{'fraction_non_alphanumeric': 0.06932202821810382, 'fraction_numerical': 0.013455544756861825, 'mean_word_length': 3.422632281013784, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 39, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A real convexity theorem for quasi-hamiltonian actions

The main result of this paper is a quasi-hamiltonian analogue of a special case of the O'Shea-Sjamaar convexity theorem for usual momentum maps. We denote by U a simply connected compact connected Lie group and we fix an involutive automorphism of maximal rank on this Lie group (such an automorphism always exists). We then denote by M a quasi-hamiltonian U-space and we prove that the image under the momentum map of the fixed-point set of a form-reversing compatible involution of M is a convex polytope, which is in fact equal to the full momentum polytope. This theorem was announced in arXiv:math/0609517v1. As an application, we obtain an example of lagrangian subspace in representation spaces of surface groups.


Introduction
In the following, M will always denote a manifold, U a compact Lie group, with Lie algebra u := Lie(U ), acting on M , and T a torus with Lie algebra t := Lie(T ). Additionally, if X is a set and α : X → X is an involution on X, we will denote its fixed-point set F ix(α) by X α . The convexity properties of the momentum map of a hamiltonian action have been extensively studied since the convexity theorem for hamiltonian torus actions of Atiyah ([Ati82]) and Guillemin and Sternberg ([GS82]), which says that the image µ(M ) ⊂ t * of the momentum map is a convex polyhedron. In [Kir84], Kirwan studied the case of hamiltonian actions of non-abelian compact connected Lie groups. There, she proved a conjecture of Guillemin and Sternberg in [GS84a] saying that the intersection µ(M ) ∩ t * + of the image of the momentum map with any closed Weyl chamber is a convex polyhedron, usually called the momentum polytope. In [Dui83], Duistermaat studied the image µ(M β ) of the fixed-point set M β of an anti-symplectic involution β : M → M . He then proved that for a hamiltonian torus action and under certain compatibility conditions of β with the action of T and the momentum map of this action, µ(M β ) was also a convex polyhedron, in fact equal to the momentum polyheron µ(M ). A convexity statement for the image µ(M β ) of the fixed-point set of an anti-symplectic involution is since then often referred to as a real convexity theorem, since in the special case where M is kählerian, M β is a totally real totally geodesic lagrangian submanifold of M . The result of Duistermaat was then extended to hamiltonian actions of non-abelian compact connected Lie groups in [OS00]. There, O'Shea and Sjamaar proved that, for an appropriate choice of a closed Weyl chamber t * + , the set µ(M β ) ∩ t * + is a convex subpolytope of the momentum polytope µ(M ) ∩ t * + , obtained by intersecting this polytope with the fixed-point set of a certain involution τ − on u * = Lie(U ) * . In particular, when τ − fixes pointwise the closed Weyl chamber, the two polytopes are equal. In all of the above convexity theorems, the momentum map µ takes its values in the (dual of the) Lie algebra u and the convexity properties of such momentum maps, the momentum maps of hamiltonian actions, are well understood. In [AMM98], Alekseev, Malkin and Meinrenken introduced the notion of quasi-hamiltonian action. There, the momentum map µ takes its values in the compact connected Lie group U and they showed that if U is in addition simply connected, the momentum map µ : M → U enjoys convexity properties similar to those of usual momentum maps: the intersection of the image µ(M ) of the momentum map with the exponential exp(W) of any closed Weyl alcove W ⊂ u is homeomorphic, via the exponential map, to a convex polyhedron of u. Their result, whose proof is based on a convexity theorem of Meinrenken and Woodward for hamiltonian loop group actions (see [MW98]), is a quasi-hamiltonian analogue of the Kirwan convexity theorem. To the best of our knowledge, there does not exist a quasi-hamiltonian analogue of the O'Shea-Sjamaar theorem. Stating and proving such a real convexity theorem for quasi-hamiltonian actions is the objective and main result of this paper (see theorem 3.3). To do this, we propose a particular construction of a symplectic slice in an arbitrary quasihamiltonian U -space (M, ω, µ : M → U ). This construction is a quasi-hamiltonian analogue of the construction of Hilgert, Neeb and Plank in [HNP94] (based on the approach to momentum convexity of Condevaux, Dazord and Molino in [CDM88]). By definition, such a symplectic slice is a hamiltonian space in the usual sense, for the action of a maximal torus T ⊂ U . The idea of our proof is then to apply the Duistermaat theorem to this symplectic slice to obtain a real convexity theorem for quasi-hamiltonian actions. This will not work directly but can serve as a guideline for the proof, as we shall see in section 4. Incidentally, the construction of the symplectic slice also enables us to give a proof of the Alekseev-Malkin-Meinrenken-Woodward convexity theorem for quasi-hamiltonian actions without using the corresponding statement for hamiltonian loop group actions. The paper is organized as follows. In section 2, we briefly recall the precise statements of the known convexity theorems in hamiltonian and quasi-hamiltonian geometry. In section 3, we likewise recall the known real convexity theorems for hamiltonian actions in the usual sense and we state our main result, which is a real convexity theorem for quasi-hamiltonian actions (theorem 3.3). In section 4, we construct a particular symplectic slice in an arbitrary quasi-hamiltonian space (M, ω, µ : M → U ) and give the proof of theorem 3.3. The explicit description of this symplectic slice as the pre-image of a cell of the closed Weyl alcove exp(W) is crucial in the proof of theorem 3.3, mostly to show that this symplectic slice contains fixed points of the involution β : M → M . The other crucial ingredient in the proof is the local-global principle of Condevaux-Dazord-Molino and Hilgert-Neeb-Plank, which we quote from [HNP94]. In section 5, the final one of this paper, we give examples of applications of the real convexity theorem for quasi-hamiltonian actions (theorem 3.3), which include obtaining an example of lagrangian subspace in representation spaces of surface groups.

Convexity properties of momentum maps
In this section, we review the usual convexity theorems for momentum maps in the hamiltonian and quasi-hamiltonian setting. In the following, M will always denote a manifold, ω a 2-form on M and U a compact connected Lie group acting on M . We will always assume that this action admits a momentum map µ : M → E. In this paper, we will consider two cases: E = u * , the dual of the Lie algebra u := Lie(U ) (hamiltonian case), and E = U (quasi-hamiltonian case). The target space E of the momentum map is also endowed with an action of U and the momentum map µ is assumed to be equivariant (if E = u, the action of U on E is the co-adjoint action, if E = U it is the conjugacy action).
2.1. Stating a convexity theorem. Stating a convexity theorem for the momentum map µ : M → E consists in showing that the image µ(M ) of the momentum map has certain convexity properties: by intersecting it with a fundamental domain D for the action of U on E, the resulting set µ(M ) ∩ D is homeomorphic to a convex subset of a vector space. In this paper, this vector space will always be finite-dimensional. In fact, it will be the Lie algebra u = Lie(U ). There are nonetheless known examples of convexity theorems for momentum maps in inifinite dimension (see for instance [MW98]). Observe that, since µ : M → E is U -equivariant, µ(M ) is a union of U -orbits. Therefore, considering the intersection µ(M ) ∩ D is equivalent to considering the projection of µ(M ) onto the orbit space E/U ≃ D: the projection p : E → E/U induces a homeomorphism between µ(M ) ∩ D and p • µ(M ) (recall that a fundamental domain D contains exactly one point of each orbit). With this approach we see that, for the notion of convexity to make sense, it is sufficient to be able to identify the orbit space E/U with a convex subset of a vector space (see in particular definition 2.3). We now give the classical examples of convexity theorems for momentum maps.
2.2. The Atiyah-Guillemin-Sternberg theorem. The following theorem, proved independently by Atiyah in [Ati82] and by Guillemin and Sternberg in [GS82], is historically the first convexity result for momentum maps. It says that the image µ(M ) of the momentum map µ : M → t * of a hamiltonian torus action is a convex polyhedron. In this case, U = T is an abelian compact connected Lie group and E = t * is the dual of the Lie algebra t = Lie(T ). The co-adjoint action is trivial and so the fundamental domain is D = t * . The original statement of the Atiyah-Guillemin-Sternberg (AGS) theorem is for M compact, in which case µ(M ) is a compact convex poyhedron (a convex polytope), whose vertices are the images under µ of fixed points of the action of T on M . This statement was later generalized to non-compact manifolds by Hilgert, Neeb and Plank in [HNP94], based on the work of Condevaux, Dazord and Molino in [CDM88] (see also [Sja98], [LMTW98] and [Ben02]). This generalization is obtained under the assumption that the momentum map is proper (in particular, closed).
Theorem 2.1 (Momentum convexity for hamiltonian torus actions). [Ati82,GS82] Let (M, ω) be a connected symplectic manifold endowed with a hamiltonian action of a torus T whose momentum map µ : M → t * is proper. Then: If M is compact then µ(M ) is a convex polytope and it is the convex hull of the images under µ of the fixed points of the action.
This theorem enjoys very nice applications, some of which in fact motivated the search for such a statement. For instance, the conjugacy action of the torus of diagonal unitary matrices T ⊂ U (n) on a conjugacy class of hermitian matrices O ⊂ H(n) (a co-adjoint orbit of U (n)) admits as a momentum map the map taking a hermitian matrix H to the collection diag(H) ∈ t * = diag(H(n)) of its diagonal entries. The AGS theorem then implies the Schur-Horn theorem: the diagonal of a hermitian matrix is a convex combination of permutations of its eigenvalues: diag(H) = σ∈Sn a σ (λ σ1 , ... , λ σn ) with (λ 1 , ... , λ n ) = Spec(H) ⊂ R n ≃ Lie(T ) and σ∈Sn a σ = 1, a σ ≥ 0. The reason why it is important to have such a statement for non-compact M is that it will later be applied to symplectic slices (also called symplectic cross-sections) to prove more general convexity results by reduction to the case of a torus action. Symplectic slices in a U -space (M, ω, µ : M → E) are symplectic manifolds N ⊂ M such that U.N = M and endowed with a hamiltonian action of a maximal torus T ⊂ U . They also satisfy µ(N ) = µ(M ) ∩ D, where D is a fundamental domain for the U -action on E. These symplectic slices are non-compact manifolds. All of this will be developped in section 4 in the case of quasi-hamiltonian spaces (M, ω, µ : M → U ). We now move on to hamiltonian U -spaces (M, ω, µ : M → u * ) with U non-abelian.
2.3. The Kirwan-Guillemin-Sternberg theorem. The following theorem, proved by Kirwan in [Kir84] (and by Guillemin and Sternberg in [GS84a] in the case where M is a Kähler manifold), deals with hamiltonian actions of compact connected groups U . In the notation of subsection 2.1, E = u * is the dual of the Lie algebra u = Lie(U ). A fundamental domain for the co-adjoint action of U on u * is homeomorphic to a closed Weyl chamber in an arbitrary Cartan subalgebra t ⊂ u, and is usually denoted by t * + . In particular, u * /Ad(U ) ≃ t * + is homeomorphic to a closed convex subset of a finite-dimensional vector space. As in subsection 2.2, the original statement of the Kirwan-Guillemin-Sternberg (KGS) theorem was for compact M and was generalized to non-compact M in [HNP94] (see also [Sja98] and [Ben02]) provided the momentum map is proper (in particular, closed). To state the theorem, it is convenient to assume that there is a given Ad-invariant non-degenerate bilinear form on u = Lie(U ), so that we can identify u * and u equivariantly and, for any subalgebra h ⊂ u, think of h as a subspace of u * . In particular, a Weyl chamber may be thought of as a subset of u * .
Theorem 2.2 (Momentum convexity for hamiltonian actions of compact groups). [Kir84] Let (M, ω) be a connected symplectic manifold endowed with a hamiltonian action of a compact connected Lie group U whose momentum map µ : M → u * is proper. Then, for any choice of a Cartan subalgebra t ⊂ u and any choice of a closed Weyl chamber t * In [Sja98], Sjamaar showed that if additionally M is compact, ⊕ u x } and that this set µ(E) is a discrete subset of t * + (this last statement being true even if M is not compact). There are now many proofs of the KGS convexity theorem, for which we refer to the above-cited papers. The one we will often use as a guiding analogue in our study of the quasi-hamiltonian case is the one given by Hilgert, Neeb and Plank in [HNP94]. The general argument, based on ideas of Guillemin and Sternberg in [GS82] and Condevaux, Dazord and Molino in [CDM88], is as follows: one constructs a symplectic slice N ⊂ M to which one applies the AGS theorem to show that µ(N ) is convex; since by construction µ(N ) = µ(M ) ∩ t * + , one gets convexity of µ(M ) ∩ t * + . The main difficulty is to be able to apply the AGS theorem to the symplectic slice N . As a matter of fact, the AGS theorem does not apply directly to N (even the version of the AGS theorem for non-compact manifolds). But the use of the local-global principle enables one to overcome this difficulty (see [HNP94]). We will see in subsection 4.3 how one can proceed in the same way in the quasi-hamiltonian case. As an application of theorem 2.2, consider the case where U = U (n) and M = O λ1 × O λ2 is a product of co-adjoint orbits. Then, for (H 1 , H 2 ) ∈ O λ1 × O λ2 , one has µ(H 1 , H 2 ) = H 1 + H 2 and µ(M ) ∩ t * + consists of the possible (ordered) spectra for the sum of two hermitian matrices. By theorem 2.2, these possible spectra form a closed convex subset of R n ≃ Lie(T ) (this is the Weyl-Horn problem; see [BS00] for a symplectic approach to this problem).
2.4. Momentum maps with values in a compact connected Lie group. We now turn to momentum maps with values in a compact connected Lie group. More precisely, the notion we shall be dealing with in the rest of this paper is the notion of quasi-hamiltonian space, which is due to Alekseev, Malkin and Meinrenken. In the notation of subsection 2.1, the momentum map µ : M → E takes its values in the Lie group U acting on the manifold M : one has E = U and a fundamental domain D for the conjugacy action of U on itself is homeomorphic to the quotient of the closed Weyl alcove W ⊂ t = Lie(T ) by the affine action of π 1 (U ) on t, where T is a maximal torus of U (see for instance [Bou82], p.45). In particular, U/Int(U ) ≃ W/π 1 (U ) is homeomorphic to the closed convex polyhedron W ⊂ t if and only if π 1 (U ) = 0. Thus, when U is simply connected, the notion of a convex subset of U/Int(U ) makes sense: Observe additionally that W is a compact polyhedron. When U is not simply connected, W/π 1 (U ) is not simply connected either, and therefore cannot be homeomorphic to a convex subset of a vector space. From now on, we will always assume that the compact connected Lie group U is simply connected. Then, there is a convenient way of understanding the identification U/Int(U ) ≃ W: the exponential map exp : u = Lie(U ) → U restricts to an homeomorphism exp | W : W → exp(W) and D := exp(W) is a fundamental domain for the conjugacy action of U on itself. A subset A ⊂ exp(W) is then called convex if exp −1 (A) ⊂ t is convex. We then have the following convexity statememt, which we quote from [AMM98] (where it is deduced from the convexity theorem for hamiltonian loop group actions of [MW98]): Remark 2.5 (On the assumption of properness of the momentum map). When U is a compact Lie group, the known examples of quasi-hamiltonian U -spaces ([AMM98], [HJS06]) are compact, so the assumption that µ be a proper map is superfluous. Nonetheless, the convexity theorem remains true under this assumption, so that it could be applied to non-compact quasi-hamiltonian spaces, should such examples be discovered.
As explained in [AMM98], although this convexity statement concerns quasi-hamiltonian actions, it can be deduced from the analoguous statement for hamiltonian loop group actions. In subsection 4.3, we will give another proof of this theorem, based on the approach to momentum convexity for usual hamiltonian actions developped in [HNP94]: we construct a particular symplectic slice N ⊂ M for any quasi-hamiltonian action and we use the local-global pinciple of [HNP94] to show that µ(N ) is convex. In subsection 4.4, we use this construction of a symplectic slice to give a proof of our main result (theorem 3.3). Before ending this section, we give an example of application of theorem 2.4, which is an analogue of the application of the KGS theorem given in subsection 2.3. Consider the case where U = U (n) and M = C λ1 × C λ2 is a product of conjugacy classes. Then, for (u 1 , u 2 ) ∈ C λ1 × C λ2 , one has µ(u 1 , u 2 ) = u 1 u 2 and µ(M ) ∩ exp(W) consists of the possible (ordered) spectra for the product of two unitary matrices. By theorem 2.4, these possible spectra form a closed convex subset of R n ≃ Lie(T ).

Real convexity theorems
The notion of real convexity theorem appears when one considers not the full image µ(M ) of the momentum map but rather the image µ(M β ) of the fixed-point set of an involution β : M → M satisfying β * ω = −ω. In the usual hamiltonian case, ω is a symplectic form and M β is a lagrangian submanifold of M . In particular when M is a Kähler manifold, M β is a totally real totally geodesic submanifold of M . This inspired the terminology real convexity theorem for convexity properties of µ(M β ).
3.1. Stating a real convexity theorem. Recall from subsection 2.1 that convexity theorems for momentum maps µ : M → E are obtained by intersecting the image of µ(M ) with a fundamental domain domain D of the action of U on E. Since the work of O'Shea and Sjamaar in [OS00], it is known that to obtain a real convexity theorem for hamiltonian actions of compact groups, one needs to assume that the Lie group U is endowed with an involutive automorphism τ and that the anti-symplectic involution β on M is compatible with the action of U and with the momentum map µ : M → u * in the sense that β(u.x) = τ (u).β(x) and µ • β = τ − • µ, where τ − denotes the involution (−dτ ) * : u * → u * . A real convexity theorem is then a statement on the convexity properties of µ(M β ), namely that µ(M β ) ∩ D is a convex polyhedron and in fact a subpolyhedron of the momentum polyhedron µ(M ) ∩ D. More precisely, for an appropriate choice of D, one has Observe that by compatibility of β with the momentum map µ : M → u * , one automatically has µ(M β ) ⊂ F ix(τ − ). In this hamiltonian context, choosing D appropriately means choosing the closed Weyl chamber t * + such that t * + ∩ F ix(τ − ) is a fundamental domain for the action of (the neutral component of) Based on this remark, a general real convexity theorem should be a statement of the following form: the intersection of the image µ(M ) of the momentum map with a fundamental domain D 0 of the action of U τ on F ix(τ − ) is a convex set, equal to µ(M β ) ∩ D 0 . This will indeed be the case in theorem 3.3. There, E = U and τ − (u) = τ (u −1 ). The fixed-point set F ix(τ − ) is not connected in general but the neutral component K of U τ acts on the connected component Q 0 of 1 in F ix(τ − ). If the closed Weyl alcove W is chosen appropriately, then exp(W) ∩ Q 0 is a fundamental domain for this action and it is natural to conjecture that if β : M → M is a form-reversing compatible involution on the quasihamiltonian space M , then µ(M β ) ∩ exp(W) is a convex subpolyhedron of µ(M ) ∩ exp(W), obtained by intersecting it with Q 0 . In theorem 3.3, we prove this conjecture under the assumption that the symmetric pair (U, τ ) is of maximal rank, meaning that there exists a maximal torus T ⊂ U fixed pointwise by τ − (such an involution always exists on an arbitrary compact connected Lie group, see for instance [Loo69]). In this case, the fundamental domain exp(W) ∩ Q 0 is simply exp(W) ⊂ T ⊂ Q 0 (for instance: if τ (u) = u on U = U (n), every symmetric unitary matrix is conjugate to a diagonal one by a real orthogonal matrix).
3.2. The Duistermaat theorem. The Duistermaat theorem ( [Dui83]) is a real version of the AGS theorem, the involutive automorphism on the (abelian) group T being τ (t) = t −1 . In particular, τ − = Id t * . The original statement was for M compact and was extended to non-compact M in [HNP94].  We will use some of the ideas of the proof of Duistermaat's theorem given in [HNP94] in subsection 4.4. As an application of theorem 3.1, let us go back to the example given in subsection 2.2. In this case, the Duistermaat theorem says: the diagonal of a real symmetric matrix is a convex combination of permutation of its eigenvalues and any such convex combination is the diagonal of a real symmetric matrix with the same spectrum.
3.3. The O'Shea-Sjamaar theorem. The O'Shea-Sjamaar theorem is a real version of the KGS theorem. The proof given in [OS00] consists, as in [Sja98], in establishing the result for projective and affine varieties and then handle the general case using a local normal form theorem.
Theorem 3.2 (A real convexity theorem for hamiltonian actions of compact groups). [OS00] Let (M, ω, mu : M → u * ) be a connected hamiltonian U -space with proper momentum map. Let τ : U → U be an involutive automorphism of U . Denote by τ − the involution (−dτ ) * : u * → u * and by q the (−1)-eigenspace of dτ : u → u. Choose a maximal abelian subspace a of q, and a Cartan subalgebra t of u containing a (such a Cartan subalgebra is globally stable under dτ ) and a closed Weyl chamber t * + ⊂ u * . Let β : M → M be an involution satisfying: As an application of this theorem, let us come back to the example given in subsection 2.3. When U = U (n) and τ : u → u, the Cartan subalgebra consisting of diagonal hermitian matrices is fixed pointwise by τ − : H → H t (the symmetric pair (U, τ ) is of maximal rank): one has a = t and therefore t * Consequently, the possible spectra of a sum S 1 + S 2 of two real symmetric matrices are the same as those of a sum H 1 + H 2 of two hermitian matrices satisfying Spec(H i ) = Spec(S i ) for i = 1, 2.
3.4. A real convexity theorem for quasi-hamiltonian actions. We can now state the main result of this paper, which is a real convexity theorem for quasi-hamiltonian actions. This result is a quasi-hamiltonian analogue of the O'Shea-Sjamaar theorem in the special case where the symmetric pair (U, τ ) is of maximal rank.
Theorem 3.3 (A real convexity theorem for quasi-hamiltonian actions of compact groups). Let U be a compact connected simply connected Lie group and let (M, ω, µ : M → U ) be a quasi-hamiltonian U -space with proper momentum map µ : M → U . Let τ : U → U be an involutive automorphism of U such that there exists a maximal torus T ⊂ U fixed pointwise by the involution τ − : u ∈ U → τ (u −1 ) and let W ⊂ t = Lie(T ) be a closed Weyl alcove. Let β : M → M be an involution satisfying: Observe that exp(W) ⊂ T ⊂ F ix(τ − ) and contains 1 ∈ U . Since exp(W) is connected, this implies exp(W) ⊂ Q 0 . Assumption (v) is of topological nature and accounts for the fact that, in a Lie group, the subset F ix(τ − ) is not necessarily connected. When F ix(τ − ) is connected (for instance if U = SU (n) and τ (u) = u, since in this case every symmetric unitary matrix is of the form exp(iS) with S a real symmetric matrix), this assumption is automatically satisfied (recall that by compatibility of the β with the momentum map one has µ(M β ) ⊂ F ix(τ − )). In the O'Shea-Sjamaar theorem there is no such assumption because F ix(τ − ) is a vector subspace of u * and is therefore connected. The rest of this paper is devoted to the proof of theorem 3.3 and to some applications. Observe that here, in contrast with subsection 3.3, we do not have an application of theorem 3.3 to the spectrum of a product of real orthogonal matrices (which would be a real version of the application of theorem 2.4 given in subsection 2.4). The reason for this is that if τ : u → u on U = U (n) and M = C 1 × C 2 is a product of two conjugacy classes, if we set β(u 1 , u 2 ) = (u 1 , u 2 ) we do not have µ • β = τ − • µ (for lack of commutativity in U ). Instead we refer to theorem 5.4 for an application of theorem 3.3 that would be a real version of the statement on the possible spectrum of a product of two unitary matrices.

Constructing a symplectic slice in a quasi-hamiltonian space
In this section, we prove the real convexity result for group-valued momentum maps stated in theorem 3.3. The idea of the proof is to apply the Duistermaat theorem to an appropriate symplectic slice N ⊂ M . The point that requires the most attention is showing that N β = ∅. The construction of the symplectic slice itself relies on adapting the ideas of [HNP94] to the quasi-hamiltonian setting. In particular, using this approach, we will be able, in subsection 4.3, to give a proof of the Alekseev-Malkin-Meinrenken-Woodward convexity theorem.
4.1. The local-global principle. We recall here the results of [HNP94] that we will later apply to the symplectic slice N ⊂ M . The following theorem is obtained by using the local normal form theorem for hamiltonian torus actions.
Theorem 4.1 (Local convexity theorem for hamiltonian torus actions). [HNP94] Let (N, ω) be a symplectic manifold endowed with a hamiltonian action of a torus T with momentum map µ : N → t * . Then for every x ∈ N , there exist an open neighbourhood V x of x ∈ N and a polyhedral cone C µ(x) ⊂ t * with vertex µ(x) such that: If in addition β is an antisymplectic involution on N satisfying β(t.x) = t −1 .β(x) and µ • β = µ, then assertion (i) above remains true for the manifold N β := F ix(β) and the same cones C µ(x) , for x ∈ N β , that is: For additional local properties, including a description of the cones C µ(x) using the local normal form of the action, we refer to [HNP94]. Conditions (ii) and (iii) above play a special role when it comes to convexity considerations insofar as they make it possible to obtain a global result from a local one (see theorem 4.3), which justifies the following definition: Definition 4.2 (Local convexity data). Let X be a connected Hausdorff space, and let V be a finite dimensional vector space. Consider a continuous map ψ : alone is said to be locally fibre-connected.
As an example, theorem 4.1 shows that the momentum map of a hamiltonian torus action gives rise to local convexity data. We then have: In particular, we see that theorem 2.1 immediately follows from theorems 4.1 and 4.3. In order to obtain a Duistermaat-like statement for our symplectic slice N ⊂ M , we will need the following two results, the first of which is a corollary of the local-global principle (these two results will be used in the proof of proposition 4.30).
Proof. A convex set is the intersection of cones containing it, so that: There is one more difficulty one is faced with when trying to apply theorem 2.1 or 3.1 to a symplectic slice. As mentioned in subsection 2.2, symplectic slices are non-compact manifolds (which in fact motivated the search for a strengthened version of the Atiyah-Guillemin-Sternberg and Duistermaat theorems in [CDM88,HNP94]) and in addition to that, the momentum map µ| N is not a proper map. In subsection 4.3, we will see how to circumvent this difficulty by applying the following result: Proposition 4.6. [HNP94] Let ψ : X → V be a map giving rise to local convexity data (V x , C ψ(x) ) x∈X . Consider any closed locally polyhedral convex subset D ⊂ V and set Y : Finally, we quote the following result from [Ben02]. It is a purely topological argument that we will use in subsection 4.3. We refer to [Ben02] for a proof and comments on the fact that the assumption that f (Ω) be convex is necessary.
The manifold N ⊂ M whose existence is guaranteed by theorem 4.8 is called a symplectic slice because it is a symplectic manifold satisfying U.N = M . Theorem 4.8 is a quasi-hamiltonian analogue of a result of Hilgert, Neeb and Plank in [HNP94]. The details of the construction, that we now set out to give, will show some remarkable similarities with the usual hamiltonian setting. 4.2.1. Structure of the Weyl alcove. We begin by giving a description of the closed Weyl alcove W ⊂ t = Lie(T ) in terms of the root system of (U, T ). We refer to [Loo69] and [Bou82] for a proof of the classical results on root systems that we shall use. The compact connected simply connected Lie group U is endowed with an Ad-invariant positive definite product (. | .) (for instance, minus the Killing form). In particular, we may identify equivariantly u * with u and t * with t. Consider now the adjoint action of the maximal torus T ⊂ U on the complexification u C of u = Lie(U ), and denote by R the corresponding root system: where for any linear form α : t → R we set: To every root α ∈ R we associate the hyperplane A connected component of t\ ∪ α∈R H α is called a Weyl chamber of the root system R. By definition, it is an open cone of t. In particular, it is convex. As we want our notation to be consistent with the statement of the KGS convexity theorem (theorem 2.2), a Weyl chamber will be denoted by t * + ⊂ t ≃ t * . Its closure will be denoted by t * + and called a closed Weyl chamber. It is a closed convex subset of t. We now choose a Weyl chamber t * + ⊂ t and denote by R + (t * + ) (or simply R + ) the set of associated positive roots: R + (t * + ) := {α ∈ R | α(X) > 0 for one and therefore all X ∈ t * + }. A positive root α ∈ R + is then said to be decomposable if it can be written as a sum α = β∈R+ n β .β where n β ≥ 0 are integers. Otherwise it is called indecomposable, or simple. We denote by ∆(t * + ) (or simply ∆) the set of simple roots in R + , also called a basis of R, since by definition every root α ∈ R is of the form β∈∆ n β .β with n β ∈ Z. The set of simple roots ∆ associated to the choice of a Weyl chamber t * + enables one to give a very nice description of the polyhedral structure of the closure t * + of the Weyl chamber, which we will recall shortly. This description is key to the proof of the momentum convexity theorem presented in [HNP94]. We will now describe in a similar way the polyhedral structure of a Weyl alcove. Generalizing the definition of the hyperplanes H α , we set, for all α ∈ R and all n ∈ Z: From now on, when we speak of a Weyl alcove, we will assume that its closure contains 0. We then have: Proposition 4.10. Let U be a compact connected simply connected Lie group and let W ⊂ u = Lie(U ) be a Weyl alcove for U . Then the set exp(W) ⊂ U is a fundamental domain for the conjugacy action of U on itself. Moreover, the exponential map exp : u → U induces a one-to-one map from the compact convex polytope W onto the closed set exp(W) ⊂ U . Consequently, we have homeomorphisms: We refer to [Bou82] (p.45) or to [Loo69] (p.37) for a proof of this result. We now wish to describe the polyhedral structure of the convex polytope W ⊂ t. We begin with the polyhedral structure of the closed Weyl chamber t * + (see [HNP94]). By definition of ∆ ⊂ R + , we have: t * + = {X ∈ t | ∀α ∈ ∆, α(X) > 0}. And: t * + = {X ∈ t | ∀α ∈ ∆, α(X) ≥ 0}. For each subset S ⊂ ∆, we set: F S := {X ∈ t | ∀α ∈ S, α(X) = 0 and ∀α ∈ ∆\S, α(X) > 0} ⊂ t * + . And we then have: One remarkable feature of the sets (F S ) S⊂∆ , which we will call the cells of t * + , is that two elements X, Y lying in a same F S have the same stabilizer U X = U Y for the (co-)adjoint action of U on u ≃ u * (see lemma 6.3 in [HNP94]). We will establish an analogous property for the closed Weyl alcove W ⊂ t * + (see proposition 4.13). We first observe that W is also a union of cells. Instead of corresponding to subsets S ⊂ ∆ of the set of simple roots, these cells correspond to subset S ⊂ R + of the whole set of positive roots. More precisely, each subset S ⊂ R + can be uniquely written S = S 1 ∪ S 2 where S 1 ⊂ ∆ and S 2 ⊂ R + \∆, and for such an S = S 1 ∪ S 2 , we set: ∀α ∈ S 1 , α(X) = 0 ∀α ∈ ∆\S 1 , α(X) > 0 and ∀α ∈ S 2 , α(X) = 1 ∀α ∈ (R + \∆)\S 2 , α(X) < 1 .
We then have the following result: Lemma 4.12.
[Bou82] (p.48) Let U be a compact connected simply connected Lie group. Then for any u ∈ U , the centralizer U u = {v ∈ U | vuv −1 = u} is connected.
We can now show that the stabilizer for the conjugacy action of an element exp(X) ∈ exp(W) depends only on the cell of W containing X: Proposition 4.13. If u, v ∈ exp W = ⊔ S⊂R+ exp(W S ) lie in a same exp(W S ), then the centralizers U u and U v are equal.
Proof. Since U is compact connected and simply connected, lemma 4.12 shows that U u and U v are compact connected subgroups of U . Therefore U u = U v if and only if their Lie algebras are equal. We then know from [Loo69] (p.7) that the Lie algebra of U u is: so that it only depends on the cell W S ⊂ W in which X lies, which proves the proposition.
Definition 4.14. For any subset S ⊂ R + , we denote by U S the stabilizer of any element u ∈ exp(W S ).
Finally, if we consider, for any integer j, the set Σ j := {u ∈ U | dim U.u = j} of points of U whose conjugacy class is of dimension j, we have: Proposition 4.15. The intersection of Σ j with exp(W) is: In addition to that, Σ j is a submanifold of U and so is every exp(W S ). For any u ∈ exp(W S ), one has: Proof. This result is a consequence of the existence of local slices for actions of compact Lie groups (see for instance [GS84b]): for any local slice L through u ∈ U , the set of elements of L whose conjugacy class has the same dimension as U.u consists precisely of elements which are fixed by U u , and for a linear action the set of such fixed points is a subspace, so that Σ j is a submanifold of U , and it is U -invariant. Now for any S ⊂ R + , exp(W S ) is a submanifold of U (the chart is given by the exponential map) and it follows from the fact that exp(W) = ⊔ exp(W S ) is a fundamental domain for the conjugacy action and from proposition 4.13 that and that exp(W S ) is a local slice through any u ∈ exp(W S ), which concludes the proof.

Inverse image of a certain cell of the Weyl alcove.
From now on, we will also call closed Weyl alcove, or even simply Weyl alcove, the subset exp(W) of U , which is homeomorphic to the actual closed Weyl alcove W via the exponential map. It will always be clear from the context which of the sets W or exp(W) we are referring to. To prove theorem 4.8, we will construct a submanifold N ⊂ M such that µ(N ) ⊂ exp(W) ⊂ U , so that the map µ : N → W is no other than µ = exp −1 •µ| N : N → t and we will show that it is a smooth map from N to t. This submanifold N ⊂ M will in fact be of the form N = exp −1 (W S ) for a certain subset S ⊂ R + . We will show that exp(W S ) is the cell of exp(W) containing the points of µ(M ) ∩ exp(W) whose conjugacy class in U is of maximal possible dimension (we will see in lemma 4.20 that such a cell is well-defined and unique). This idea is inspired by the usual hamiltonian setting (see [HNP94] and also [LMTW98]). Studying points of µ(M ) whose conjugacy class is of maximal possible dimension is a natural thing to do as the set of such points is dense in µ(M ) and that we are ultimately interested in studying µ(M ). To begin with, let us define: Recall that Σ q is a submanifold of U (see proposition 4.15). Define then: so that µ(M q ) is exactly the set of points of µ(M ) whose conjugacy class in U is of maximal possible dimension. The first thing to observe is that M q is an open, connected, and dense subset of M . Let us first show that it is open. For any x ∈ M q , there exists an open neighbourhood V of µ(x) in U such that for all u ∈ V, dim U.u ≥ dim U.µ(x). Since µ is continuous, we have µ(y) ∈ V for all y in some open set U of M containing x. Since dim U.µ(x) is maximal, we necessarily have dim U.µ(y) = q for all y ∈ U, so that U ⊂ M q . We now want to prove that M q is dense and connected. To that end, we introduce the set M reg of points of M whose orbit under U is of maximal possible dimension: Indeed, since µ is a momentum map, one has Im T x µ ≃ u ⊥ x (see [AMM98]), and therefore: max In particular, µ is of constant rank on M reg . Now, to show that M q is dense and connected in M , since we have M reg,q ⊂ M q ⊂ M , it is enough to prove that M reg,q is dense and connected in M . First, we observe that this is locally true in the following sense: Proof. Since µ is of constant rank on M reg , there exists an open connected neighbourhood V x of x in M reg such that µ(V x ) is a (connected) submanifold of U (of dimension equal to rk T x µ) and such that µ| Vx : V x → µ(V x ) is a locally trivial submersion onto a connected manifold with connected fibres (the constant rank theorem says that µ is locally equivalent to the linear projection (x 1 , . . . , x n ) → (x 1 , . . . , x r , 0, . . . , 0)). Since µ is equivariant and continuous, we have u.µ(y) ∈ µ(V x ) for y ∈ V x and u sufficiently close to 1 in U . Therefore, µ(V x ) is a union of connected open subsets of conjugacy classes of U . Since in addition to that µ(V x ) is connected, we have, if we set (observe that q x is not necessarily equal to dim U.µ(x)) and that Ω x is an open, connected, and dense subset of µ(V x ). Now, since Ω x is an open dense and connected subset of µ(V x ) and since µ| Vx : V x → µ(V x ) is a locally trivial submersion with connected fibres over the connected manifold µ(V x ), we have that (µ| Vx ) −1 (Ω x ) = µ −1 (Ω x ) ∩ V x is an open dense and connected subset of V x (recall that the submersion µ| Vx is equivalent to (x 1 , . . . , x n ) →  (x 1 , . . . , x r , 0, . . . , 0)). Moreover, if x, y ∈ M reg , we can join them by a path c : is open, connected, and dense in V z . By compactness, we can coverĉ by a finite number of such V z :ĉ = V z1 ∪ . . . ∪ V zp with z 1 = x and z p = y. If V zi ∩ V zj = ∅, then by density and openness, R zi ∩ R zj = ∅. Therefore, for w ∈ R zi ∩ R zj , the conjugacy class of µ(w) has dimension q zi = q zj , whence we get q x = q y , so that q x is the same for all x ∈ M reg . As M reg ∩ M q = ∅, one necessarily has q x = q for all x ∈ M reg . Therefore µ −1 (Ω x ) ∩ V x = M reg,q ∩ V x , which proves the lemma.
We now go back to our global study: so that x is the limit of a sequence of points of M reg,q , which proves that M reg ⊂ M reg,q . Let us now prove that M reg,q is connected. Take x, y ∈ M reg,q . As M reg is connected, there exists a path c : [0, 1] → M reg joining x to y in M reg , and we setĉ := c([0, 1]). Then, as in the proof of lemma 4.16, there exists a finite open coverĉ = V z1 ∪ . . . ∪ V zp with z 1 = x and z p = y. Then V z1 ∩ V zi = ∅ for some i ≥ 2 and by density one has M reg,q ∩ (V z1 ∩ V zi ) = ∅. By connectedness of V z1 ∩ M reg,q , any z ′ i ∈ M reg,q ∩ (V z1 ∩ V zi ) can be joined to z 1 by a path in M reg,q . Likewise, this z ′ i can be joined to z i by a path in M reg,q , therefore z 1 can be joined to z i by a path in M reg.q . Repeating this, we obtain a path from z 1 = x to z p = y in M reg,q . Finally, we have proved that M reg,q is connected in M , and we have M reg,q ⊂ M q ⊂ M = M reg,q , which proves that M q is connected in M .
Instead of describing all of µ(M q ), what we are really interested in is describing µ(M q ) ∩ exp(W) ⊂ Σ q ∩ exp(W). Recall that W is a convex polyhedron of t, which can de described entirely in terms of roots of (U, T ) (see 4.2.1). Moreover, we know from proposition 4.15 that the intersection of Σ q with exp(W) is a finite disjoint union of submanifolds of U : (where U S is the stabilizer of any element in exp(W S ), see definition 4.14), so that we have: and we now want to study points in each exp(W S (i) ) which lie in the image of µ. To that end, we set, for all i ∈ {1, . . . , m}: By definition we have M S (i) ⊂ M q , and since M q is U -stable, we have u.x ∈ M q for all x ∈ M S (i) and all u ∈ U . Proof. Recall that exp(W S (i) ) is a submanifold of Σ q and that for all u ∈ exp(W S (i) ), one has (see proposition 4.15): Moreover, M S (i) = µ −1 (exp(W S (i) )), where µ is seen as a map µ : M q → Σ q . Hence for all x ∈ M S (i) : But: where the equality follows from the equivariance of µ), so that, for all x ∈ M S (i) : which means that the map µ is transverse to the submanifold exp(W S (i) ). By the transversality theorem, M S (i) = µ −1 (exp(W S (i) )) is therefore a submanifold of M q , hence of M , and one has, for all x ∈ M S (i) : Consequently, for all v ∈ T x M = T x M q , one has: Then one has T x µ.v 2 = ξ 2 ∈ T µ(x) (exp(W S (i) )) and therefore: And therefore: One may observe that this sum is generally not a direct sum, since ker T x for X satisfying Ad µ(x).X = X, see [AMM98]). Equality , the other inclusion following from the facts that M S (i) ⊂ M q and that M q is U -invariant. Consider an element x ∈ M q . Then µ(x) ∈ Σ q . But there exists u ∈ U such that u.µ(x) ∈ exp(W), so that µ(u.  From now on, we simply denote S (i0) by S. The submanifold M S := µ −1 (exp(W S )) will end up being our symplectic slice. We first prove the following result: (2) S and u 1 , u 2 ∈ U such that u 1 .x 1 = u 2 .x 2 , hence u −1 2 u 1 .x 1 = x 2 . But then, by lemma 4.22, u −1 2 u 1 ∈ U S , which is connected by proposition 4.12. Therefore, there is a path (u t ) joining 1 to u −1 2 u 1 in U S , hence u t .x 1 is a path joining x 1 to x 2 in M S (observe that U S acts on M S = µ −1 (exp(W S )), as µ is an equivariant map), which contradicts the fact that We now want to study precisely the relation between µ(M S ) and µ(M ) ∩ exp(W), which was our initial motivation. Recall that µ(M S ) ⊂ exp(W S ) ⊂ exp(W), the latter being closed in U . Proof. Take µ(x) ∈ µ(M ) ∩ exp(W). Since M q = U.M S is dense in M by corollary 4.21, there exist a sequence (x j ) j∈N of elements of M q and a sequence (u j ) j∈N of elements of U such that x = lim x j and u j .x j ∈ M S . Since U is compact, we may assume that (u j ) is convergent and denote its limit by u := lim u j . Then: In particular, u.µ(x) ∈ exp(W), so that u.µ(x) = µ(x), since exp(W) is a fundamental domain. Observe that lemma 4.24 is a consequence of corollary 4.21 and of the fact that µ(M S ) ⊂ exp(W). This last point also means that under the identification U/Int(U ) ≃ W, the map is simply µ = exp −1 •µ| MS . As a matter of fact, it follows from the definition of M S that µ(M S ) lies in the submanifold exp(W S ) of U , which is diffeomorphic to W S under exp −1 , so that µ = exp −1 •µ| MS is a smooth map from M S to t: We now compute the differential of µ, which is defined to be the composed map d µ := pr • T µ of the tangent map T µ : T M S → T t ≃ t × t and the projection pr : T t ≃ t × t → t onto the second factor.
Lemma 4.25. The differential d µ of µ is equal to the t-valued 1-form µ * θ on M S , where θ is the Maurer-Cartan 1-form on T , that is, the t-valued 1-form defined for t ∈ T and ξ ∈ T t T by Proof. Recall that the tangent map to the exponential map exp : u → U is given, for all X ∈ u and all ξ ∈ T X u = X + u, by: where 1−e −ad X ad X is the endomorphism of u given, for all ζ ∈ u, by: and where exp(X).ζ denotes the effect on tangent vectors ζ ∈ u = T 1 U of the left translation of element exp(X) in U . In the present case, we have to consider exp : t → T with T abelian, since for x ∈ M S , we have µ(x) ∈ exp(W S ) ⊂ T , so that: 1 − e −ad X ad X .ζ = ζ as (ad X) k−1 .ζ = 0 as soon as k − 1 ≥ 1. Then, for all X ∈ t and all ξ ∈ T X t = X + t: Therefore, for all x ∈ M S and all v ∈ T x M S , we have: So that: We may now prove theorem 4.8: Proof of theorem 4.8 (Existence of a symplectic slice). Let us set N := M S , where M S = µ −1 (exp(W S )) is the submanifold of M constructed above.
(i) Lemma 4.23 shows that N is connected. (ii) Since N = M S = µ −1 (exp(W S )) with µ equivariant, and since the conjugacy action of T on exp(W S ) is trivial (as T is abelian), we have that N is T -stable.
(iii) Let us show that ω| MS is a symplectic form. We denote by i the inclusion map i : But µ • i = µ| MS is T -valued and χ| T = 0 as T is abelian. Therefore, d(i * ω) = 0. Second, let us show that i * ω is non-degenerate. Take x ∈ M S and v ∈ T x M S such that for all w ∈ T x M S , ω x (v, w) = 0. In particular, v ∈ (T x M S ) ⊥ω ⊂ T x M . But we know from lemma 4.18 (see (1)) that: Hence: And therefore: Since (M, ω, µ : M → U ) is a quasi-hamiltonian space, we have (see [AMM98]): Take now X ∈ u such that v = X # x . Then, by equivariance of µ: (where X † denotes the fundamental vector field associated to X ∈ u by the conjugacy action of U on itself, and where the last equality follows from proposition 4.15). Hence v ∈ ker T x µ. But by equality (3), one has ker T x µ ⊂ (T x (U.x)) ⊥ω , therefore: And we know from proposition 4.18 (see (2)) that: Therefore v ∈ (T x M ) ⊥ω = ker ω x . Then v ∈ ker T x µ ∩ ker ω x , which is always equal to {0} (see [AMM98]). (iv) Let us now show that the action of T on M S is hamiltonian with momentum map µ = exp −1 •µ : M S → t. Denore by θ L = u −1 .du and θ R = du.u −1 the left and right Maurer-Cartan 1-forms of U . Take X ∈ t. Since (M, ω, µ : M → U ) is a quasi-hamiltonian space, we have: Therefore, for all x ∈ M S and all v ∈ T x M S = (T x µ) −1 (T µ(x) exp(W S )): But T x µ.v ∈ T µ(x) exp(W S ) with exp(W S ) ⊂ T , so that, since T is abelian: where θ is the Maurer-Cartan 1-form of T . Hence: where the last equality follows from lemma 4.25. Denote by ( µ | X) the function: We then have: Therefore, for all X ∈ t: That is: the hamiltonian vector field associated to the function ( µ | X) is the fundamental vector field X # , which shows that the action of T on M S is hamiltonian.

4.3.
Momentum convexity for quasi-hamiltonian actions. In the previous subsection, we gave a detailed explicit construction of a symplectic slice in an arbitrary quasi-hamiltonian U -space (M, ω, µ : M → U ), with U simply connected. We did so because, in the next subsection, having a symplectic slice of the form M S = exp −1 (W S ) where W S is a cell of the closed Weyl alcove W will be crucial to our proof of the real convexity theorem 3.3. Incidentally, it enables us to give a proof of the convexity theorem 2.4. This proof is a quasi-hamiltonian analogue of the proof of the Kirwan theorem given in [HNP94]. It rests on the following lemma: Proof. By theorem 4.1, the map µ| MS gives rise to local convexity data (V x , C µ(x) ) x∈MS . But we cannot conclude immediately that µ(M S ) is convex because µ| MS is not a proper map in general (as M S = µ −1 (exp(W S )) is in general not closed in M ). Instead, we proceed as in [HNP94]: write the convex set W S as an increasing sequence of closed locally polyhedral convex subsets (D n ) n∈N . Then: Therefore, proposition 4.6 applies to the closed sets Y n := µ −1 (exp(D n )) and µ| Yn gives rise to local convexity data (V x , C µ(x) ∩R + .(exp(D n )\{µ(x)})) x∈Yn . Additionally, since Y n is closed in M S , µ| Yn is a proper map. Since M S is connected and is an increasing union of closed subsets M S = ∪ n∈N Y n , we can find an ascending sequence (Z n ) n∈N of connected components of the (Y n ) n∈N such that M S = ∪ n∈N Z n . Each Z n is closed in Y n , so that µ| Zn is a proper map which gives rise to local convexity data (V x , C µ(x) ∩ R + .(exp(D n )\{µ(x)})) x∈Zn . Therefore, by theorem 4.3, µ(Z n ) is a convex polytope. We then have that µ(M S ) is an increasing union µ(M S ) = ∪ n∈N µ(Z n ) of convex subpolytopes of exp(W) ≃ W, which implies that it is a convex polytope.
We can now prove theorem 2.4: We can now give a proof of theorem 3.3. The idea is to apply Duistermaat's theorem 3.1 to the symplectic slice M S = exp −1 (W S ) ⊂ M constructed in subsection 4.2. This will not work directly for, as we already noted in subsection 4.3, the momentum map µ| MS is not proper in general. But this idea will guide in the following. To apply Duistermaat's theorem, we need to verify the following: (1) the involution β leaves the symplectic slice M S globally stable.
(2) the involution β S := β| MS is anti-symplectic and compatible with the torus action on M S and with the momentum map of this action.
(3) the fixed-point set of β| MS is non-empty. The proof of these three facts relies very strongly, as we shall see, on the following two assumptions: (H1) the symmetric pair (U, τ ) is of maximal rank. In particular, there exists a closed Weyl alcove satisfying exp(W) ⊂ F ix(τ − ). (H2) the symplectic slice M S ⊂ M is the pre-image of a face of such a closed Weyl alcove. Observe that the construction of the symplectic slice M S = µ −1 (exp(W S )) depends on the choice of the Weyl alcove W ⊂ t, hence on the choice of a maximal torus T . From now on, we will always choose such a maximal torus T ⊂ U to satisfy T ⊂ F ix(τ − ). In particular, τ (t) = t −1 for all t ∈ T , so that for all x ∈ M S and all t ∈ T one has β(t. , which means that β| MS is compatible with the action of T on M S and with the momentum map of this action in the sense of theorem 3.1. The fact that β| MS is anti-symplectic is immediate since β * ω = −ω on M and the symplectic form on M S is ω| MS . Thus, the only thing left to verify is that F ix(β| MS ) = ∅. Recall that we assumed, in the statement of theorem 3.3, that As a matter of fact, since here the compact group U is assumed to be simply connected, the group U τ is necessarily connected (see for instance [Loo69]), but this is not very important here. Recall from subsection 4.2 that q = max x∈M {dim U.µ(x)} and that M q = {x ∈ M | dim U.µ(x) = q}. One then has: q ′ . Further, the momentum map µ is of constant rank on the set Λ ∩ L (K) q ′ , since all U -orbits of points in this set have the same dimension and since rk T x µ = dim u ⊥ x = dim U.x (see [AMM98]). We can then proceed exactly as in lemmas 4.16 and 4.17 and obtain that Λ ∩ L (K) q ′ , and therefore L (K) q ′ , is dense in L. We will conclude the proof by showing that L (K) q ′ = L ∩ M q . Take x ∈ L. Then µ(x) ∈ Q 0 . Since the symmetric pair (U, τ ) is of maximal rank, exp(W) is a fundamental domain for the action of K on Q 0 . This implies that if w ∈ Q 0 , then dim K.w is maximal if and only if dim U.w is maximal. Indeed, dim K.w = dim K − dim (U S ′ ) τ , where S ′ is the uniquely defined set such that K.w ∩ exp(W S ′ ) = ∅ and U S ′ is the stabilizer of any point in exp(W S ′ ), and dim U.w = dim U − dim U S ′ , for the same S ′ since U.w ⊃ K.w and the fundamental domains of the U action on U and the K-action on Q 0 are the same. Therefore, here, dim K.µ(x) = q ′ if and only if dim U.µ(x) = q, which means that This concludes the proof of the proposition, as we have seen that L Lemma 4.27 means that the subset of points of L whose image is regular in µ(M ) is (non-empty and) dense in L. A similar result was established earlier in lemma 4.17 for points of M whose image is regular. Observe that the fact that (U, τ ) is of maximal rank is crucial: in the terminology of appendix B of [OS00], the root system of (U, T ) and the system of restricted roots of (U, τ ) are the same (recall that we assumed T ⊂ F ix(τ − )) and therefore dim K.w is maximal if and only if dim U.w is maximal, for any w ∈ Q 0 . Let us now carry on with the proof of theorem 3.3.
Lemma 4.28. One has: M β S = ∅ and L q = K.M β S . Proof. Lemma 4.27 shows that L q = ∅. Take now x ∈ L q . Then µ(x) ∈ Q 0 . Since the symmetric pair (U, τ ) is of maximal rank, exp(W) is a fundamental domain for the action of K on Q 0 , hence the existence of a k in K such that kµ(x)k −1 ∈ exp(W) ∩ µ(M q ) = exp(W S ), where the last equality follows from lemma 4.20. Hence (k.x) ∈ µ −1 (exp(W S )) = M S . Moreover, β(k.x) = τ (k).β(x) = k.x, hence (k.x) ∈ M βS S , which is therefore non-empty, and we have indeed: Recall that µ(M βS S ) ⊂ exp(W) ⊂ T ⊂ Q 0 by construction of the symplectic slice M S . Since K is connected and M βS S is not connected in general, we observe that L q is not connected in general (this could be seen from the proof of lemma 4.27: the set Λ constructed there had no reason to be connected in general). We can now prove an analogue of lemma 4.24 (or equivalently, of statement (v) of proposition 4.8): Lemma 4.29. If µ is a closed map (in particular, if µ is proper), one has: µ(M βS S ) = µ(L) ∩ exp(W).
Thus, µ(M βS S ) is almost the whole of µ(L) ∩ exp(W). This is interesting because we can now relate µ(M βS S ) to µ(M S ) (the latter being dense in µ(M ) ∩ exp(W) by lemma 4.24) in the following way, which is essentially Duistermaat's statement applied to the symplectic slice M S : Proposition 4.30. Assume that µ : M → U is a proper map. Then, in the above notations: µ(M βS S ) = µ(M S ). As in the proof of lemma 4.26, we cannot apply theorem 3.1 directly to the symplectic slice M S , for µ| MS is in general not proper. But we may work with the ascending sequence (Z n ) n∈N introduced in the proof of lemma 4.26: W S is an ascending union of closed convex subsets W S = ∪ n∈N D n and M S = µ −1 (exp(W S )) is an ascending union of closed connected sets Z n ⊂ µ −1 (D n ). By proposition 4.6, the map µ| Zn is a proper map which gives rise to local convexity data (V x , C e µ(x) ) x∈Zn and, by theorem 4.3, the set µ(Z n ) is convex. We then observe the following fact: Proof. First, observe that such an n ∈ N always exists since M S = ∪ n∈N Z n and M βS S = ∅ by lemma 4.28. Second, observe that W is closed in Z n . As µ| Zn is a closed map, µ(W ) is a closed connected subset of t. Take then x ∈ W . It follows from point (iii) of the local convexity theorem 4.1 that µ(x) ) x∈Zn is the local convexity data of the map µ| Zn . Further, µ(W ) is contained in the convex set µ(Z n ), therefore in where the last equality follows from theorem 4.3. Hence: so that µ(W ) is convex by corollary 4.4. In fact, corollary 4.4 also shows that for all x ∈ W : We may now prove proposition 4.30: Proof of proposition 4.30. Consider n ∈ N such that Z n ∩ M βS S = ∅ and let W be a connected component of Z n ∩ M βS S . Then we know from lemma 4.31 that µ(W ) ⊂ µ(Z n ) is an inclusion between two convex sets of a finite-dimensional vector space. Additionally, by comparing (4) and (5), we obtain: Therefore, lemma 4.5 applies and we get µ(W ) = µ(Z n ). Consequently, µ(Z n ∩ M βS S ) = µ(Z n ). Since M S = ∪ n∈N Z n , one has M βS S = ∪ n∈N (Z n ∩ M βS S ) and therefore: Since µ(M S ) is contained in exp(W) and µ = exp −1 •µ| MS , the above equality is equivalent to µ(M S ) = µ(M βS S ).
We can now prove the real convexity theorem 3.3: Proof of the real convexity theorem 3.3. Since µ is a proper map, one may apply successively lemma 4.29, proposition 4.30 and lemma 4.24 and obtain: where L is any connected component of M β satisfying µ(L) ∩ Q 0 = ∅.
As an immediate corollary of the above real convexity theorem, we point out the following result, which we will use later in applications (see section 5). Proof. Assume that µ −1 ({t}) = ∅. Then t ∈ µ(M ) ∩ exp(W) = µ(M β ) ∩ exp(W) by theorem 3.3, so that there exists y ∈ M β satisfying µ(y) = t. The converse implication is obvious.
Observe that, by equivariance of µ, the above statement is still true for t ∈ K. exp(W) = Q 0 . It can obviously not be true for any t ∈ U since one necessarily has µ(M β ) ⊂ F ix(τ − ).

Applications
In this final section, we give some applications of the real convexity theorem 3.3. The first application is the construction of lagrangian submanifolds in quasi-hamiltonian quotients. The quasihamiltonian quotient associated to a quasi-hamiltonian space (M, ω, µ : M → U ) is the set M//U := µ −1 ({1})/U . Alekseev, Malkin and Meinrenken showed in [AMM98] that if the action of U on µ −1 ({1}) is free then M//U is a symplectic manifold (and that if the action is locally free then M//U is a symplectic orbifold). To make the presentation simpler, we will always assume in the following that M//U is a smooth manifold. To obtain a lagrangian submanifold of M//U , one may for instance construct an anti-symplectic involution σ on M//U . Then, if F ix(σ) is non-empty it is a lagrangian submanifold of M//U . To obtain such a σ, one may start from an involution β on the quasi-hamiltonian space M . Proof. Assume first that U is simply connected. The involution β satisfies the assumptions of theorem 3.3. Consequently, by corollary 4.32, one has µ −1 ({1})∩F ix(β) = ∅, which implies F ix(β) = ∅. When U is semi-simple, we refer to [Sch07a] to see how to reduce this situation to the case of the universal cover U of U (which is still compact as U is assumed to be semi-simple).
There is nonetheless somehing to be careful about in this approach: usually the representation space Hom C (π g,l , U )/U is not a manifold. But it is a stratified space in the sense of Lerman and Sjamaar in [LS91]. In particular, it is a disjoint union of symplectic manifolds called strata. Then, as a generalization of proposition 5.1, a compatible form-reversing involution β on M = (U × U ) g × C 1 × · · · × C l induces an involution F ix(β) whose fixed-point set is non-empty (since proposition 5.1 showed that we had in fact the stronger statement µ −1 ({1}) ∩ F ix(β) = ∅) and is a union of lagrangian submanifolds of some of the strata. We refer to [Sch07a] for a proof of this result and we now focus on the construction of such an involution β on (U × U ) g × C 1 × · · · × C l . This construction relies on the notion of decomposable representation of π g,l into U . A representation of π g,l into U can be thought of as a (2g + l)-tuple (a 1 , b 1 , ... , a g , b g , c 1 , ... , c l ) ∈ (U × U ) g × C 1 × · · · × C l satisfying g i=1 [a i , b i ] l j=1 c j = 1. To keep the exposition simple, we restrict ourselves here to the case where g = 0, but the notion of decomposable representation, and therefore theorem 5.3, are valid for arbitrary g ≥ 0 (see [Sch07a]). When U = U (n), the notion of decomposable representation is due to Falbel and Wentworth (see [FW06], and see [Sch07b] and [Sch07a] for arbitrary U and arbitrary g ≥ 0). In the g = 0 case and for arbitrary U , the representation (c 1 , ... , c l ) ∈ C 1 × · · · × C l is said to be decomposable if there exist elements w 1 , ..., w l ∈ F ix(τ − ) ⊂ U satisfying c 1 = w 1 w −1 2 , ... , c l = w l w −1 1 . To characterize such decomposable representations, the idea is to introduce an involution β on M = C 1 × · · · × C l (see [Sch07b] for an explanation on the origin of this idea). This involution is: Definition 5.2.
When U is simply connected, the proof of theorem 5.3 is an immediate application of the real convexity theorem. We refer to [Sch07a] for the case where the compact connected Lie group U is not simply connected. We now end this paper with an application to linear algebra of the real convexity theorem 3.3: Theorem 5.4. [Sch07a] Consider λ 1 , ... , λ l ∈ R n . Then the following statements are equivalent: (i) There exist l special unitary matrices u , ... , u l ∈ SU (n) such that: Spec u j = exp(iλ j ) and u l ...u l = 1.
Proof. Let C j be the conjugacy class of exp(iλ j ) ∈ SU (n). Assume first that A 1 , ... , A l ∈ SU (n) satisfy A t j A j ∈ C j and A 1 ...A l = 1. Set u l = A t l A l , u l−1 = A t l (A t l−1 A l−1 )(A t l ) −1 , ... , and u 1 =