0 = G0 exp(iΩ0)(K0)C/(K0)C denotes the complex crown of

G0/K0, then (3.22) yields a holomorphic G0-equivariant embedding

Ξ0 → Ξ.

˜ As described in Section 3.3.1, there exists a map loga ˜0 : Ξ0 = (N0)C(A0)C(K0)C → (a ) with loga ˜ (1) = 0. Note that loga ˜ = loga ˜| . 0 C 0 0 Ξ˜0

Let σ denote the Cartan involution on g0. We also write σ for the doubly complex linear extension of σ to gC. Likewise θ also stands for the complex linear extension of θ to gC. We will be interested in the involution τ = θ ◦ σ = σ ◦ θ on gC (which is the complex linear extension of the complex conjugation on g with respect to g0).

All these involutions on gC can be lifted to involutions on GC/KC and on BC, and we use the same letters to denote the lifts. ˜ Recall the deﬁnition of the symplectic manifold Ma = b(Ka) from Section 3.3.2.

Notice that Ma is τ-invariant. The connected component of the τ-ﬁxed point set ˜ which contains a is given by Qa = b(K0a). We also have a Hamiltonian action by the torus T0 = exp(ja0) on Ma.

The following lemma asserts T -τ-compatibility of the actions of T0 and τ on Ma.

Lemma 3.16. Consider the Hamiltonian torus action of T0 = exp(ja0) on Ma with

∗ moment map Φ: Ma → t0. Then the following assertions hold:

1. Qa is a Lagrangian submanifold of Ma.

−1 2. t ◦ τ = τ ◦ t for all t ∈ T0.

3. Φ ◦ τ = Φ.

77 ˜ Proof. 1. Consider U, V ∈ k0, k0 ∈ K0 and b = b(k0a) ∈ Qa. From the formula

(3.17) for the symplectic form ω on Ma we get

˜ ˜ −1 −1 ωb(Ub, Vb) = hpr (Ad(b )U), Ad(b )V i. kC

−1 −1 Now, both pr (Ad(b )U) and Ad(b )V lie in g0 + ig0. But for general ele- kC

ments X1,X2,Y1,Y2 ∈ g0 we have

¯ ¯ ¯ ¯ hX1 + iX2,Y1 + iY2i = <κ(X1 + jX2,Y1 + jY2) − <κ(X1 + jX2, Y1 + jY2)

¯ ¯ ¯ ¯ = <κ(X1,Y1) − <κ(X2,Y2) − <κ(X1, Y1) + <κ(X2, Y2)

¯ ¯ ¯ ¯ +<κ(X1, jY2) + <κ(jX2,Y1) − <κ(X1, jY2) − <κ(jX2, Y1)

= 0

The last equality is due to the invariance of κ and the fact that X1,X2,Y1,Y2 ∈ ˜ ˜ g0. This shows that ωb(Ub, Vb) = 0, i.e. Tb(Qa) is isotropic.

˜ 2. For b = b(ka) ∈ Ma and t ∈ T0,

t.τ(b) = t.˜b(τ(k)a) = ˜b(tτ(k)a) = ˜b(τ(t−1k)a) = τ(t−1.b).

3. We need to introduce some additional notation.

Let M M M n+ := gα ⊆ n, n− := g−α, and n0 := gα. + + α∈Σ+\Σ α∈Σ0 α∈Σ0 0 We denote by N +, N − and N 0 the analytic subgroups of G with Lie algebras C C C C n+, n− and n0 , respectively. Notice that n = n0 +n+ and therefore N = N +N 0. C C C C C C It is important to observe that τ(N +) = N + but τ(N 0) ∩ N 0 = {1}. C C C C ˜ ˜ Write t for a τ-invariant complement of a0 + ia0 in aC. Then aC = a0 + ia0 + t.

78 ˜ ˜ Let now x = ka for some k ∈ K. Then x, τ(x) ∈ NCACKC can be written

0 0 0 0 0 x = n+n0btk, τ(x) = n+n0b t k (3.23) with elements n , n0 ∈ N +, n , n0 ∈ N 0, b, b0 ∈ exp(a + ia ), t, t0 ∈ exp(˜t) + + C 0 0 C 0 0 0 and k, k ∈ KC. Clearly, (2) will be proven if we can show that b2 = (b0)2 (which forces b = b0 by the comments in Section 3.3.1). This will be established now.

It follows from (3.23) that

0 0 0 0 0 −1 τ(x) = n+n0b t k = τ(n+)τ(n0)bt τ(k).

Since τ leaves KC invariant and θ ﬁxes each element of KC, we obtain

−1 0 0 0 2 0 −1 0 −1 τ(x)θ(τ(x)) = n+n0(b ) θ(n0) θ(n+) (3.24)

2 −1 −1 = τ(n+)τ(n0)b θ(τ(n0)) θ(τ(n+)) . (3.25)

0 0 2 0 −1 2 −1 Notice that n0(b ) θ(n0) and τ(n0)b θ(τ(n0)) belong to the reductive group

+ + + − ZG (A0), and recall that τ(N ) = N and θ(N ) = N . Hence (3.24) com- C C C C C bined with the Bruhat decomposition of GC with respect to the parabolic sub-

+ 0 group ZG (A0)N forces that n = τ(n+). But then we have C C +

0 0 2 0 −1 2 −1 n0(b ) θ(n0) = τ(n0)b θ(τ(n0)) in Z (A ). The components of A , the center of Z (A ), on both sides must GC 0 0,C GC 0 coincide, therefore

(b0)2 = b2.

79 With this result at hand we are now able to prove the complex convexity result for non-complex groups. Write W0 for the Weyl group of Σ0.

Theorem 3.17. Let G0 be a non-compact connected semisimple Lie group with Lie algebra g0. Fix an element X ∈ Ω0. Then

= loga ˜0(K0 exp(iX)) = conv(W0.X) . (3.26)

Proof. Deﬁne a = exp(iX). The left hand side in equality (3.26) coincides with

Φ(Qa) where Φ = = ◦ log ◦a˜ is the moment map on Ma. Lemma 3.16 says that conditions (1)-(3) in Theorem 2.14 are satisﬁed. Therefore,

= loga ˜0(K0 exp(iX)) = Φ(Qa) = conv(Φ(Fix(Qa))).

˜ Standard structure theory shows that Fix(Qa) = b(W0 exp(iX)). This implies

Φ(Fix(Qa)) = W0.X, and ﬁnishes the proof.

80 APPENDIX

POISSON LIE GROUPS AND MANIN TRIPLES

We recall several deﬁnitions and facts about Poisson Lie groups and their relation with Manin triples (without proofs). Much of what follows is based on the exposition in [3], adapted to the case of smooth manifolds.

Deﬁnition. A smooth manifold M is called a Poisson manifold if there is a bracket operation {, } : C∞(M) × C∞(M) → C∞(M) such that

1. {, } is bilinear and antisymmetric,

2. For f, g, h ∈ C∞(M),

{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 (Jacobi identity),

3. For f, g, h ∈ C∞(M),

{f, gh} = {f, g}h + g{f, h} (Leibniz rule).

The Poisson bracket {, } gives rise to a bivector ﬁeld u on M: For m ∈ M, f, g ∈

C∞(M),

um(df(m), dg(m)) = {f, g}(m). (A1)

81 ∗ ∗ At each point m ∈ M the bivector u(m) is antisymmetric on TmM × TmM . Here

∗ TmM denotes the tangent space of M at m and TmM its dual space.

The Jacobi identity for {, } translates into a certain cocycle identity for u. On the other hand, an antisymmetric bivector ﬁeld u satisfying that cocycle identity deﬁnes a Poisson structure via (A1).

∞ Due to the Leibniz rule each f ∈ C (M) deﬁnes a vector ﬁeld Xf via

∞ Xf (g) = {f, g} ∀ g ∈ C (M).

The vector ﬁeld Xf is called Hamiltonian and f its Hamiltonian function. The Jacobi

∞ identity implies that Xf Xg − XgXf = X{f,g} for f, g ∈ C (M). Therefore, the set of all Hamiltonian vector ﬁelds is a Lie algebra.

At every point m ∈ M the Hamiltonian vector ﬁelds span a subspace Sm(M) =

∞ S {Xf (m): f ∈ C (M)} of the tangent space TmM. The subset S(M) = m∈M Sm(M) of the tangent bundle T M defines a differentiable generalized distribution. It can be shown that the flow of a Hamiltonian vector field leaves the Poisson bracket invariant.

This and the fact that the Hamiltonian vector ﬁelds form a Lie algebra imply that

Frobenius’ theorem can be applied. It asserts that S(M) is integrable. This means for every m ∈ M there exists a submanifold N containing m such that TnN = Sn(M) for every n ∈ N. The Poisson structure induces a symplectic structure ωN on N by the prescription ωN (Xf (m), Xg(m)) = {f, g}(m). If N is maximal connected then it is called a symplectic leaf of the Poisson manifold (M, {, }). Summarizing this gives the following theorem.

82 Theorem. A Poisson manifold (M, {, }) admits a foliation into (maximal connected) symplectic submanifolds, the so called symplectic leaves. The tangent space at each point of a leaf is generated by the Hamiltonian vector ﬁelds. The symplectic structure ω on each leaf is induced by the Poisson bracket {, } via ω(Xf , Xg) = {f, g}.

A map φ : M1 → M2 between two Poisson manifolds is called a Poisson map if it

∞ preserves the brackets, i.e. if for all f, g ∈ C (M2),

∗ φ {f, g}2 = {f ◦ φ, g ◦ φ}1.

Here, φ∗ denotes the pullback (of a function), i.e. (φ∗h)(m) = h(φ(m)) for h ∈

∞ C (M2), m ∈ M1.

Given two Poisson manifolds (M1, {, }1) and (M2, {, }2) one can deﬁne a Poisson

∞ bracket {, } on the cartesian product M = M1 × M2. For f, g ∈ C (M), m =

(m1, m2) ∈ M,

{f, g}(m) = {f(·, m2), g(·, m2)}1(m1) + {f(m1, ·), g(m1, ·)}2(m2).

We now consider a special class of Poisson manifolds.

Deﬁnition. A Lie group H carrying (as a manifold) a Poisson structure {, } is called Poisson Lie group if the multiplication mapping µ : H × H → H is Poisson.

Note that this deﬁnition does not make a statement about the inversion map inv : H → H. One can show that in a Poisson Lie group the inversion map is always anti-Poisson, i.e. inv∗{, } = −{, }.

83 In a Lie group H the tangent space ThH at any point h ∈ H can be identiﬁed with the Lie algebra h = Lie(H) = TeH using left translation lh : x 7→ hx. In this way the bivector u induced by {, } as in (A1) can be identiﬁed with a map ψ : H → ∧2h.

A direct computation shows how the Jacobi identity of {, } translates into a property of ψ:

Let H be a Lie group with Poisson bracket {, }. Then the following are equivalent.

• H is a Poisson Lie group

• ψ(ab) = b−1ψ(a) + ψ(b) ∀ a, b ∈ H. (A2)

Here, the action of H on ∧2h is induced by the coadjoint action.

Consider now the differential dψ(e): h → ∧2h. Its dual defines an antisymmetric product on h∗. One can show that it even defines a Lie algebra product.

Thus, both h and h∗ are Lie algebras. We denote both brackets by [, ]. Each Lie algebra acts on the other one by coadjoint action, e.g. for X,Y ∈ h, α ∈ h∗, hX.α, Y i =

−hα, [X,Y ]i.

Theorem. Let H be a Poisson Lie group with Lie algebra h. The formula

[X + α, Y + β] = [X,Y ] + X.α − α.X + β.Y − Y.β + [α, β] for X,Y ∈ h, α, β ∈ h∗ deﬁnes a Lie algebra structure on h ⊕ h∗.

The bilinear form

∗ ∗ h, i :(h ⊕ h ) × (h ⊕ h ) → R, h(X, α), (Y, β)i = α(Y ) + β(X)

84 is symmetric, non-degenerate and invariant. Moreover, h and h∗ are maximal isotropic subspaces with respect to h, i.

Deﬁnition. A triple (g, h, k) of Lie algebras is called a Manin triple if

1. h and k are Lie subalgebras of g, and g = h ⊕ k as vector spaces, and

2. There is a non-degenerate, symmetric and invariant bilinear form h, i on g with

respect to which h and k are maximal isotropic subspaces.

So the last theorem means that a Poisson Lie group H determines a Manin triple

(h ⊕ h∗, h, h∗). It is not diﬃcult to derive from (A2) that the Poisson structure on H is determined completely by this Manin triple.

Deﬁnition. A triple (G, H, K) of Lie groups is called Manin triple if the corresponding triple of Lie algebras (g, h, k) is a Manin triple.

We now want to see how a Manin triple (G, H, K) gives rise to a bracket {, } that turns H into a Poisson Lie group.

Let prh : g → h denote the projection with kernel k. Since h and k are maximal isotropic with respect to the bilinear form h, i from the Manin triple (g, h, k), there is a natural identiﬁcation h∗ ' k. The bracket {, } on H is now deﬁned as follows. For f, g ∈ C∞(H), x ∈ H,

−1 −1 {f, g}(x) = hprhAd(x )df(x), Ad(x )dg(x)i. (A3)

Here, df(x) and dg(x) are viewed as elements of k under two identiﬁcations k ' h∗ '

85 ∗ TxH . It is immediate that {, } is bilinear, antisymmetric and satisﬁes the Leibniz rule. From the Jacobi identity on the Lie algebra g it can be derived that the Jacobi identity also holds for {, }. Finally, the map ψ : H → ∧2h associated to the Poisson bracket {, } satisﬁes (A2).

We summarize the last paragraph.

Theorem. Let (G, H, K) be a Manin triple. The bracket {, } on H as deﬁned in (A3) turns H into a Poisson Lie group.

Finally, we want to describe the symplectic leaves of the Poisson Lie group H arising from the Manin triple (G, H, K). Consider the restriction p : H → G/K of the quotient map for the right action of K on G. Since p is a local diffeomorphism, the left K-action on G/K translates into a local action of K on H. In particular, each element X ∈ k defines a vector field on H. One can verify that, at a point h ∈ H, the tangent vectors generated by k are exactly the tangent vectors generated by the

Hamiltonian functions of H. From the ﬁrst theorem in the Appendix we conclude:

Proposition. Let H be a Poisson Lie group deﬁned as above by the Manin triple

(G, H, K).

The symplectic leaves of H are exactly the connected components of the K-orbits in

H. The symplectic form ω of such a leaf at the point x ∈ H is given by

−1 −1 ωx(X,Y ) = hprhAd(x )X, Ad(x )Y i ∀ X,Y ∈ k. (A4)

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