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Coadjoint orbitopes

We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of a compact Lie group. We show that all the faces of such an orbitope are exposed. The face structure is studied by means of the momentum map and it is shown that every face is again a coadjoint orbitope. Up to conjugation the faces are completely determined by the momentum polytope and can be described in a simple way in terms of root data. Finally we consider the complex geometry of the coadjoint orbit and we prove that the submanifolds of the orbit that are extreme sets of a face are exactly the closed orbits of parabolic subgroups.


Introduction
Let K be a compact Lie group and let K → GL(V ) be a finite-dimensional representation. An orbitope is by definition the convex envelope of an orbit of K in V (see [23]). An interesting class of orbitopes is given by the convex envelope of coadjoint orbits. We call these coadjoint orbitopes. The case of an integral orbit has been studied in [6], where it was realised that a remarkable construction introduced by Bourguignon, Li and Yau [8] in the case of complex projective space can be generalized to arbitrary flag manifolds. This allowed to show that the convex envelope of an integral coadjoint orbit is equivariantly homeomorphic to a Satake-Furstenberg compactification. This homeomorphism is constructed by integrating the momentum map, but unfortunately it is not explicit and its nature is not yet wellunderstood. On the other hand, the Satake-Furstenberg compactifications admit a very precise combinatorial description going back to Satake [24].
The first purpose of this paper is to give a precise description of the face structure of coadjoint orbitopes without the integrality assumption and without relying on the connection with Satake-Furstenberg compactifications. In the second place we establish the combinatorial description of the orbitope in terms of the notions put forward by Satake to describe his compactifications. This description is established directly, without need to mention the compactifications, and is valid for all orbits integral or not.
We describe the contents of the paper. In §1 we recall some elementary facts from convex geometry. In §2 we fix the notation and recall some well-known results.
In §3 we study the face structure of coadjoint orbitopes using the symplectic properties of coadjoint orbits and the structure theory of compact Lie groups. Let O denote the coadjoint orbit and let O be the convex hull of O, i.e. the coadjoint orbitope. To describe a face F of O it is enough to describe the set ext F ⊂ O that consists of the extreme points of F . Therefore it is important to study the subsets O ′ ⊂ O of the form O ′ = ext F , i.e. those subsets O ′ ⊂ O such that the convex hull of O ′ is a face of O. It turns out that for exposed faces ext F is a submanifold and it is best studied in terms of standard properties of the momentum map. The main results are the following. In the first place every face of a coadjoint orbitope is a coadjoint orbitope itself (Theorem 23). In particular faces of coadjoint orbitopes are orbitopes (see also Remark 26). In the second place, a coadjoint orbitope has only exposed faces (Theorem 36). This answers Question 1 of [23] for this particular class of orbitopes. These two properties are quite remarkable and show that coadjoint orbitopes are quite special among general orbitopes.
If we fix a maximal torus T of K, then the orthogonal projection of O onto t coincides with the intersection P = O ∩ t and is given as the convex hull of a Weyl group orbit or alternatively as the image of the momentum map with respect to the action of T on O (this is Kostant theorem). We show in §4 that the faces of the orbitope O are completely determined by the faces of the momentum polytope P . In fact up to conjugation the faces of O are in one-to-one correspondence with the faces of the momentum polytope. In particular there are a finite number of faces up to conjugation. This is applied in §5 where it is shown that the face structure yields a natural stratification of the boundary of a coadjoint orbitope. The strata are smooth fibre bundles over flag manifolds naturally associated with the faces.
In §6 we give a description of the faces of the coadjoint orbitope in terms of root data. We use the formalism of x-connected subset of simple roots developed by Satake [24]. This gives a combinatorial way to generate all the faces of the orbitope.
In §7 we completely characterize the submanifolds O ′ ⊂ O of the form O ′ = ext F . The characterization is best expressed using the complex structure of the coadjoint orbit. The main result asserts that the submanifolds of the form O ′ = ext F are exactly the closed orbits of parabolic subgroups of G = K C .
In the last section we prove that if O is an integral orbit, the faces of O are also orbitopes of integral orbits. This is equivalent to saying that if O comes from irreducible representation, also its faces are associated to irreducible representations.

Preliminaries from convex geometry
It is useful to recall a few definitions and results regarding convex sets (see e.g. [25]). Let V be a real vector space and E ⊂ V a convex subset. The relative interior of E, denoted relint E, is the interior of E in its affine hull. A face F of E is a convex subset F ⊂ E with the following property: if x, y ∈ E and relint[x, y] ∩ F = ∅, then [x, y] ⊂ F . The extreme points of E are the points x ∈ E such that {x} is a face. If E is compact the faces are closed [25, p. 62]. If F is a face of E we say that relint F is an open face of E. A face distinct from E and ∅ will be called a proper face.
Assume for simplicity that a scalar product , is fixed on V and that E ⊂ V is a compact convex subset with nonempty interior. Definition 1. The support function of E is the function If u = 0, the hyperplane H(E, u) := {x ∈ E : x, u = h E (u)} is called the supporting hyperplane of E for u. The set is a face and it is called the exposed face of E defined by u or also the support set of E for u.
In using the notation F u (E) we will tacitly assume that the affine span of E is V . Hence by definition an exposed face is proper. We notice that in general not all faces of a convex subsets are exposed. A simple example is given by the convex hull of a closed disc and a point outside the disc: the resulting convex set is the union of the disc and a triangle. The two vertices of the triangle that lie on the boundary of the disc are non-exposed 0-faces.
Proof. It is immediate that F ∩ ext E ⊂ ext F . The converse follows from the definition of a face. Lemma 4. If G is a compact group, V is a representation space of G and G · x is an orbit of G, then conv(G · x) contains a fixed point of G. Moreover any fixed point contained in conv(G · x) lies in relint conv(G · x).
Proof. Just pick x ∈ O and setx where dg is the Haar measure. Thenx is G-invariant and belongs O. Now let y be any fixed point of G that lies in O. By Theorem 6 there is a unique face F ⊂ O such that y belongs to relint F . Since O is G-invariant and y is fixed by K, it follows that a · F = F for any a ∈ G. So F is G-invariant, and hence also ext F is G-invariant. Since ext F ⊂ ext(conv(G · x)) ⊂ G · x, it follows that ext F = G · x and hence that F = G · x.
Proposition 5. If F ⊂ E is an exposed face, the set Proof. Let u 1 , u 2 ∈ C F and λ 1 , λ 2 ≥ 0 and set u = λ 1 u 1 + λ 2 u 2 . We need to prove that if at least one of λ 1 , λ 2 is strictly positive, then F = F u (E). Assume for example that . This proves the first fact. To prove the second, pick any vector u ∈ C F and apply the previous lemma to the orbit G · u ⊂ C F : this yields a G-invariantū ∈ C F . Theorem 6 ([25, p. 62]). If E is a compact convex set and F 1 , F 2 are distinct faces of E then relint F 1 ∩ relint F 2 = ∅. If G is a nonempty convex subset of E which is open in its affine hull, then G ⊂ relint F for some face F of E. Therefore E is the disjoint union of its open faces. Lemma 7. If E is a compact convex set and F E is a face, then dim F < dim E.
Proof. If dim F = dim E, then relint F is open in the affine span of E, so relint F ⊂ relint E. By the previous theorem this implies that F = E. Lemma 8. If E is a compact convex set and F ⊂ E is a face, then there is a chain of faces which is maximal, in the sense that for any i there is no face of E strictly contained between F i−1 and F i .
Proof. If F = E there is nothing to prove. Otherwise put F 0 := F . If there is no face strictly contained between F 0 and E, just set F 1 = E. Otherwise we find a chain F 0 F 1 F 2 = E. If this is not maximal, we can refine it. Repeting this step we get a chain with k + 1 elements. Since dim F i−1 < dim F i , k ≤ n. Therefore the chain gotten after at most n steps is maximal.

Coadjoint orbits
Throught the paper we will use the following notation. K denotes a compact connected semisimple Lie group with Lie algebra k. If T ⊂ K is a maximal torus and Π ⊂ ∆(k C , t C ) is a set of simple roots, the Weyl chamber of t corresponding to Π is defined by B is the Killing form of k C and , = −B| k×k is a scalar product on k. By means of , we identify k with k * . Lemma 10. Let T ⊂ K be a maximal torus, let ∆ be the root system of (k C , t C ) and let Π ⊂ ∆ be a base. Define H α ∈ t C by the formula B(H α , ·) = α(·) and choose a nonzero vector X α ∈ g α for any α ∈ ∆. For α ∈ ∆ + set Then it is possible to choose the vectors X α in such a way that [X α , X −α ] = H α and so that the set {u α , v α |α ∈ ∆ + } be orthonormal with respect to For a proof see e.g. [18, pp. 353-354]. Set Then If O is an adjoint orbit of K and x ∈ O, then Since we identify k ∼ = k * we may regard O as a coadjoint orbit. As such it is equipped with a K-invariant symplectic form ω, named after Kostant, Kirillov and Souriau, and defined by the following rule. For u, v ∈ k See e.g. [17, p. 5]. ω is a K-invariant symplectic form on O and the inclusion O ֒→ k is the moment map. If T ⊂ K is a maximal torus, we denote by W (K, T ) or simply by W the Weyl group of (K, T ). We let π : k → t denote the orthogonal projection with respect to the scalar product , = −B. Its restriction to O is denoted by Φ T : O → t; it is the momentum map for the T -action on O. P := Φ T (O) is the momentum polytope. The following convexity theorem of Kostant [20] is the basic ingredient in the whole theory.
Theorem 12 (Kostant). Let K be a compact connected Lie group, let T ⊂ K be a maximal torus and let O be a coadjoint orbit. Then P is a convex polytope, ext P = O ∩ t and ext P is a unique W -orbit.
There is a unique K-invariant complex structure J on O such that ω be a Kähler form. It can be described as follows (see [16, p. 113] for more information). Fix a maximal torus T and a system of positive roots in such that a way x belongs to the closure of the positive Weyl chamber. Then the complex structure on T x O is given by the formula Set G = K C . The action of K on O extends to an action G × O → O which is holomorphic. If v O denotes the fundamental vector field induced by v ∈ g = k C , this implies that  Proof. Fix x ∈ O. Let M denote the affine hull of O in k and let V be the associated linear subspace, i.e. M = x + V . We claim that M contains the origin. Since O is Ad K-invariant, so are M and V . Hence V is an ideal and V ⊥ is an ideal as well. Write i.e. gx 1 = x 1 . This means that x 1 is a fixed point of the adjoint action. Since K is semisimple, x 1 = 0, x ∈ V and M = V as desired. Let K i , i = 1, . . . , r be the simple factors of K. Since V is an ideal, V = i∈I k i for some subset I of Let H be a compact connected Lie group (not necessarily semisimple) and let O ⊂ h be an orbit. There is a splitting of the algebra where z is the center of h and k i are simple ideals. Let K i be the closed connected subgroups of H with Lie K i = k i . So H = Z · K 1 · · · K r , where Z is the connected component of the identity in the center of H. Any two of these subgroups have finite intersection. We can reorder the factors in such a way that K i acts nontrivially on O if and only if 1 ≤ i ≤ q for some q between 1 and r. Set By construction there is a decomposition Any two factors in this decomposition have finite intersection.

Lemma 18.
For any x ∈ O, there is a unique decomposition x = x 0 + x 1 with x 0 ∈ z and x 1 ∈ l. Moreover the affine span of O is x 0 + l and x 0 belongs to relint O.
Proof. Write x = x 0 +x 1 +x 2 with x 0 ∈ z, x 1 ∈ l and x 2 ∈ l ′ . Since L ′ ·x = x, the component x 2 is fixed by L ′ . Since L ′ is semisimple, this forces x 2 = 0. It follows immediately that H · x = x 0 + L · x 1 . By definition all simple factors of L act nontrivially on L · x 1 , hence the orbit L · x 1 is full in l by The statement about the affine span is equivalent to L · x 1 being full in l. Therefore after possibily replacing K by L and translating by x 0 we can assume for most part of the paper that O is full.
We are interested in the facial structure of O and we start by considering the structure of exposed faces. We start the analysis of the face structure of O by looking at the esposed faces. At the end of the section we will prove that all faces are exposed.
Let u be a nonzero vector in k and let Φ u : O → R be the function Φ u (x) := x, u . Set Φ u is just the component of the moment map along u. Then for x ∈ O and u, v ∈ k This implies that x ∈ O is a critical point of Φ u if and only if x ∈ z k (u), i.e.
Lemma 20. Let H = Z K (u) be the centraliser of u in K and let F u ( O) be the exposed face of O defined by u. Then Proof. By Atiyah theorem [2] the level sets of Φ u are connected. In particular Lemma 21. Fix a maximal torus T ⊂ K, a nonzero vector u ∈ t and a point x ∈ O ∩ t. Then x ∈ Crit(Φ u ) and x is a maximum point of Φ u if and only if there is a Weyl chamber in t whose closure contains both x and u.
Proof. By assumption x ∈ t ⊂ z k (u) and z k (u) = Crit(Φ u ). To check the second assertion recall that Φ u is a Morse-Bott function with critical points of even index (this is Frankel theorem, see e.g. [ (One can prove the same formula much more generally and by a more geometric argument, see [15,Prop. 2.5].) Thus the quadratic form D 2 Φ u (x) is negative semidefinite if and only if the operator ad u • f −1 is positive semidefinite. This operator preserves each Z α and its restriction to Z α is just multiplication by α(u)/α(x). Hence it is positive semidefinite iff and only iff α(u)α(x) ≥ 0 for any α ∈ ∆. This is equivalent to the condition that x and u lie in the closure of some Weyl chamber (see e.g. [14, p. 11]).
The computation above goes back to [10] and to Heckman's thesis [14].
Then (a) S is a nontrivial torus and fixes F pointwise, Proof. Since u = 0 by the definition of exposed face, S is a nontrivial torus.
such that for any i there is no face strictly contained between F i−1 and F i . This is possible by Lemma 8. We will prove (a)-(c) by induction on k. If k = 1, then F is a maximal proper face. Since any face is contained in an exposed face, F is necessarily exposed. Thus (a)-(c) follow from the previous lemma. We proceed with the induction. Let k > 1 and assume that the theorem is proved for faces contained in a maximal chain of length k − 1. Fix F with a maximal chain as above of lenght k. By the inductive hypothesis the theorem holds for F 1 , so that there is a nontrivial subtorus S 1 ⊂ K which pointwise fixes F 1 . Moreover if we set H 1 = Z K (S 1 ) and h 1 = Lie H 1 = z k (s 1 ), then F 1 ⊂ h 1 and ext F 1 is an orbit of H 1 . In particular if we choose a point x ∈ ext F ⊂ ext F 1 , then ext (17) and write x = x 0 + x 1 as in Lemma 18, with x 0 ∈ z = z(h 1 ) and x 1 ∈ l, so that ext Since u ∈ l and l ⊂ h 1 , u commutes with s 1 . So S 1 and S 2 commute and generate a torus S. Set H := Z K (S). If g ∈ H, then g commutes with S 1 , This proves (b). Next observe that the previous lemma also ensures that F ′ ⊂ z l (u) and that z l (u) ⊂ h. Since x 0 ∈ h too, we conclude that F = F ′ + x 0 ⊂ h. This proves (c). By definition h = z k (s), so S fixes any point of h and in particular it fixes pointwise F . Thus (a) is proved.
We remark that the inductive argument used in the previous proof does not imply that all faces are exposed, since being an exposed face is not a transitive relation.
Proof. Let S and H be as in Theorem 23. Then ext F ⊂ O ∩ h is an H-orbit. The result follows directly from Lemma 19.

Proof. A maximal torus of H is also a maximal torus of K.
Remark 26. The above results shows that every face of O is a coadjoint orbitope for some subgroup H ⊂ K. One might wonder if a similar property holds for all orbitopes: if a group K acts linearly on V and O is an orbit, one might ask if every face of O is an orbit of some subgroup of K. The answer is negative in general. Counterexamples are provided e.g. by convex envelopes of orbits of S 1 acting linearly on R n . These are called Carathéodory orbitopes, since their study goes back to [9]. In [26] there is a thorough study of the 4-dimensional case (see also [4]). It turns out (see Theorem 1 in [26]) that there are many 1-dimensional faces whose extreme sets are not orbits of any subgroup of K. Therefore the fact that we just established, namely that the faces of a coadjoint orbitope are all orbitopes of the same kind, seems to be a rather remarkable property.
The subgroups S and H in Theorem 23 are not unique. Later in Theorem 36 (d) we will show that there is a canonical choice. Now we wish to show that one can always assume that S = Z(H) 0 .
Corollary 27. In Theorem 23 we can assume that Z(H) acts trivially on F and that S = Z(H) 0 .
Proof. Let p : k → h denote the orthogonal projection. H acts on O in a Hamiltonian way with momentum map p| O . If x ∈ ext F , then H · x = ext F is a symplectic orbit by Corollary 24. Therefore H x = H p(x) , see e.g. [13,Thm. 26.8,p. 196]. Since p(x) ∈ h, the stabilizer H p(x) contains the center of H. So Z(H) ⊂ H x . This proves the first statement. Next set S ′ = Z(H) 0 . Then S ′ is a positive dimensional torus. To prove the second fact it is enough to show that changing S to S ′ does not change the centralizer, i.e. that The following is an immediate consequence of Lemma 18.
Lemma 28. Let F be a face of O, H ⊂ K a connected subgroup and assume that ext F is an H-orbit and that F ⊂ h. Decompose H as in (17), i.e. L is the product of the simple factors of (H, H) that act nontrivially on F , while L ′ is the product of those factors that act trivially. If x ∈ ext F , then Now we fix a maximal torus T and we use the notation of p. 6. We wish to show that the T -stable faces of O and the faces of the momentum polytope are in bijective correspondence. This will be used to prove that all faces of O are exposed. The relation between the T -invariant faces of O and the faces of P will be studied further in the next section.
The following lemma is a consequence of Kostant convexity theorem. See [11,Lemma 7] for a proof in the context of polar representations.
Lemma 29. Let K be a compact connected Lie group, T ⊂ K be a maximal torus and let π : k → t be the orthogonal projection.
Lemma 30. Let T ⊂ K be a maximal torus and let F ⊂ O be a nonempty T -invariant face. Set σ := π(ext F ). Then σ = π(F ) = F ∩ t. Moreover σ is a nonempty face of the momentum polytope P .
Proof. We prove this lemma in the same way as Kostant theorem is deduced from the Atiyah-Guillemin-Sternberg theorem. By Corollary 24 ext F is a symplectic submanifold of O. T acts on ext F with momentum map given by the restriction of π to ext F . By definition σ = π(ext F ) is the momentum polytope for this action. By the Atiyah-Guillemin-Sternberg theorem This means first of all that σ is convex. Since π is linear it follows that π(F ) = conv π(ext F ) = σ. On the other hand, since π(ext F ∩t) = ext F ∩t, we get Thus the first part is proven. In particular we can apply this with F = O, and we get that P = O ∩t. That F ∩t is a face of P now follows directly from Lemma 9 without assuming that F be T -invariant. To check that σ = ∅, recall that if a torus acts on a compact Kähler in a Hamiltonian way, then it has some fixed points. So (ext F ) T = ext F ∩ t = ∅ and σ = ∅.
From these two inclusions we get that aff σ = x 0 + (l ∩ t). Therefore σ ⊥ is We already know (from Lemma 30) that σ is a nonempty face of P . By Theorem 23 z = {0}, so σ ⊥ = {0}, aff σ = t and σ P . This shows that σ is a proper face.
Corollary 34. Let F 1 , F 2 be a proper faces of O, let H 1 , H 2 be corresponding subgroups as in Theorem 23 and let T be a maximal torus of K which is contained in both H 1 and H Proof. Set σ := F i ∩ t. Recall from (31) that ext σ ⊂ ext F i and pick x ∈ ext σ. Then we can apply the previous lemma to both faces and we get ext The following is the main result of this section.
Theorem 36. All proper faces of O are exposed. More precisely, if F is a Proof. We start by proving (a) under the assumption that the maximal torus T is contained in some subgroup H that has the properties listed in Theorem 23. By Lemma 33 σ := F ∩ t = F ∩ P is a proper face of P . Since all faces of a polytope are exposed [25, p. 95], there is a vector u ∈ t such that σ equals the exposed face of P defined by u, i.e. σ = F u (P ). Since u ∈ t and P = π(O), h P (u) = max x∈O u, . It is also immediate that F ′ ∩ t = σ. So we have two faces F and F ′ with F ∩ t = F ′ ∩ t = σ. Set H ′ := Z K (u). By Lemma 20 ext F ′ = Max(Φ u ) is an H ′ -orbit and H ′ satisfies (a)-(d) of Theorem 23 for F ′ . Clearly T ⊂ H ′ since u ∈ t, and by hypothesis also T ⊂ H. We can therefore apply Corollary 34 and we get F = F ′ . In particular F = F u ( O) is an exposed face. We have thus proved (a) under the assumption that T ⊂ H for some H as in Theorem 23. Next we show that the vector u can be chosen inside z F . The subgroup H F ⊂ K is compact and preserves both O and F . By Proposition 5 there is a vector u ∈ C F that is fixed by H F . Note that H F is of maximal rank since H ⊂ H F . If T is a maximal torus contained in H F , then u is is fixed by T , so u ∈ t ⊂ h F . It follows that u ∈ h F and since H F fixes u it follows that u ∈ z F . Thus (b) is proved. To prove (c) assume that u ∈ z F and that by Lemma 20. Therefore Z K (u) preserves F and therefore Z K (u) ⊂ H F by definition. So H F = Z K (u) and (c) is proved. (d) follows from Lemma 22 and the fact that H F = Z K (u). Now we know that H F itself has the properties of Theorem 23. Hence (a) holds for any torus T ⊂ H F .
Remark 37. In general the faces of an orbitope are not necessarily exposed. For example 4-dimensional Carathéodory orbitopes have non-exposed faces, see [26, Thm. 1(5b)] (note that the author uses the word "facelet" for face and "face" for exposed face). It is important to understand whether an orbitope has only exposed faces. Indeed this is Question 1 in [23]. The previous theorem shows that this is always the case for coadjoint orbitopes.
Proof. Set F = conv(O ′ ). From the fact that O is contained in a sphere, it follows as in Lemma 14 that ext F = O ′ . Therefore the statement follows immediately from Lemma 20 and the fact that every face of O is exposed. This is a first characterization of the submanifolds that appear as ext F for some face F . In §7 we will see that this characterization becomes much more transparent using the complex structure of O. An explicit characterization in terms of root data will be given in §6.
Various results about the faces have been established using some subgroup H satisfying the properties stated in Theorem 23. Now we know that H F does satisfy these properties. Hence we can state those results more cleanly. This is done in Theorem 40 below. Next in Lemma 42 we will make precise the possible freedom in the choice of the group H. First of all decompose H F as in Lemma 28: Z F is defined in (35), K F is the product of the simple factors of (H F , H F ) that act nontrivially on ext F and K ′ F is the product of the remaining factors. Theorem 40. Let T ⊂ K be a maximal torus. a) If F ⊂ O is a proper T -invariant face, then σ := F ∩ t = π(F ) = π(ext F ) is a proper face of the momentum polytope P and ext F is a and To prove the converse, assume that σ 1 ⊂ σ 2 and pick x ∈ σ 1 . Then Proof. Necessarily F = ∅. Since ext F is an H-orbit, H preserves ext F , hence F . So H ⊂ H F by definition (35). To prove the opposite inclusion, split as usual H = Z · L · L ′ and write x = x 0 + x 1 as in Lemma 28. The orbit L · x 1 ⊂ l is full, so the affine span of F is x 0 + l. Since also H F has the properties stated in Theorem 23 we can repeat the same reasoning for H F instead of H. Thus we get that the affine span of F is x 0 + k F . Therefore l = k F . So L and K F are connected subgroups of K with the same Lie algebra and therefore coincide. This implies K F = L ⊂ H. Given a Hermitian matrix u = 0 we wish to study the face We can assume that u be tangent to H 1 , i.e. Tr u = 0. Let C n+1 = V 1 ⊕ · · · ⊕ V s be its eigenspace decomposition, i.e. u | V i = µ i Id V i . Since u = 0 and Tr u = 0 s > 1. We assume µ 1 < µ 2 < · · · < µ s . Let follows that this is the set of projectors onto lines that are contained in some of the V i 's, i.e. Crit(Φ u ) = P(V 1 ) ⊔ · · · ⊔ P(V s ). For the same reason Z SU(n+1) (u) = S(U(V 1 ) × · · · × U(V s )).
Let v i be a non zero vector of V i and let P v i denote the orthogonal projection onto the complex line Cv i . Then the maximum of Φ u is equal to µ s and it is attained exaclty on P(V s ). This means that . In particular in this example C F is much larger than z F ∩ C F . The above computation shows that to each face corresponds a subspace, namely V s . Viceversa, given a subspace W ⊂ C n+1 , let w be the Hermitian operator such that Therefore faces of O ′ are in one-to-one correspondence with the subspaces of C n+1 .

The role of the momentum polytope
Consider a full orbit O ⊂ k, a maximal torus T ⊂ K and the momentum polytope P . In this section we will study in detail the relation between the faces of O and those of P . Denote by F ( O) the set of proper faces of O and by F (P ) the proper faces of the polytope P . If F is a face of O and a ∈ K, then a · F is still a face, so K acts on F ( O). Similarly W = W (K, T ) acts on F (P ). We wish to show that F ( O)/K ∼ = F (P )/W . Lemma 44. If F is a face of O, there is a T -stable face F ′ which is conjugate to F , i.e. F ′ = a · F for some a ∈ K. F ′ is unique up to conjugation by elements of N K (T ).
Proof. By Corollary 25 F is preserved by some maximal torus S ⊂ K. There is a ∈ K such that S = a −1 T a. Hence F ′ = a · F is preserved by T . To prove uniqueness assume that F 1 and F 2 be T -stable faces of O and that F 2 = a·F 1 for some a ∈ K. Then H F 2 = aH F 1 a −1 . In particular both T and aT a −1 are contained in H F 2 , so there is b ∈ H F 2 , such that aT a −1 = bT b −1 .
. Denote by W F and W ′ F the Weyl groups of (K F , K F ∩ T ) and (K ′ F , K ′ F ∩ T ) respectively. W F and W ′ F can be considered as subgroups of W . They commute and have the following sets of invariant vectors:

Lemma 45. Let T ⊂ K be a maximal torus and let F be a nonempty
Proof. Recall that ext F = x 0 +K F ·x 1 . By Kostant theorem σ = π(ext F ) = π(x 0 + K F · x 1 ) = x 0 + conv(W F · x 1 ) = conv(W F · x). Hence W F preserves σ. Moreover σ ⊂ z F ⊕ (t ∩ k F ) hence W ′ F fixes σ pointwise and (i) follows. Similarly, since σ ⊂ z F ⊕ k F , Z F · K ′ F fixes σ pointwise. Therefore H F · σ = K F · σ. By Lemma 29 K F · (σ − x 0 ) is convex and the same is true of On the other hand σ ⊂ F and F is H F -invariant, so also H F · σ ⊂ F . This establishes (ii).
If σ is a face of P set Lemma 46. If σ ∈ F (P ) there is a vector u ∈ t that is fixed by G σ and such that σ = F u (P ). If u is any such vector and F := F u ( O), then F ∩ t = σ, G σ = W F × W ′ F , z F = t Gσ and F does not depend on u but only on σ. Proof. The existence of u follows directly from Lemma 5. By Lemma 45 (ii) W F × W ′ F ⊂ G σ , so u ∈ t W F ×W ′ F = z F and using Theorem 36 it follows that H F = C K (u). Therefore the subgroup of W that fixes u is the Weyl group of (H F , T ) i.e. W F × W ′ F . It follows that W F × W ′ F = G σ . From this it follows that z F = t Gσ , that H F = C K (z F ) = C K (t Gσ ) and in particular that H F and hence ext F and F only depend on σ. Proof. Let σ be a face of P . Choose u ∈ t Gσ such that σ = F u (P ). Then . So ϕ • ψ is the identity. It follows immediately from Theorem 40 (c) that ϕ is injective. Hence it is a bijection and ψ = ϕ −1 .

Smooth stratification
As we saw in the previous section the group K acts on F ( O), which is the set of faces of O and this action has a finite number of orbits, which are in one-to-one correspondence with the orbits of the Weyl group on the finite set F (P ). Let B denote one of the orbits of K on F ( O). We call B a face type. The set There is an obvious map p : S B → B which maps a point x ∈ S B to the unique face F such that x ∈ relint F . To study S B it is expedient to fix an element F ∈ B. Thus B = {g · F : g ∈ K} ∼ = K/H F and K → K/H F is a right principal bundle with structure group H F . Let be the associated bundle gotten from the action of H F on relint F . Note that E F → K/H F is a homogeneous bundle in the sense that the left action of K on K/H F lifts to an action of K on E F that is given by the following rule is the point in the associated bundle.) Proposition 49. Let B be a face type and let F ∈ B be a representative.
Define a map Then f is a smooth K-equivariant embedding of E F into k with image S B . Therefore S B is a smooth embedded submanifold of k. Moreover p : S B → B is a smooth fibre bundle.
Proof. It is straightforward to check that f is well-defined, smooth and equivariant. It is also clear that f (E F ) = S B . We proceed by showing that f is injective. Recall from Theorem 6 that if F 1 and F 2 are different faces, This shows that f is injective. Next we show that f is an immersion. Denote by V the fibre of E F over the origin of K/H F . Since E F is a homogeneous bundle and f is equivariant, it is enough to show injectivity of df p at points p ∈ V , i.e. at points of the form p = [e, x], x ∈ relint F . At such points Indeed T p V is the vertical space, while U is the tangent space at p of a local section of K → K/H F . The injectivity of df p will follow from the following three facts: x ∈ relint(g · F ) ∩ relint F , so g · F = F by Theorem 6 and g ∈ H F . Therefore K x ⊂ H F and k x ⊂ h F , as claimed. Now let u be an element of U . By definition there is v ∈ k such that Then

(The bracket on right is the Lie bracket in
. Thus (c) is proved and f is an immersion. In order to prove that it is an embedding we shall prove that f is proper as a map f : E F → S B = f (E F ). Let {y n } be a sequence in S B converging to some point y ∈ S B . Set [g n , x n ] := f −1 (y n ). We wish to show that {[g n , x n ]} admits a convergent subsequence. Since K is compact by extracting a subsequence we can assume that g n → g. Then y n = f ([g n , x n ]) = g n · x n . Therefore x n = g −1 n · y n → x := g −1 · y. Since y ∈ S B , y ∈ relint(g −1 F ) and x ∈ relint F . Therefore [g n , x n ] → [g, x] as desired.
Lemma 51. If B is the face type of F , then We introduce a partial order on the face types, as follows: B 1 B 2 if for some (and hence for any) choice of representatives F i ∈ B i there is some g ∈ K such that gF 1 ⊂ F 2 . This is a partial order. We write Proof of Theorem 48. We already know that the strata are smooth embedded submanifold of k. In particular they are locally closed subsets both of k and of O. By Prop. 49 Since any face F is the disjoint union of all proper faces contained in F To conclude we need to show that dim S C < dim S B if C ≺ B. Fix representatives F ∈ B and G ∈ C such that G F . By the previous lemma it is enough to show that dim In the case of equality, we would get Z F · K ′ F = Z G · K ′ G , so Z F = Z G , H F = H G and hence ext F = ext G and F = G.
Example 52. We shall describe the strata of the orbitope O ′ studied in Example 43. We saw there that the faces of O ′ are in one-to-one correspondence with subspaces of C n+1 . Two subspaces are interchanged by an element of SU(n + 1) if and only if they have the same dimension. So the orbit types are indexed by the dimension. Let W ⊂ C n+1 be a subspace of dimension k, let F = conv(P(W )) be the corresponding face and let B be the orbit type of F . Then Therefore B is simply the Grassmannian G(k, n + 1). Since relint F = {A ∈ F : rank A = k}, it follows that In fact this is a bundle over the Grassmannian of k-planes. Finally, notice that H F acts on relint F simply by the adjoint action of SU(W ).

Satake combinatorics of the faces
In this section we describe the faces of O and the faces of the momentum polytope in terms of root data. The description uses the notion of x-connected subset of simple roots, which was introduced in [24]. In that paper Satake introduced certain compactifications of a symmetric space of noncompact type (the Satake-Furstenberg compactifications). The notion of x-connected subset was used in the study of the boundary components of these compactifications. It is no coincidence that faces of O and boundary components admit a description in terms of the same combinatorial data: in fact it was shown in [6] that the Satake compactifications of the symmetric space K C /K are homeomorphic to convex hulls of integral coadjoint orbit of K. Here we do not use the link with the compactifications. Instead we show directly how to construct all the faces of O (up to conjugation) starting from the root data. This is accomplished for a general coadjoint orbit with no integrality assumption.
Fix a maximal torus T of K and a system of simple roots Π ⊂ ∆ = ∆(k C , t C ). As usual we identify k C with its dual using the Killing form B. The roots get identified with elements of it.
Definition 53. A subset E ⊂ it is connected if there is no pair of disjoint subsets D, C ⊂ E such that D ⊔ C = E, and x, y = 0 for any x ∈ D and for any y ∈ C.
(A thorough discussion of connected subsets can be found in [22, §5].) Connected components are defined as usual. For example the connected components of Π are the subsets corresponding to the simple roots of the simple ideals in k.
Definition 54. If x is a nonzero vector of t, a subset I ⊂ Π is called xconnected if I ∪ {ix} is connected.
Equivalently I ⊂ Π is x-connected if and only if every connected component of I contains at least one root α such that α(x) = 0.
Definition 55. If I ⊂ Π is x-connected, denote by I ′ the collection of all simple roots orthogonal to {ix} ∪ I. The set J := I ∪ I ′ is called the xsaturation of I.
The largest x-connected subset contained in J is I. So J is determined by I and I is determined by J. Given a subset E ⊂ Π we will use the following notation: We denote by T E , H E , K E the corresponding connected subgroups. Note that H E is the subgroup associated to the subset E ⊂ Π, while H F is the subset associated to the face F ⊂ O. This should cause no confusion.
Lemma 56. Let O be a full coadjoint orbit and let F ⊂ O be a proper face. Assume that u ∈ C F and that v ∈ there is some s ∈ (0, 1) such that α(u s ) = 0. Since u s ∈ C F and α(v) = 0, this would contradict (a).
Denote by C + the positive Weyl chamber associated to Π. The following is immediate and well-known.
Theorem 58. Let O be a full coadjoint orbit and let x be the unique point in O ∩ C + . a) If I ⊂ Π is x-connected and J is its x-saturation, then is a face of O. If u ∈ t J and −iα(u) > 0 for any α ∈ Π − J, then b) Given an arbitrary subset E ⊂ Π, denote by I the largest x-connected subset contained in E and by J the x-saturation of I. Then H E · x = H I · x = H J · x. c) Any face of O is conjugate to one of the faces constructed in (a).
More precisely, given a face F and a maximal torus T ⊂ H F there are a base Π ⊂ ∆(k C , t C ) and a subset I ⊂ Π with the following properties: (i) if C + is the positive Weyl chamber corresponding to Π, We claim that x ∈ F . Indeed x and u belong to C + , so by Lemma 21 x is a maximum point of Φ u , i.e. x ∈ ext F . By Lemma 20 ext F = Z K (u) · x, so F = conv(H J · x). This proves that conv(H J · x) is indeed a face of O. By Lemma 42 K F ⊂ H J = Z K (u) ⊂ H F and K F = K I . Pick v ∈ C F ∩ z F (this exists by Theorem 36). By Lemma 56 −iα(v) ≥ 0 for every α ∈ ∆ + , i.e. v ∈ C + . By Theorem 36 (c) and Lemma 57 F . This entails on the one hand that [Z α , k I ] = 0, i.e. α ⊥ I; on the other hand that Z α fixes x, i.e. α(x) = 0. This means in fact that α ∈ I ′ . Hence E = J as claimed and (59) follow. (b) Split E in connected components: E = E 1 ⊔ · · · ⊔ E r . We can assume that E j is x-connected iff j ≤ q for some q between 1 and r. Then . Then ext F = Max(Φ u ), so there is a Weyl chamber C + such that x, u ∈ C + . Let Π be the base corresponding to C + . By Theorem 36 (c) H F = Z K (u). Since u ∈ C + , Lemma 57 says that H F = H E with E = {α ∈ Π : α(u) = 0}. Let I and J be as in (b). Then I is x-connected and using (b) we get ext F = H F · x = H E · x = H J · x. Thus F = conv(H J · x) as desired.
Remark 60. In the proof of (c) we have in fact that E = J. Indeed from (a) H F = H J , so H E = H J i.e. E = J.
Example 61. Let K = SU(n + 1), n ≥ 4, and let x ∈ su(n + 1) be the diagonal matrix x = diag(i(n − 1), i(n − 1), −2i, . . . , −2i). The coadjoint orbit through x is the momentum image of the Grassmannian G(2, n + 1). Let t be the set of the diagonal matrices and denote by Π = {α 1 , . . . , α n } the standard set of simple roots, i.e. α i (diag(x 1 , . . . , The vector x lies in the closure of the positive Weyl chamber containg x and α i (x) = 0 if and only if i = 2. Therefore the x-connected subsets of Π are the following: is the image of the embedding is the complex Grassmannian G(2, k + 1). The stratum corresponding to F 1 k is a fiber bundle over SU(n + 1)/S(U(k + 1) × U(n − k)) = G(k + 1, n + 1) .
is the image of the embedding is a complex projective space P k−1 (C). The strata corresponding to F 2 k is a fiber bundle over the flag manifold SU(n + 1)/S(U(1) × U(k) × U(n − k)).

Complex geometry of the faces
Then p E = h C E ⊕ u E is a parabolic subalgebra. Denote by U E and P E the corresponding connected subgroups of G. Then P E is a parabolic subgroup, U E is its unipotent radical and H C E is a Levi factor. In particular As O ′ is compact, the compact form H E must be transitive on O ′ . This concludes the proof. (d) ⇒ (a) First observe that O ′ = H E · x is a complex submanifold since it is a connected component of the fixed point set of the torus T E . Therefore H C E preserves O ′ . By assumption there is x ∈ C + ∩ O ′ . By Lemma 13 the stabilizer G x contains the negative Borel subgroup, so U E fixes x. If Therefore P E preserves O ′ which is therefore a compact P E -orbit.
We notice that in condition (d) the set E can be chosen to be the xsaturation of the maximal x-connected subset I ⊂ E as shown in Theorem 58 (c).
The above result establishes a one-to-one correspondence between two rather distant classes of objects: on the one side the faces of the orbitope O, on the other side the closed orbits of parabolic subgroups of G inside O. To illustrate this correspondence recall the following fact.
Lemma 64. If P ⊂ G is a parabolic subgroup, in O there is only one orbit of P which is closed.
Proof. Since the action is algebraic and O is a compact manifold, there is at least one orbit which is closed. Let O ′ ⊂ O be a closed P -orbit and let B ⊂ P be a Borel subgroup. Then O ′ is B-invariant, so it contains a closed B-orbit. But the B-orbits in O are just the Schubert cells and the only one which is closed is the fixed point of B. Hence any closed P -orbit contains this fixed point and this implies that the closed P -orbit is unique.
The above uniqueness statement can also be considered from the point of view of the orbitope, as can be seen from the proof of the implication (c) ⇒ (d) in the previous theorem. Indeed, if P is a parabolic subgroup, we write it as P = P E for some E ⊂ Π. Then there is a unique orbit of H E that is of the form ext F , namely the orbit H E · x for x ∈ O ∩ C + . Alternatively this orbit can be described as follows: choose u ∈ t E = z(h E ) such that −iα(u) > 0 for α ∈ E. Then the closed P -orbit is Max(Φ u ). In a sense to fix a parabolic subgroup P E is equivalent to fixing H E and the vector u.
Proof. f (G) is just the orbit of G through the point p = [O ′ ] ∈ H, while f (K) is the orbit of K through p. The subgroup P defined in (63) is just the stabilizer G p . Therefore f (G) ∼ = G/P . It follows immediately that the three conditions are equivalent to P being parabolic, so they are implied by (c). Conversely, if they are satisfied, P is parabolic. By assumption O ′ is an orbit of some subgroup L ⊂ K. Then L ⊂ P and O ′ is a P -orbit, thus (c) holds.
This condition is not enough either. In fact Crit(Φ u ) will contain at least two orbits, one for the maximum and one for the minimum. These are "good" orbits, in the sense that they correspond to orbits, namely to F u ( O) and F −u ( O) respectively. The orbits in between in general do not generate faces. Consider the following example. Let O ⊂ su(3) be the momentum image of the flag manifold of pairs (L 1 , L 2 ) where L 1 ⊂ L 2 ⊂ C 3 and dim L i = i. Let u = i diag(1, 1, , −2). Set V = C 2 × {0}. Then Crit(Φ u ) has the following three connected components: Each component is an orbit of Z K (u) = S (U (4) × U (1)). Let P i denote the stabilizer of C i for the action of G = SL(3, C). Then P 2 = {g ∈ SL(5, C) : g(V ) = V } and P 3 = {g : ge 5 = e 5 }. These two subgroups are parabolic. So C 2 and C 3 correspond to faces, by Prop. 62. On the other hand we claim that P 1 is the subgroup of SL(5, C) of matrices of the form g = A 0 0 λ A ∈ SL(2, C), λ ∈ C * .
Since v ∈ V − {0} is arbitrary it follows that w = 0. The claim is proved, hence P 1 is not parabolic and conv(C 1 ) is not a face of O. Another way to express integrality of O is the following. Fix a maximal torus T ⊂ K and choose a point x ∈ O ∩ t. Recall that a linear functional λ ∈ (it) * is an algebraically integral weight if λ, α |α| 2 = λ(H α ) |H α | 2 ∈ Z for any root α ∈ ∆(k C , t C ), see e.g. [18, p. 265]. Then O is integral if and only if λ = ix, · is an algebraically integral weight. (For all this see [17,Chapter 1] or [19].)

The case of an integral orbit
Theorem 68. Let O ⊂ k be an integral coadjoint orbit and let F be a face of O. Write ext F = x 0 + K F · x 1 as in (41). Denote by , F the scalar product on k F induced by the Killing form of k F . Define x ′ 1 ∈ k F by the following rule: for all y ∈ k F . Then K F · x ′ 1 is an integral coadjoint orbit in k F . Proof. This fact can be proved in a variety of ways using the various characterizations of integrality. One simple way is using the definition, i.e. the condition on the integrality of the Kostant-Kirillov-Souriau form. Let ω F be the KSS form of K F · x ′ 1 ⊂ k F . Let µ ∈ k * F be the functional µ(y) = x 1 , y = x ′ 1 , y F . The stabilizers (for the adjoint action) of x 1 and x ′ 1 are the same, because both coincide with the stabilizer of µ (for the coadjoint action). Moreover the stabilizers in K F of x and of x 1 coincide since x = x 0 + x 1 and x 0 is fixed by K F . Summing up we get that the stabilizers in K F of x ′ 1 and x coincide. Hence the map j : K F · x ′ 1 ֒→ k g · x ′ 1 → j(g · x ′ 1 ) := g · x is an embedding of K F ·x ′ 1 onto ext F = K F ·x ⊂ k. We claim that j * ω = ω F . By equivariance it is enough to check that j * ω = ω F at x ′ 1 . Take X, Y ∈ k F and set u = [X, Then and similarly dj x ′ . This proves that indeed ω F = j * ω and thus [ω F ]/2π is integral if [ω]/2π is.
Remark 70. Since the various definitions of integrality are equivalent, this theorem ensures that if ix, · is an integral weight, then ix 1 , · F is integral as well. Since integral weights give rise to representations, to each face F of an integral coadjoint orbitope is attached an irreducible representation of K F . If one fixes root data and F is the face corresponding to an x-connected subset I ⊂ Π as in §6, then the representation corresponding to F is the representaion V I originally described by Satake [24, p. 89] (see also [7, p. 67]. Remark 71. If O is an integral orbit, then O is the momentum image of a flag manifold M provided with an invariant Hodge metric lying in a polarization L → M . The space H 0 (M, L) is an irreducible representation τ of K. Out of these data one can construct a Satake-Furstenberg compactification X τ S of the symmetric space K C /K and it is possible to define a homeomorphism (named after Bourguignon-Li-Yau) between this compactification and the orbitope O. This was accomplished in [6]. Since this homeomorphism respects the boundary structure, some properties of the faces of O can be deduced in this way. The arguments in the present paper apply also to the non-integral case, give much more information and are more direct and geometric, since no use is made of the Bourguignon-Li-Yau map.