K¨ahlerian structures of coadjoint orbits of semisimple Lie groups and their orbihedra
DISSERTATION
zur Erlangung des Doktorgrades der Naturwissenschaften an der Fakult¨atf¨urMathematik der Ruhr-Universit¨atBochum
vorgelegt von Patrick Bj¨ornVilla aus Bochum im Januar 2015 Contents
List of Figures ii
Introduction iii
1 Preliminaries 1 1.1 Convex geometry ...... 1 1.1.1 Polyhedra, Polytopes and Cones ...... 4 1.2 Complexification of irreducible real representations ...... 8 1.3 Almost complex structures ...... 10 1.3.1 Integrability ...... 10 1.3.2 Almost complex structures on homogeneous spaces ...... 10 1.4 Compatible subgroups ...... 12 1.5 Compact Centralizer ...... 13 1.6 Compact Cartan subalgebras ...... 15 1.6.1 Root decomposition ...... 15 1.6.2 Quasihermitian Lie algebras ...... 22 1.6.3 Hermitian Lie algebras ...... 24
2 Coadjoint orbits as K¨ahlermanifolds 26 2.1 Invariant almost complex structures ...... 27 2.1.1 The special case H = T ...... 27 2.1.2 The case T ⊂ H ⊂ K ...... 28 2.2 Integrable invariant almost complex structures ...... 29 2.3 Parabolics q− and Q− ...... 33 2.4 The K¨ahler structure ...... 35 2.5 Open embedding ...... 39
3 Coadjoint orbihedra 42 3.1 Momentum map ...... 42 3.1.1 Connected Fibers ...... 42 3.2 Face structure ...... 46 3.2.1 Faces as orbihedra ...... 46 3.2.2 All faces are exposed ...... 53 3.3 Correspondences of faces ...... 60 3.3.1 The correspondence of the faces of the polyhedron and the orbihedron 60 3.3.2 Complex geometry related to the faces ...... 63
i List of Figures
1 Nonexposed faces of a convex set E ...... 2 2 Root systems ...... 17 + 3 Choices of ∆p ...... 17 4 The maximal and minimal cone ...... 19 ∼ 5 The adjoint SL(2, R) = Sp(2, R)-orbits ...... 21 6 Sketch of the connecting curve c ...... 45 7 The orbit O and its projection ...... 56 8 The projection in the Sp(4, R)-case ...... 57 9 The projection in the SU(2, 1)-case ...... 57 10 Faces of the polyhedron up to equivalency ...... 62 11 Sketch of the general Sp(4, R)-orbihedron and its faces ...... 63 12 One-dimensional projections PG and PU ...... 69 13 Two-dimensional projections PG and PU ...... 70 14 Three-dimensional projections PG and PU ...... 71
ii Introduction
In 1923, Schur ([Schu]) proved that the diagonal entries of a Hermitian n × n-matrix with eigenvalues λ = (λ1, . . . , λn) are contained in the convex hull of
Sn · λ = {(λi1 , . . . , λin )|ij ∈ {1, . . . , n}, ij 6= ik for j 6= k} ,
n where Sn denotes the symmetric group acting on C by permuting the coordinates and is in a more general context to be known as the Weyl group. In 1954, Horn ([Hor]) proved that each point in the convex hull of Sn ·λ is the diagonal of a Hermitian matrix whose eigenvalues are λ. Kostant ([Ko]) generalized in 1973 the result of Schur and Horn to arbitrary compact Lie groups. In particular he formulated their result on Hermitian matrices as a property of an adjoint orbit O of the unitary group. More precisely, he proved that for an element x ∈ t, where t is a maximal abelian subspace in a compact Lie algebra k, one has a convex polytope given by
µ(Ad K · x) = conv(W · x),
where the group K is the connected analytic Lie group associated to k, the map µ : k → t is the orthogonal projection with respect to the Killing form and W is the Weyl group associated to (kC, tC). In 1982 Atiyah ([At]) and Guillemin and Sternberg ([GS]) gave independently a far reaching interpretation of Kostant’s result in terms of images of a momentum map with respect to a Hamiltonian torus action. A momentum map for a symplectic action of a Lie group G on a symplectic manifold (X, ω) is given by a G-equivariant map µ : X → g∗ from X to the dual of the Lie algebra g such that
ξ dµ = ιξX ω holds for all ξ ∈ g. Here the function µξ : X → R with µξ(x) := hµ(x), ξi is the component X of µ along ξ and ιξX ω is the contraction of the symplectic form ω with the vector field ξ on X induced by ξ ∈ g. If such a map exists the action of G on X is said to be Hamiltonian. Momentum maps admit considerable convexity properties of its images which have been studied extensively. The general result of Atiyah and Guillemin and Sternberg is that the set µ(X) is a convex polytope in t∗ where X is a compact connected symplectic manifold with a Hamiltonian torus action of a compact torus and µ : X → t∗ is the momentum map. Furthermore its vertices are given by images of T -fixed points in X. This symplectic convexity theorem was proven applying Morse theory on components of the momentum map. Furthermore they used the necessary fact that the symplectic manifold is compact. In the case of noncompact symplectic manifolds with a Hamiltonian torus action one can show similar results under the assumption that the momentum map is proper. In 1994, Hilgert, Neeb and Plank ([HNP]) used this to prove a convexity theorem for noncompact coadjoint orbits.
iii In this case one needs a Cartan subalgebra t ⊂ k in the Lie algebra g, where k is a maximal ∗ ∗ compact subalgebra in g. For any element α ∈ t the projection µ : Oα → t is proper if the convex envelope Obα := conv(Oα) contains no affine lines. Then one has
P := µ(Oα) = conv Wk · α + Cmin,
where Wk := NK (t)/ZK (t) is the Weyl group of t with respect to k and Cmin is a closed convex cone which is computed by the roots with respect to (gC, tC). In the same year Prato [Pr] proved a similar result using the properness of components of µ rather the properness of µ. These results show that the image P of the momentum map is a polyhedron, which generalizes the theorem in the compact setting In this work we are interested in the geometry of the convex hull of coadjoint orbits and its relation to the geometry of their momentum polyhedra P . The momentum map will play a major role in describing the geometry of the convex hull of coadjoint orbits. We will consider coadjoint orbits O of a reductive Lie group G such that its Lie algebra is quasihermitian and their momentum map µ is proper. Their convex hulls Ob := conv(O) will be studied extensively by means of µ. We will see that this convex body has a very rich boundary structure. The class of quasihermitian Lie algebras generalizes the class of compact Lie algebras in a natural way. In the case where G is a reductive Lie group its Lie algebra g is quasihermitian if and only if every simple ideal in g is either compact or hermitian. These types of Lie algebras admit a compact Cartan subalgebra t which is an essential ingredient in this work. Considering special elliptic coadjoint orbits (see Section 1.6.2) with a proper momentum map we prove the following theorem.
Theorem (3.35). All proper faces of Ob are exposed.
This theorem shows that every face of the orbihedron Ob is given by a supporting hyperplane. Moreover we will study the faces by means of the momentum map and we will show that every face has the same structure as Ob. More precisely the extremal points of any face are given as an orbit of a reductive and quasihermitian subgroup of G. This theorem generalizes the result of Biliotti, Ghigi and Heinzner in [BGH, Theorem 38] for compact Lie groups and their orbitopes, i.e. compact orbihedra. In order to prove this theorem one has to study components of the momentum map and their maxima. It becomes apparent that this property is more difficult in the noncompact setting than in the compact setting since not every continuous function on a noncompact set has a maximum. Using the result of [HNP] and [Pr] the image P of the t-momentum map of the orbit O is a polyhedron, which has only exposed proper faces. We will verify a strong connection between the faces of the orbihedron and its polyhedron. For this let F(P ) denote the set of faces of the polyhedron P and similarly let F(Ob) denote the set of faces of Ob. The Weyl group Wk acts on the set of faces F(P ) and the Lie group G acts on F(Ob). With respect to the actions on the set of faces we get the following theorem on the quotient spaces.
iv Theorem (3.37). The set of equivalence classes of faces F(P )/Wk and F(Ob)/G are in bijective correspondence.
This theorem draws a connection between the faces of the orbihedron Ob and the faces of its momentum image P . More precisely up to conjugation every face of Ob is completely determined by the faces of its momentum polyhedron P . In the special case where G is a compact Lie group, we obtain the result of Biliotti, Ghigi and Heinzner in [BGH, Theorem 49]. The coadjoint orbits we consider admit another property. In Chapter 2 we will study invariant almost complex structures to a high extent. In particular we will investigate which coadjoint orbits admit a homogeneous complex structure J such that their natu- ral symplectic structure ω is a K¨ahlerform. These complex structures relate to specific parabolic subalgebras in gC using the results of Fr¨olicher ([F]). Then following similar ideas of Bordemann, Forger and R¨omer([BFR]) any complex structure J relates to a closed set + C C of roots ∆J ⊂ ∆(g , t ) which corresponds to Weyl chambers in a subalgebra of t. + If the coadjoint G-orbit is K¨ahlerianthen the set of roots ∆J contains a set of positive + noncompact roots ∆p which is invariant under the Weyl group W . These sets of roots play a major role in describing the coadjoint orbits which are K¨ahlerian.In the case where G is semisimple, a coadjoint G-orbit may be identified with an adjoint G-orbit O through a point x ∈ t. Then the orbit O admits a K¨ahlerstructure (O, ω, J) if and only if there exist + + positive, Wk-invariant noncompact roots ∆p such that −iα(x) is positive for any α ∈ ∆p . It should be underlined that the complex structure J is the unique complex structure such that the orbit O can be embedded holomorphically into a generalized flag manifolds of the compact form U. This generalizes the case where the coadjoint orbit is a hermitian symmetric space and thus can be embedded into its compact dual (see [He, Proposition 7.14]). Moreover the K¨ahlerianorbits have a proper momentum map such that the previous theorems hold. In the compact case, i.e. G = K, every vector in g defines an exposed face to the orbitope and one can find a well defined representative by averaging over a compact group. Moreover, the complexification KC of the compact group acts on O as well. A further result in [BGH] is the correspondence of the faces to closed orbits of parabolic subgroups in O. These facts do not hold in the case where the Lie group G is noncompact. In particular in the noncompact setting the complexification does not act on O. Hence one can not expect to have a similar result. Nevertheless one may use the above open embedding of the orbit OG of G into an orbit OU of the compact form U. Using the complex structure J of Chapter 2 we may embed OG holomorphically into OU . Since the natural symplectic forms ωG and ωU are quite different due to the G- resp. U-invariance the induced momentum maps and their images also differ. Nonetheless the face structure of the momentum polyhedron PG and the momentum polytope PU are quite similar. Moreover we may describe the exposed faces of ObG and ObU in relation to each other with components of their momentum maps. The last goal of this monograph is to draw a connection between the faces of ObG and ObU . This is described in the following theorem.
v Theorem (3.46). There is a one to one correspondence between the set of faces of ObG and ObU with respect to the corresponding actions of G and U. Hence we get the isomorphisms on the quotients: ∼ ∼ ∼ F(ObG)/G = F(PG)/Wk = F(PU )/Wu = F(ObU )/U.
In this theorem the isomorphisms on the left and right hand side are given by the previous theorem or the respective result in the compact case. The isomorphism in the middle exposes an entirely new connection between the orbits OG and OU which shows that they behave similarly with respect to their convex hulls and their face structure.
Acknowledgements
First of all, I would like to thank my supervisor Prof. Dr. Peter Heinzner for introducing me to the research field, for many helpful and encouraging discussions during the last years and for critically reading the manuscript. Next I would like to thank my friends and members of the chair of complex analysis in Bochum, in particular Fabian Heising, Hannah Bergner, Dr. Valdemar Tsanov and Lisa Knauss. From each of them I have learned a lot. Moreover I thank Prof. Dr. Leonardo Biliotti and Dr. Alessandro Ghigi for several fruitful discussions while they were here in Bochum or I was visiting them in Parma. I am very grateful for the support of the Sonderforschungsbereich SFB/TR 12 of the Deutsche Forschungsgemeinschaft and the Faculty of Mathematics at Bochum. Last but not least I would like to thank my family for their invaluable help and all those who supported me personally while I worked on this thesis.
vi 1 Preliminaries
In this chapter we will review the most important facts and basic notations concerning the notions of convex geometry, almost complex structures and Lie groups which admit a compact Cartan subalgebra t in their Lie algebra. Most of the results in convex geometry are taken from [Schn]. I will assume the reader is familiar with the concepts of general Lie theory. Most of these facts are taken from [Kn] and [He]. For more details concerning Lie algebras admitting compact Cartan subalgebras the reader is referred to [Ne].
1.1 Convex geometry First we will recall some definitions and results of convex sets.Most of which can be found in [Schn]. In this section we denote by V a real vector space equipped with a fixed scalar product h , i and by E ⊂ V a convex set, i.e. a set E such that for any two points x, y ∈ E the closed segment [x, y] := {(1 − t)x + ty|0 ≤ t ≤ 1} is also contained in E. The relative interior of E, denoted relint(E), is the interior of E in its affine hull.
Definition 1.1 (Faces). A face F of E is a convex subset of E with the following property: if x, y ∈ E and relint([x, y]) ∩ F 6= ∅, then [x, y] ⊂ F . The extreme points of E are the points x ∈ E such that {x} is a face. The set of all extreme points of E is denoted by ext(E). An extreme ray of E is a ray that is a face of E. By extr(E) we denote the union of the extreme rays of E. A face distinct from E and ∅ will be called a proper face.
If E is a nonempty, closed and convex set then the faces are closed [Schn, p. 62]. For the description of a certain type of faces one needs to discuss hyperplanes in the vector space V .A hyperplane of V can be written in the form
Hu,α := {x ∈ V |hx, ui = α} with u ∈ V \{0} and α ∈ R. We say that u is a normal vector of Hu,α. The hyperplane Hu,α bounds the two closed halfspaces
− Hu,α := {x ∈ V |hx, ui ≤ α} ,
+ Hu,α := {x ∈ V |hx, ui ≥ α} .
Definition 1.2 (Hyperplanes/exposed faces). The hyperplane Hu,α supports E at x if − x ∈ E ∩ Hu,α and E ⊂ Hu,α. The hyperplane Hu,α is called the supporting hyperplane of E for u. If Hu,α is a supporting hyperplane then the set
Fu(E) := E ∩ Hu,α
is a face and is called the exposed face of E defined by u. Extreme points of E which are also exposed will be called exposed points and are denoted by exp E.
1 Figure 1: Nonexposed faces of a convex set E
Example 1.3. Figure 1 shows a convex set E which has faces that are not exposed. The set E is the convex hull of a circle and a square. The two intersection points are extreme points of E but are not exposed.
Lemma 1.4 ([BGH, Lemma 5]). If F is a face of a convex set E, then ext(F ) = F ∩ext(E).
Proposition 1.5 ([BGH, Proposition 7]). If F is an exposed face of a convex set E, the set CF := {u ∈ V : F = Fu(E)} is a convex cone. If G is a compact subgroup of O(V ) that preserves both E and F , then CF contains a fixed point of G.
Theorem 1.6 ([Schn, Theorem 2.1.2.]). If E is a nonempty closed convex set and F1,F2 are distinct faces of E then relint(F1)∩relint(F2) = ∅. To each nonemtpy relatively open convex subset A of E there is a unique face F of E with A ⊂ relint(F ). Therefore E is the disjoint union of the relative interiors of its faces.
The following two Lemmas taken from [BGH] refer to compact convex sets but the previous theorem only uses that E is nonempty and closed. Therefore these lemmas, which proofs base on the last theorem, extend to the case of closed convex sets.
Lemma 1.7 ([BGH, Lemma 9]). If E is a nonempty closed convex set and F ( E is a face, then dim F < dim E.
Lemma 1.8 ([BGH, Lemma10]). If E is a nonempty closed convex set and F ⊂ E is a face, then there is a chain of faces
F0 = F ( F1 ( ... ( Fk = E which is maximal, in the sense that for any i there is no face of E strictly contained between Fi−1 and Fi.
Lemma 1.9 ([BGH, Lemma 11]). If E is a convex subset of Rn, M ⊂ Rn is an affine subspace and F ⊂ E is a face, then F ∩ M is a face of E ∩ M.
The importance of the extreme points and extreme rays of a closed convex set becomes evident with the following theorem. There the closed convex set needs to be line-free, i.e. does not contain any line. Otherwise the set does not simply have any extreme points or rays.
2 Theorem 1.10 ([Schn, Theorem 1.4.3.]). Each line-free closed convex set A ⊂ Rn is the convex hull of its extreme points and extreme rays;
A = conv(ext(A) ∪ extr(A)).
Recall that for two sets A ⊂ V1,B ⊂ V2 the cartesian product A × B is convex if and only if A and B are convex. The following lemma shows how to compute the exposed points and the extreme points of the cartesian product.
Lemma 1.11. For two convex sets A, B we have
i) ext(A × B) = ext(A) × ext(B), ii) exp(A × B) = exp A × exp B.
Proof. i) For x = (x1, x2), y = (y1, y2) and z = (z1, z2) ∈ A × B it is straight forward that z ∈ relint([x, y]) if and only if zi ∈ relint([xi, yi]) for i = 1, 2. Hence i) follows. ii) We may assume that the exposed points will be zero in the respective vector space V1,V2 or V = V1 × V2. The scalar product on the product space V1 × V2 is the sum of the scalar products on each space V1 and V2. Furthermore let u1 ∈ V1, u2 ∈ V2 and u ∈ V1 × V2 be the respective normal vectors defining the exposed point. With the relation u = (u1, u2) we conclude ii) since
hu, (x1, x2)i = hu, (x1, 0)i + hu, (0, x2)i = hu1, x1i + hu2, x2i
for all x1 ∈ A, x2 ∈ B holds. If each summand on the right hand side is bigger than zero then the same holds for the left hand side. Otherwise if the left hand side is bigger than zero and u is a defining normal vector to the exposed point 0 then each summand is bigger than zero in the above equation.
3 1.1.1 Polyhedra, Polytopes and Cones In this section we will consider special types of convex sets, namely polyhedra, polytopes and cones. We will see that if we impose the finiteness of the sets ext(A) and extr(A) in Theorem 1.10 we get exactly these sets which are line-free.
Definition 1.12 (Cones). A cone C in Rn is a subset such that for any positive scalar λ the equation λC = C holds. In the following we distinguish some special types of cones.
a) A polyhedral cone is a subset C ⊂ Rn which is the intersection of finitely many closed linear half-spaces.
n b) For a finite number of vectors b1, . . . , bk ∈ R a set which is given by
( k )
X C := cone(b1, . . . , bk) := λibi λi ≥ 0 i=1 is called a finitely generated cone C.
c) The dual cone C∗ of a subset C in Rn is the set C∗ = {y| hy, xi ≥ 0 for all x ∈ C}. d) A cone C is said to be generating if C contains a basis of Rn. Remark 1.13. In fact C∗ is a closed convex cone for any subset C. If C is indeed a cone itself then C∗ is given by the set of elements which define supporting hyperplanes to C.
The concept of a polyhedral cone and being finitely generated are equivalent and this statement goes back to Minkowski ([Min]) and Weyl ([We]).
Theorem 1.14. A convex cone is polyhedral if and only if it is finitely generated.
Definition 1.15 (Polytopes and Polyhedra). a) A polyhedron is a subset P ⊂ Rn which is the intersection of finitely many closed affine half-spaces. b) A polytope is the convex hull of a finite set of points in Rn. In convex geometry it is a well known fact that the concepts of polyhedra and of polytopes are related. Again this statement goes back to Minkowski ([Min]) and Weyl ([We]).
Theorem 1.16. A set P is a polytope if and only if P is a bounded polyhedron.
Definition 1.17. For two subsets P,Q ⊂ Rn the set P + Q := {p + q| p ∈ P, q ∈ Q} is the Minkowski sum of P and Q.
4 Definition 1.18. a) The recession cone of a convex set P ⊂ Rn is the cone n rec(P ) := {y ∈ R | x + λy ∈ P for all x ∈ P, λ ≥ 0} . b) The lineality space of a polyhedron is the linear subspace
n lin(P ) := rec(P ) ∩ (− rec(P )) = {y ∈ R | x + λy ∈ P for all x ∈ P, λ ∈ R} . c) A polyhedron is pointed if lin(P ) = {0}. Lemma 1.19 ([Schr, p. 100-101]). Let P be a polyhedron. a) Any polyhedron P is the Minkowski sum of its lineality space and a pointed polyhedron, i.e. P = lin(P ) + ((lin(P ))⊥ ∩ P ). b) P + rec(P ) = P . c) P is bounded, i.e. a polytope, if and only if rec(P ) = {0}. d) If P = Q + C, with Q a polytope and C a polyhedral cone, then C = rec(P ). e) P has only finitely many faces. f) Each face of P is a polyhedron. g) If F is a face of P and F 0 ⊂ F , then F 0 is a face of P if and only if F 0 is a face of F . With the previous lemma and definitions the following theorem by Motzkin ([Mo]) states that any polyhedron can be decomposed into a sum of a polytope, a cone and a linear subspace. Theorem 1.20. A set P is a polyhedron if and only if P = Q+C for some polytope Q and some polyhedral cone C. More precisely any polyhedron P has a minimal representation
P = conv{x1, . . . xk} + cone{y1, . . . , yl} + lin(P )
where x1, . . . xk, y1, . . . , yl are orthogonal to lin(P ) and k, l are minimal. If P is pointed then x1, . . . , xk are the vertices of P and y1, . . . , yl are the generators of the extreme rays of rec(P ), which are unique up to scaling. Lemma 1.21. Let P = Q + C be a pointed polyhedron where Q is a polytope and C is a convex polyhedral cone. An element u defines a supporting hyperplane to P if and only if it supports C. Proof. Let u define a supporting hyperplane to P . Hence the number α := max hx, ui is x∈P defined and Hu,α supports the polyhedron P with normal vector u. After a translation the cone C lies in P and hence β := max hx, ui is also well defined. Thus the hyperplane x∈C Hu,β supports C with normal vector u. Conversely let u define a supporting hyperplane to C. Hence max hx, ui ≤ max hx, ui + max hx, ui is defined and induces a supporting x∈P x∈Q x∈C hyperplane.
5 The previous Lemma shows that it suffices to know which vectors support the recession cone of a polyhedron in order to know which vectors support the polyhedron. Now we want to show that every face of a polyhedron is exposed. Although this is a well known property of polyhedra the following proofs are based on [Schn, Chapter 2.4], which discusses polytopes, i.e. compact polyhedra.
n Proposition 1.22. Let P ⊂ R be a polyhedron, F1 an exposed face of P and F an exposed face of F1. Then F is an exposed face of P .
Proof. We may assume that P is pointed. Otherwise we may quotient Rn by the lineality space lin P . Later the hyperplanes in the quotient can be extended by lin P to acquire a n hyperplane in R . Furthermore we may assume that 0 ∈ F ⊂ F1. Since F1 is exposed there exists a support plane Hu,0 to P with normal vector u such that Hu,0 ∩ P = F1 and − P ⊂ Hu,0 hold. Considering F as an exposed face of F1 there exists a support plane H to F1 with H ∩ F1 = F inside the hyperplane Hu,0. Furthermore there is a vector v ∈ Hu,0 − with H = {x ∈ Hu,0| hx, vi = 0} and F1 ⊂ H = {x ∈ Hu,0| hx, vi ≤ 0}. Since P is a pointed polyhedron, i.e. it is line-free, we have P = conv(ext(P ) ∪ extr(P )) due to Theorem 1.10 where ext(P ) and extr(P ) are finite sets. For every extreme ray Ri ∈ extr(P ) we choose a point xi in Ri which is not an extreme point of P . In order to construct the supporting hyperplane to P and the face F we need the following numbers. Define hx, vi η0 := max − x ∈ ext(P ) \ ext(F ) hx, ui and hxi − x, vi η1 := max − xi ∈ Ri ∈ extr(P ) \ extr(F ), x extreme point of Ri and hxi − x, ui= 6 0 . hxi − x, ui
Note that hxi − x, ui ≤ 0 since Hu,0 supports P . Now we will show that the hyperplane Hηu+v,0 with η > max{η0, η1} supports P in F . In order to prove that we will show that every extreme point and every extreme ray of F lies − in Hηu+v,0. Furthermore any other extreme point or extreme ray of P will be in Hηu+v,0 but not in Hηu+v,0. • The extreme points:
– If x ∈ ext(P ) \ ext(F1) we have hx, ui < 0 hence
hx, ηu + vi < η0 hx, ui + hx, vi < 0.
– If x ∈ ext(F1) \ ext(F ) we have hx, ui = 0 and hx, vi < 0 hence
hx, ηu + vi = hx, vi < 0.
6 – If x ∈ ext(F ) we have hx, ui = 0 and hx, vi = 0 hence
hx, ηu + vi = 0.
• The extreme rays:
– If xi ∈ Ri ∈ extr(P ) \ extr(F1), x the extreme point to Ri with hxi − x, ui < 0 and λ > 0 we have
hx + λ(xi − x), ηu + vi < hx, ηu + vi + λ(η1 hxi − x, ui + hxi − x, vi) < 0.
– If xi ∈ Ri ∈ extr(P ) \ extr(F1), x the extreme point to Ri with hxi − x, ui = 0 and λ > 0 we have
hx + λ(xi − x), ηu + vi < hx, ηu + vi + λ hxi − x, vi < 0.
– If xi ∈ Ri ∈ extr(F1) \ extr(F ), x the extreme point to Ri and λ > 0 we have
hx + λ(xi − x), ηu + vi = hx + λ(xi − x), vi < 0.
– If xi ∈ Ri ∈ extr(F ), x the extreme point to Ri and λ > 0 we have
hx + λ(xi − x), ηu + vi = 0.
− Thus ext(F ), extr(F ) ⊂ Hηu+v,0 whereas ext(P ) \ ext(F ), extr(P ) \ extr(F ) ∈ int Hηu+v,0.
Corollary 1.23. Each proper face of a polyhedron is exposed.
Proof. We will prove this by induction on the dimension of the polyhedron. For zero- dimensional polyhedra there is nothing to prove. Assume that P is a polyhedron of dimen- sion n and that the assertion has been proved for polyhedra of smaller dimension. Let F be a proper face of P . Since every face is contained in some exposed face of P , there exists an exposed face F1 which contains F . Furthermore F is a face of F1. Now either F = F1 and then F is an exposed face of P , or F is a face of F1 and then an exposed face of F1, by the induction hypothesis. By Proposition 1.22, F is also an exposed face of P .
7 1.2 Complexification of irreducible real representations Let G be a Lie group and ρ : G → GL(V ) be an irreducible real representation on the vector space V . Moreover let J be a G-invariant complex structure on V i.e. g ◦ J = J ◦ g for all g ∈ G and J is a linear map on V with J 2 = −id. The aim of this section is to understand the representation of G on the complexified space V C. The space V C decomposes into two eigenspaces of J. The eigenspace to the eigenvalue i is V i = {X − iJX| X ∈ V } and the eigenspace to the eigenvalue −i is
V −i = {X + iJX| X ∈ V } .
The eigenspaces V i and V −i are complex, G-invariant subspaces of V C with V C = V i ⊕V −i.
Lemma 1.24. The induced representations of G on V i and V −i are irreducible.
Proof. The invariance of the eigenspaces with respect to G follows from the linearity of the representation and the invariance of the complex structure J. The map V → V i,X 7→ X − JX is R-linear, bijective and G-equivariant. Hence these vector spaces are isomorphic as G-vector spaces. Since the representation on V is irreducible the vector space V i resp. V −i has no real proper subspaces which are invariant under G. In particular it has no complex invariant proper subspaces.
Remark 1.25. The map V i → V −i,X − iJX 7→ X + iJX is C-linear, bijective and G-equivariant.
Now we will be interested in the G-invariant real resp. complex endomorphisms on V C G C G resp. V . Thus we will consider the map EndR(V ) → EndC(V ) . Under the assumption that the subspaces V i and V −i are non-isomorphic as complex representations of G we get the following theorem.
C G C Theorem 1.26. Let f ∈ EndC(V ) be a G-equivariant endomorphism of V . Then there exist complex numbers λ, µ ∈ C such that f(v + w) = λv + µw, for all v ∈ V i, w ∈ V −i.
Proof. The image f(V i) (resp. f(V −i)) is a G-invariant subspace of V C. Since the repre- sentation of G on V i and V −i is irreducible and V C = V i ⊕ V −i holds we conclude that the image f(V i) resp. f(V −i) lies in {0,V i,V −i}. The assumption V i V −i obstructs the case f(V i) = V −i resp. f(V −i) = V i. Hence we have f(V i) ⊂ V i and f(V −i) ⊂ V −i. Since we consider irreducible complex representations we can use the Lemma of Schur and get
f|V i = λIdV i und f|V −i = µIdV −i for complex numbers λ, µ ∈ C.
8 C G C Corollary 1.27. The ring EndC(V ) of G-equivariant endomorphisms on V is commu- tative.
Remark 1.28. The complex structure J is of the form J = iIdV i − iIdV −i . An important example where we complexify an irreducible representation will be de- scribed in the following example. The majority of these cases will be of this type. Example 1.29. Let V be the real vector space R2 with the usual representation of S1 given cos φ − sin φ by multiplication of 2×2 matrices of the form A = . This representation φ sin φ cos φ 1 0 −1 is irreducible of complex type, i.e. End ( 2)S = holds. Here J = is a S1- R R C 1 0 invariant complex structure on R2. The complexification of R2 is C2 with the decomposition into the following subspaces i x −y x 2 1 V = − i ∈ R = z z ∈ C y x y −i and −i x −y x 2 z 1 V = + i ∈ R = z ∈ C = z z ∈ C . y x y iz i
Moreover we have the following equations V i = V −i and V C = V i⊕V −i. The representation of S1 on V i resp. V −i is given as follows: z z A · = eiφ φ −iz −iz and z z A · = e−iφ . φ iz iz Here we see that V i and V −i are not isomorphic as complex representations.
C G Corollary 1.30. Let f ∈ EndC(V ) be a G-equivariant endomorphism, i.e. it is of the form f = λIdV i + µIdV −i , where λ, µ ∈ C. Then the restriction of f on V is G-equivariant if and only if λ = µ holds. Proof. For all X ∈ V we have the following continued equation. 1 f(X) = f(X − iJX + X + iJX) 2 1 = (λX − iJλX + µX + iJµX) 2 1 = ((λ + µ) X − iJ (λ − µ) X) . 2 Then V is invariant under the map f if and only if λ + µ ∈ R und λ − µ ∈ iR. This is exactly the case when λ = µ.
9 G Corollary 1.31. The algebra EndR(V ) of G-equivariant endomorphisms on V is isomor- phic to C.
G Proof. Every endomorphism f ∈ EndR(V ) may be complexified to an endomorphism on C V . Then f is of the form f = λIdV i + λIdV −i and it is completely determined by the complex number λ.
1.3 Almost complex structures An almost complex structure on a manifold can be seen as a complex structure on every tangent space of that manifold which varies smoothly. In this chapter we will consider the integrability of almost complex structures, i.e. if the manifold is a complex manifold. Fur- thermore we will discuss homogeneous almost complex structures on homogeneous spaces.
1.3.1 Integrability Definition 1.32. A smooth manifold M is called an almost complex manifold if it admits an almost complex structure J : TM → TM which is a smooth vector bundle isomorphism on the tangent bundle TM with J 2 = −id.
In general an almost complex manifold M does not need to be a complex manifold. But the following theorem (cf. [NN]) by Newlander and Nirenberg states that an almost complex manifold is indeed a complex manifold if the almost complex structure satisfies a certain torsion condition.
Theorem 1.33 (Newlander-Nirenberg). An almost complex structure J is integrable, i.e. is a complex structure, if and only if the torsion of J which is given by
NJ (X,Y ) = [X,Y ] + J [JX,Y ] + J [X,JY ] − [JX,JY ]
vanishes for all vector fields X,Y ∈ X(M) on M.
1.3.2 Almost complex structures on homogeneous spaces Let M be a manifold and G be a Lie group that acts transitively on M. The goal of this section is to understand almost complex structures on M which are compatible with the action of the Lie group G. These almost complex structures are called homogeneous and are completely determined in one base point. Let X0 ∈ M be a fixed point in M. The X0 stabilizer H = G of X0 is a Lie subgroup of G such that we can identify M with the quotient space G/H.
Theorem 1.34 ([F, Chapter 20]). The existence of a homogeneous complex structure on M = G/H is necessary and, if Ad H is connected, also sufficient if there exist in the complexified Lie algebra gC two complex conjugate Lie subalgebras which span gC and intersect each other in hC.
10 Here the complexified Lie algebra gC is equal to g ⊕ ig, where g (resp. ig) is the 1- eigenspace (resp. −1-eigenspace) of the conjugation. To get an idea of the proof of this theorem we start by identifying the manifold M with the homogeneous space G/H. Since the almost complex structure is assumed to be invariant under the action of the Lie group G it suffices to analyze the tangent space at eH. This tangent space can be identified with a subspace m of g which is complementary to h. So the existence of an integrable almost complex structure on G/H reduces to the existence of a linear map J on g with
J = 0 on h, (1) 2 m is J-invariant with J|m = − Idm, (2) [X,Y ] + J [JX,Y ] + J [X,JY ] − [JX,JY ] ∈ h for all X,Y ∈ g. (3)
(cf. [F, 19.15]). Here condition (3) is the torsion of J, i.e. corresponds to the integrability condition of Theorem 1.33, in terms of the Lie algebra g. Given two Lie subalgebras q, q0 ⊂ gC with q + q0 = gC, q ∩ q0 = hC and ¯q = q0 the complex structure is obtained in the following way:
J(X) = 0 for X ∈ hC, (4) J(X) = iX for X ∈ mC ∩ q (5) J(X) = −iX for X ∈ mC ∩ q0. (6)
The linear map J satisfies conditions (1) - (3) which is proven in [F, Chapter 20].
11 1.4 Compatible subgroups In general and for setting the notation for further chapters if G is a Lie group with Lie algebra g and E,F ⊂ g we set
EF := {η ∈ E| [η, ξ] = 0, for all ξ ∈ F }
and GF := {g ∈ G| Ad gξ = ξ, for all ξ ∈ F } . If F = {X} consists of only one element we simply write EX and GX instead of E{X} and G{X}. Let U be a compact Lie group and U C be its universal complexification in the sense of [Ho] which is a linear reductive complex algebraic group. Furthermore we denote by θ the conjugation map θ : uC → uC with respect to u and the corresponding group isomorphism θ : U C → U C. The map θ is then the Cartan involution in the sense of [Kn, Proposition VI.6.14]. Let φ : U × iu → U C be the diffeomorphism φ(g, ξ) = g · exp ξ.
Definition 1.35. A closed subgroup G ⊂ U C is said to be compatible with respect to U if for K := G ∩ U and p := g ∩ iu the equation φ(K × p) = G holds.
If G is a compatible subgroup then the restriction of φ to K × p is a diffeomorphism onto G. Furthmore the statements in this section apply to any compatible subgroups but in this work we are mostly concerned with a compatible Lie group G which is also a real form of U C = GC. Especially the compact subgroup K is the maximal compact subgroup of G such that the relations g = k ⊕ p,[k, p] ⊂ p and [p, p] ⊂ k hold. Note that G has finitely many connected components, is a closed linear group and is given as a set of fixed points of an involution on GC denoted by σ cf. Chapter 1.6. Moreover g is a real reductive Lie algebra, hence g = z(g) ⊕ [g, g]. Let Gss be the analytic subgroup of G with Lie algebra [g, g] then Gss is 0 0 closed and G = Z(G) ·Gss ([Kn, Corollary 7.11]), where Z(G) is the connected component of the centralizer of G containing the identity. Furthermore let B be the Killing form of g. Then by Bθ(·, ·) := −B(·, θ·) we define a positive definite, symmetric, bilinear form on g (see [Kn, 6.13, p. 355]).
Lemma 1.36 ([BGH2, Lemma 2.7]). Let G ⊂ U C be a compatible subgroup then the following assertions hold. a) If H ⊂ G is closed and θ-invariant, then H is compatible if and only if it has only finitely many connected components. b) The subgroup Gss is compatible. c) If E ⊂ g is a θ-invariant subalgebra, then GE is compatible.
12 1.5 Compact Centralizer In this section we will describe properties of elements X ∈ g which admit a compact centralizer GX . This leads for example to the fact that the induced torus exp(RX) is also compact. Later in this thesis we will see that the compactness of the centralizer does impose a restriction to the existence of an almost complex structure on the homogeneous space G/GX . In this section G denotes a connected semisimple real Lie group with finite center.
Definition 1.37. Let G, resp., H denote connected real Lie groups with Lie algebras g, resp., h and ρ : G → H a Lie group homomorphism. An element X ∈ g is called ρ-compact if exp dρ(RX) ⊂ H is compact. We are mostly interested in the case where ρ = Ad but the following proposition shows that it suffices to look at ρ = Id. Hence we drop the notion and call elements compact if they are Ad-compact or Id-compact.
Proposition 1.38. An element X ∈ g is Ad-compact if and only if it is Id-compact.
Proof. Since G is semisimple with finite center the map Ad : G → Ad(G) is a finite covering. This shows that Ad(M) = Ad(M) holds and the claim follows.
Definition 1.39. The set comp(g) := {X ∈ g|GX is compact} is the set of elements with compact centralizers.
Since GX is a closed subgroup of G with exp(RX) ⊂ GX , an element with compact centralizer is compact in the sense of Definition 1.37. Now let g = k ⊕ p be the Cartan decomposition of g and K be the maximal compact subgroup of G with Lie algebra k.
Lemma 1.40. An element X ∈ g is compact, if and only if there exists an element g ∈ G with Ad g X ∈ k.
Proof. Suppose that the Torus T = exp RX is compact. By Theorem 2.1, Chapter VI in [He], there exists a g ∈ G with gT g−1 ⊂ K. So the inclusion Ad g RX ⊂ k follows on the level of Lie algebras and we have proven the claim. Conversely we can assume that X lies in k. Using Proposition 1.38 we see that the inclusion Ad(exp RX) ⊂ Ad(exp RX) ⊂ Ad K holds. Since K is compact, every set in this chain of inclusion is compact. It shows that X is Ad-compact and with Proposition 1.38 the proof follows.
Lemma 1.41. If X ∈ g is compact, then the centralizer GX is connected and is the subgroup generated by exp gX .
Proof. A more general statement is proven in [Ne, Theorem VII.1.10].
13 Lemma 1.42. Let X ∈ k. Then the following conditions are equivalent.
1) X ∈ comp(g).
2) gX ∩ p = {0}.
3) ad X|p : p → p is an isomorphism. Proof. First we show the equivalence of 1) and 2). Let X ∈ comp(g), i.e. GX is compact. By Theorem 2.1, Chapter VI in [He], there exists a g ∈ G with GX ⊂ gKg−1. Hence gX ⊂ Ad g k. Since Ad g k is a compact Lie subalgebra of g, i.e. the Killing form is negative definite on Ad g k, and the Killing form is positive definite on p the identity p ∩ Ad g k = {0} holds. Therefore gX ∩ p = {0}. X X Conversely suppose g ∩ p = {0}. Let Y = Yk + Yp ∈ g with Yk ∈ k and Yp ∈ p. Since 0 = [X,Y ] = [X,Yk] + [X,Yp] with [X,Yk] ∈ k and [X,Yp] ∈ p we have 0 = [X,Yk] = [X,Yp]. X X Since g ∩ p = {0} the element Yp must be 0. Therefore g ⊂ k and X is compact. Thus GX ⊂ K because of Lemma 1.41 and we have shown that GX is compact. X The properties 2) and 3) are equivalent because of g ∩ p = ker(ad X|p).
14 1.6 Compact Cartan subalgebras Let G be a connected real semisimple Lie group with Lie algebra g. We say that the Lie algebra g contains a compact Cartan subalgebra t if there exists a Cartan decomposition g = k ⊕ p such that t is a Cartan subalgebra which lies in k. From now on we will assume that the Lie algebra g is always equipped with such a compact Cartan subalgebra t.
1.6.1 Root decomposition In this chapter we will state some results regarding roots of Lie algebras which will be needed for the construction of almost complex structures on homogeneous spaces. These roots will give us a tool to describe the Lie subalgebras discussed in Chapter 1.3.2. For a basic introduction into root systems we refer the reader to [Hu2].
Example 1.43. Our main example will be the Lie algebras sp(2n, R) and su(p, q) where we have compact Cartan subalgebras. The description of their Cartan subalgebras t ⊂ k of g are as follows. 0 D AB t = : D diagonal ⊂ k = : A = −At,B = Bt −D 0 −BA AB ⊂ sp(2n, ) = : B = Bt,C = Ct . R C −At t = {iD : D diagonal, T r(D) = 0}⊂ k A 0 = : A ∈ u ,B ∈ u , T r(A) + T r(B) = 0 0 B p q ⊂ su(p, q) AC = t : A ∈ u ,B ∈ u , T r(A) + T r(B) = 0 . C B p q
Let gC be the complexification of g. Then tC is a Cartan algebra in gC. We consider the involution σ of gC with Fixσ(gC) = g. Here σ(X + iY ) = X − iY , where X,Y ∈ g. ∗ For a linear functional α ∈ tC , we define the root space
C C C gα = X ∈ g |[H,X] = α(H)X for all H ∈ t
∗ C C C C C and write ∆ = ∆(g , t ) := α ∈ t \{0}|gα 6= 0 for the set of roots of g with respect C C C to t . We further put g[α] := gα ⊕ g−α ∩ g, where [α] := {α, −α}.
15 Lemma 1.44. The following assertions hold:
C C L C (i) g = t ⊕ α∈∆ gα.
C (ii) dim gα = 1 and Rα ∩ ∆ = {±α}.
(iii) α(t) ⊂ iR for all α ∈ ∆.
C C (iv) If Z ∈ gα, then σ(Z) ∈ g−α and [Z, σ(Z)] ∈ it.
C C C (v) g[α] = gα ⊕ g−α. For the proof of the previous lemma and more detail on the topic of roots we refer the reader to [Kn, Chapter II.4, II.5 and VI.7] and [Ne, Theorem VII.2.2, Lemma VII.2.3.]. We can show the first property of Lemma 1.44 (iv) in a more general sense. With the following lemma it follows that also the involution θ, given by the Cartan decomposition g = k ⊕ p, acts on the root spaces.
Lemma 1.45. Let τ be an involution, i.e. a Lie algebra homomorphism with τ 2 = Id, on C ¯ C C g which is C-linear and fixes t. Then τ(gα) = g−α for all α ∈ ∆.
Proof. Since τ fixes t and is C¯-linear it follows directly that τ(t) = t¯ for all t ∈ tC. C Now let ξ ∈ gα with [t, ξ] = α(t)ξ. Using τ on that equation we get [τ(t), τ(ξ)] hom.= τ[t, ξ] = τ (α(t)ξ) C−=lin. α(t)τ(ξ).
Inserting τ(t) instead of t = t1 + it2 we get [t, τ(ξ)] = α(τ(t))τ(ξ)
= α(t1 − it2)τ(ξ) α,C−lin. = (α(t1) − iα(t2))τ(ξ) 1.44(iii) = −α(t1) − iα(t2)τ(ξ) = −α(t)τ(ξ).
C C Definition 1.46. A root α in ∆ is called compact (resp. noncompact) if gα ⊂ k (resp. C C gα ⊂ p ) holds. The set of compact roots is denotet by ∆k and the set of noncompact roots by ∆p. Lemma 1.47 ([Ne, p. 234-236]). A root is compact (resp. noncompact) if and only if C α([Zα, σ(Zα)]) < 0 (resp. α([Zα, σ(Zα)]) > 0) holds for all Z ∈ gα. Furthermore the subspace k = t ⊕ L g is the unique maximal compact subalgebra of α∈∆k [α] g containing t. The corresponding Cartan decomposition is then given by g = k ⊕ p with p = L g . α∈∆p [α]
16 (a) The root system of Sp(4, R) (b) The root system of SU(2, 1) Figure 2: Root systems
Remark 1.48. Let α in ∆ be a root. Then we denote by tα ∈ t with −iα(X) = Bθ(tα,X) ∗ for all X ∈ t the coroot of −iα ∈ t with respect to Bθ. By abuse of notation we may call coroots just roots when in t.
Example 1.49. Figure 2a and 2b show the compact and noncompact roots of Sp(4, R) resp. SU(2, 1). The red points describe the noncompact roots and the blue points describe the compact roots. The lines are hyperplanes through zero which are orthogonal to some root in ∆. Definition 1.50. a) A subset ∆+ ⊂ ∆ is called a positive system if there exists an element X0 ∈ t such that + ∆ = {α ∈ ∆ : −iα(X0) > 0}
and α(X0) 6= 0 for all α ∈ ∆.
+ + + b) A positive system ∆ is called adapted if ∆p := ∆ ∩ ∆p is Wk-invariant, where Wk is the Weyl group with respect to t and k.
+ + (a) A set ∆p for Sp(4, R) (b) A set ∆p for SU(2, 1) + Figure 3: Choices of ∆p
+ Example 1.51. Figure 3a and 3b show possible choices of ∆p for an adapted positive + + root system in Sp(4, R) resp. SU(2,1). The only choices for ∆p in these cases are ∆p and + − −∆p =: ∆p . Any other choice of defining noncompact positive roots are not invariant under the Weyl group Wk. In these cases the group Wk is just given by a single reflection with respect to the line orthogonal to the compact root.
17 Recall that the Weyl chamber with respect to a positive root system ∆+ is defined as the set C := {X ∈ t| − iα(X) ≥ 0 for all α ∈ ∆+}. These Weyl chambers are in a natural one to one correspondence to positive root systems. Furthermore the respective Weyl group permutes the Weyl chambers and acts simply transitive on the set of Weyl chambers.
C Lemma 1.52. A root α is compact if and only if hZα, σ(Zα)i < 0 for all Zα ∈ gα. C Analogously α is noncompact if and only if hZα, σ(Zα)i > 0 for all Zα ∈ gα.
C Proof. Let α be a compact root, i.e. 0 > α([Zα, σ(Zα)]) holds for all Zα ∈ gα. Furthermore let tα be the coroot to −iα with respect to Bθ, i.e. −iα(X) = Bθ(tα,X) holds for all X in t, where B is the Killing form. Thus we get the following equation:
0 > α([Zα, σ(Zα)]) = −iα(i[Zα, σ(Zα)]) = Bθ (tα, i[Zα, σ(Zα)]) = − htα, i[Zα, σ(Zα)]i = −i h[tα,Zα], σ(Zα)i = −i hα(tα)Zα, σ(Zα)i = −iα(tα) hZα, σ(Zα)i = Bθ(tα, tα) hZα, σ(Zα)i . | {z } >0
Since Bθ is positive definite on g we have proven the lemma. For a noncompact root α the prove is analogous with 0 < α([Zα, σ(Zα)]). Lemma 1.53. Let t ⊂ k be a compact Cartan subalgebra of g. Then [ comp(g) ∩ t = t \ ker α.
α∈∆p Proof. Using Lemma 1.42 and Lemma 1.47 it follows that X ∈ comp(g) ∩ t if and only if X ∈ t and ad X is injective on p = L g . α∈∆p [α] X L X Remark 1.54. If X ∈ t we get g = t⊕ g[α]. Hence the centralizer g is completely α∈∆ α(X)=0 determined by the roots which vanish on X. The special case of a compact element X is described in the following lemma. Lemma 1.55. If X ∈ comp(g) ∩ t then
X M g = t ⊕ g[α].
α∈∆k α(X)=0 Proof. The inclusion “⊃” follows from X ∈ t and from the definition of roots. Conversely X P C suppose Y ∈ g with the root decomposition Y = Z + Zα with Z ∈ t and Zα ∈ gα. P α∈∆ Then 0 = [X,Y ] = α∈∆ α(X)Zα. By Lemma 1.53 we have α(X) 6= 0 for all α ∈ ∆p and thus we conclude Zα = 0 for all ∆p.
18 Definition 1.56. A Lie algebra g with compact Cartan subalgebra t and its respective roots ∆ is said to have cone potential if for any noncompact root α ∈ ∆p and any nonzero C vector Zα ∈ gα the inequality [Zα, Zα] 6= 0 holds.
The following cones Cmin and Cmax will play an important part in describing the mo- mentum polyhedra in the later chapters.
Definition 1.57. For an adapted positive system ∆+ of roots we define the cones
+ + Cmax := Cmax(∆p ) := {X ∈ t : −iα(X) ≥ 0 for all α ∈ ∆p } and + + gα Cmin := Cmin(∆p ) := cone{i[Zα, Zα]: α ∈ ∆p ,Zα ∈ C}, where for a subset M of a vector space, the set cone(M) denotes the smallest closed convex cone containing M.
Example 1.58. See Figure 4a for an example of Cmax and Cmin in the case of Sp(4, R) and see Figure 4b for an example of Cmax and Cmin in the case of SU(2, 1). The choices of + the noncompact positive roots ∆p are shown in Figure 3. Note that the negatives of these cones denote other maximal and minimal cones with respect to the negative set of roots − ∆p .
(a) Cones for Sp(4, R) (b) Cones for SU(2, 1) Figure 4: The maximal and minimal cone
0 D Example 1.59 (Sp(2n, ), see also Example 1.43). Given X ∈ n, let X = ∈ R R t −D 0 t be that element where diag D = X. Identifying t with Rn in this way, the Weyl group is given by all permutations of the entries of X ∈ Rn. The center of k is spanned by the vector (1,..., 1)t. The noncompact root vectors generate the convex cone Cmin = {Xt : Xj ≥ 0, 1 ≤ j ≤ n}, which is a positive orthant. Since an orthant is self-dual we have Cmax = Cmin using Theorem 1.76. Figure 4a describes the case of n = 2.
19 Example 1.60 (SU(p, q), see also Example 1.43). Let
p+q X X E = {(λ1, . . . , λp, µ1, . . . , µq) ∈ R : λi = µj}, 1 E = {(λ, µ) ∈ E : λi + µj ≥ 0 for all i, j}, 0 E = {(λ, µ) ∈ E : λi, µj ≥ 0}.
A 0 For X = (λ, µ) ∈ E denote X = ∈ t that element where diag A = −iλ t 0 B and diag B = iµ holds. The Weyl group acting on t consists of all permutations of the λi, together with all permutations of the µj separately. The center of k is spanned 0 by (1/p, . . . , 1/p, 1/q, . . . , 1/q)t. Then the noncompact root vectors generate Cmin = Et , 1 whose dual cone is Cmax = Et . Figure 4b describes the case of p = 2 and q = 1. For more details to these examples and for further classical examples the reader is referred to [Pa].
C Remark 1.61. For any X ∈ t and Zα ∈ gα we get the following continued equation:
Bθ(X, i[Zα, Zα]) = −i α(X)Zα, Zα = Bθ(X, tα) Zα, Zα = Bθ(X, tα Zα, Zα ).
Since Bθ is nondegenerate the equality i[Zα, Zα] = tα Zα, Zα holds. Hence for any non- compact root α the vector i[Zα, Zα] and the coroot tα differ only by the positive scalar
Zα, Zα using Lemma 1.52. We have shown that Cmin is just the closed convex cone spanned by the coroots of the positive noncompact roots.
Definition 1.62. For an adapted positive system ∆+ of roots we define the cones
Wmax := {X ∈ g|πt(OX ) ⊂ Cmax}
and Wmin := {X ∈ g|πt(OX ) ⊂ Cmin},
where OX is the adjoint G-orbit of X and πt is the projection onto the torus.
Example 1.63. Figure 5 shows the different types of adjoint orbits of the group Sp(2, R) in its Lie algebra sp(2, R). There are the trivial orbit which is the point zero, the two nilpotent orbits which are the upper and lower part of the cone, the one-sheeted hyperboloids and the connected components of the two-sheeted hyperboloids. The compact torus t is one-dimensional, equals the maximal compact subalgebra k of sp(2, R) and is represented by the vertical axis. As we have seen in Example 1.59 Cmin = Cmax is an orthant in t, i.e. in this case a half ray. Hence in the symplectic case we get Wmax = Wmin. In the case of Sp(2, R) the cone Wmax = Wmin is the convex hull of the closure of either the upper or the lower nilpotent orbit.
20 ∼ Figure 5: The adjoint SL(2, R) = Sp(2, R)-orbits
Theorem 1.64 ([HLV, p. 136, Theorem 5.3]). Let g be a finite dimensional real Lie algebra which contains a pointed generated invariant cone C0. Then there exists a compact Cartan subalgebra t ⊂ g and an adapted positive root system ∆+ satisfying the following conditions.
1. The chain of inclusions Cmin ⊆ C0 ∩ t ⊆ Cmax hold.
2. The invariant pointed generating cones in g containing Cmin are in one-to-one cor- respondence via C ↔ C ∩ t with those cones in t which are pointed and generate t, contain Cmin, are contained in Cmax and are invariant under the action of the Weyl group Wk. 3. If g is semisimple, then the largest and smallest pointed generating invariant cones Wmax and Wmin contain Cmin and they satisfy Wmax ∩t = Cmax and Wmin ∩t = Cmin.
21 1.6.2 Quasihermitian Lie algebras Let g be a real Lie algebra, k ⊂ g a maximally compact subalgebra and t ⊂ k a compact Cartan subalgebra of g.
Proposition 1.65 ([Ne, Proposition VII.2.14]). The following are equivalent:
(1) There exists an adapted positive root system.
(2) zg(z(k)) = k. Definition 1.66. A Lie algebra which satisfies one of the properties in Proposition 1.65 is called quasihermitian.
Since any two maximal compact subalgebras are conjugate under inner automorphisms, the condition zg(z(k)) = k is satisfied by one maximal compact subalgebra if and only if it is satisfied by all of them.
Lemma 1.67. Let g be a quasihermitian Lie algebra and Y ∈ k. Then gY is quasihermitian. More generally for any vector space V ⊂ k we get that gV is quasihermitian.
Y Y Proof. We have to show that zgY (zkY (k )) = k . The inclusion ”⊃ ” follows from the Y Y definition of the centralizer. On the other hand we have g ⊃ zgY (zkY (k )) ⊂ zgY (zk(k)) ⊂ Y Y zg(zk(k)) = k where we have used that zk(k) ⊂ zkY (k ) since Y ∈ k. Now zgY (zkY (k )) ⊂ gY ∩ k = kY . The second statement follows analogously. Later we will examine orbits of centralizers inside a G-orbit, where the the Lie algebra g is quasihermitian. Thus the lemmas in this chapter allow us to make similar conclusion on the orbits of centralizers inside the G-orbit.
Lemma 1.68. Let L be a Lie group, g ⊂ l a quasihermitian Lie algebra and g0 ∈ L. Then Ad g0 g is quasihermitian. Proof. The following calculation shows the lemma:
zAd g0 g (zAd g0 k(Ad g0 k)) = {Ad g0 ξ : ξ ∈ g, [Ad g0 ξ, zAd g0 k(Ad g0 k)] = 0} = {Ad g0 ξ : ξ ∈ g, [Ad g0 ξ, Ad g0 zk(k)] = 0} = {Ad g0 ξ : ξ ∈ g, [ξ, zk(k)] = 0} = Ad g0 {ξ : ξ ∈ g, [ξ, zk(k)] = 0} = Ad g0 zg(zk(k)) = Ad g0 k.
22 Lemma 1.69. Let g be a reductive quasihermitian Lie algebra with the decomposition z⊕gss into its center and its semisimple part and t a compact torus. Then the Lie algebra gss is quasihermitian.
Proof. We know that the Lie algebra g satisfies the condition zg(z(k)) = k. We want to show that the semisimple part gss has the same property. Since z ⊂ t we also get the 0 0 0 decomposition k = z ⊕ k . Furthermore the inclusion k ⊂ zgss (z(k )) together with the following continued equality prove the lemma:
0 0 0 0 z ⊕ zgss (z(k )) = zg(z(k )) = zg(z ⊕ z(k )) = zg(z(k)) = k = z ⊕ k .
0 0 Thus we get zgss (z(k )) = k . Remark 1.70. Among the simple real Lie algebras the compact Lie algebras, i.e. g = k, are trivially quasihermitian. The non-compact simple Lie algebras which are quasihermitian are those where z(k) is not trivial which then turns out to be one-dimensional. These kind of Lie algebras are called hermitian Lie algebras. They will be studied more in the next chapter. In the reductive case a Lie algebra is quasihermitian if and only if all its simple ideals are either compact or hermitian (see [Ne, p. 241]). Hence we get an ideal direct sum decomposition
n M g = z(g) ⊕ gj, j=1
where gj are either compact or hermitian simple ideals. Thus the classification of reductive quasihermitian Lie algebras corresponds to the classification of compact Lie algebras (which is the same as simple complex Lie algebras) and the classification of hermitian simple Lie algebras. Proposition 1.71 ([Ne, Propositions VIII.1.18, VIII.3.7]). Let g be a reductive quasi- hermitian Lie algebra, t ⊂ g a compact Cartan algebra and ∆+ an adapted positive root system. Then the following assertions hold:
i) The cone Cmin is pointed.
ii) Cmin ⊆ Cmax.
Fact ii) is equivalent to the existence of a closed convex generating invariant cone Wmax ⊂ g such that Wmax ∩ t = Cmax. Definition 1.72. A generating invariant convex subset C ∈ g is called elliptic if C0 ⊂ comp(g).
Example 1.73. In Example 1.63 where G = Sp(2, R) we have seen that the cone Wmax contains exactly one connected component of every two-sheeted hyperboloid. These are exactly the orbits which have T = K or conjugates of this group as their isotropy. Hence 0 Wmax ⊂ comp(sp(2, R)) holds and Wmax = Wmin is an elliptic cone in sp(2, R).
23 Theorem 1.74 ([Ne, Theorem VIII.3.10]). A Lie algebra g contains an invariant elliptic cone if and only if it contains a compact Cartan subalgebra t and there exists an adapted + positive root system ∆ with Cmin ⊆ Cmax. If this condition is satisfied and W ⊂ g is an invariant elliptic cone, then there exists a unique adapted positive system ∆+ such that W ⊆ Wmax. Then we also have Cmin ⊆ Cmax. In particular, the cones Wmax are maximally elliptic.
1.6.3 Hermitian Lie algebras A semisimple Lie algebra is hermitian if each of its simple ideals is hermitian, i.e. semisim- ple quasihermitian with no compact ideals. A hermitian Lie algebra g with Lie group G admits a hermitian symmetric space G/K. The reader is referred to [Se], [Vin] for the following result.
Theorem 1.75 (Kostant-Vinberg). A semisimple Lie algebra g contains a pointed, gen- erating invariant cone if and only if it is hermitian.
Moreover it is clear that any invariant convex cone in a semisimple Lie algebra g is contained in the direct sum of invariant cones in the simple summands. That is why we will mostly be concerned about the simple case. The next theorem describes more properties of generating invariant cones in hermitian simple Lie algebras. Therefore let Z0 ∈ k be an element which spans the one-dimensional center of k. Then Z0 ∈ t, by maximal commutativity of t, and there exists an adapted + + positive root system ∆ such that −iα(Z0) ≥ 0 for all α in ∆ (see [He, Corollary 7.3, 7.13]).
Theorem 1.76 ([Pa] or [HLV, p. 137, Theorem 6.2]). Let g be a hermitian simple Lie algebra. Then
∗ i) Cmin contains Z0 in its algebraic interior, and Cmax = (Cmin) , where the dual is taken in t.
ii) Each pointed generating invariant cone in g contains either Z0 or −Z0 in its interior. iii) For each pointed generating invariant cone C, we have (C ∩ t)∗ = C∗ ∩ t, where the first dual cone is computed in t and the second in g, both with respect to Bθ(·, ·).
iv) The cone Wmin is the smallest pointed generating invariant cone containing Z0 and ∗ Wmax = (Wmin) . Theorem 1.77 ([Ne, Theorem A.V.1]). If Π is a base of the root system ∆ of a complex simple Lie algebra g and β ∈ Π is a base root for which the coefficient of the highest root is 1, then there exists an involution σ : X 7→ X∗ on g turning g into an involutive Lie algebra with root decomposition such that for Πk = Π \{β} we have
+ ∆k = (spanZΠk) ∩ ∆ and ∆p = (β + spanZΠk) ∩ ∆
24 is contained in an adapted positive root system. Each real simple hermitian Lie algebra is isomorphic to one of the following:
(An) gR = su(p, q), g = sl(p + q, C), n = p + q − 1
(Bn) gR = so(2n − 1, 2), g = so(2n + 1, C), n ≥ 2
(Cn) gR = sp(2n, R), g = sp(2n, C), n ≥ 3
(Dn) gR = so(2n − 2, 2), g = so(2n, C), n ≥ 4 ∗ (Dn) gR = so (2n), g = so(2n, C), n ≥ 4
(E6) gR = e(6,−14),
(E7) gR = e(7,−25). Furthermore, we have the exceptional isomorphisms ∼ ∼ ∼ ∼ sl(2, R) = so(1, 2) = su(1, 1) = sp(2, R), so(2, 2) = sl(2, R) ⊕ sl(2, R), ∗ ∼ ∼ so (4) = sl(2, R) ⊕ so(3, R), so(3, 2) = sp(4, R), so(4, 2) ∼= su(2, 2), so∗(6) ∼= su(1, 3) and so∗(8) ∼= su(6, 2). Theorem 1.78 ([He, Chapter X §6.3]). The classification of all irreducible Hermitian symmetric spaces G/K are given in the second column in the following table. Their compact dual U/K is given in the third column.
Type G/K U/K Dimension A III SU(p, q)/S(Up × Uq) SU(p + q)/S(Up × Uq) 2pq BD I SO0(p, 2)/SO(p) × SO(2) SO(p + 2)/SO(p) × SO(2) 2p D III SO∗(2n)/U(n) SO(2n)/U(n) n(n − 1) CI Sp(2n, R)/U(n) Sp(2n)/U(n) n(n + 1) E III (e(6(−14)), so(10) + R)(e(6(−78)), so(10) + R) 32 E VII (e(7(−25)), e6 + R)(e(7(−133)), e6 + R) 54
In this work most examples we will consider are the groups SU(p, q) and Sp(2n, R).
25 2 Coadjoint orbits as K¨ahlermanifolds
∗ Let α ∈ g then the coadjoint orbit O := Oα := G · α may be identified with the corre- sponding homogeneous space G/Gα. For ξ ∈ g we consider the fundamental vector field O O d ξ defined by ξ (x) = dt 0 exp(tξ) · x for every x ∈ O. Note that the tangent space to the orbit at any point is given by evaluation of the fundamental vector fields in that point, in O particular we have Tβ(Oα) = ξ (β)|ξ ∈ g . It is a well known theorem that for any Lie group G every coadjoint orbit is a symplectic manifold. This is described in the following theorem. For a more detailed introduction into the topic of coadjoint orbits the reader is referred to [Be, Chapter 2.5] or [Sil, Part VIII]. A proof of the well known theorem is given for example in [Mi, Chapter VII.31.14].
Theorem 2.1 (Kirillov, Kostant, Souriau). If G is a Lie group, then a coadjoint orbit Oα carries a canonical symplectic structure ω which is invariant under the coadjoint action of G and is given in the point α by
O O ωα ξ (α), η (α) = α ([ξ, η])
for all ξ, η ∈ g.
Since G is semisimple we can identify g with its dual space g∗ via the Killing form B(·, ·) = h·, ·i. Hence we consider adjoint orbits in g rather than coadjoint orbits in g∗. If X ∈ t corresponds to α ∈ t∗ then the symplectic form turns into
O O ωX ξ (X), η (X) = hX, [ξ, η]i
for all ξ, η ∈ g. The goal of this chapter is to show that, given a connected semisimple Lie group G with finite center and a compact Cartan subalgebra t in g, the adjoint orbit OX where X lies in comp(g) is a complex manifold with complex structure J and under certain conditions on X is also K¨ahlerianwith respect to the natural symplectic form ω. The Riemannian metric is then defined by g(·, ·) := ω(·.J·). Thus the centralizer GX of X X is compact because the Riemannian metric is G-invariant and hence G lies in Iso(gX ), the isometry group of gX . Due to this fact we will only consider compact centralizers, hence X ∈ comp(g) and up to conjugation the chain of inclusions T ⊂ H := GX ⊂ K ⊂ G holds.
26 2.1 Invariant almost complex structures In this section we consider homogeneous almost complex structures on homogeneous spaces G/H. We will first discuss the case where H = T and then go to the more general case where H ⊂ K which contains T . Either way we are interested in how many different homogeneous almost complex structures the space G/H admits.
2.1.1 The special case H = T In this section we look at the generic case where the centralizer H := GX of an element X ∈ g is given by a compact torus T . So we are interested in the number of G-invariant almost complex structures J on the orbit O ∼= G/T . Since J is G-invariant it suffices to ∼ ∼ know the number of T -invariant complex structures on the tangent space TX O = g/t = L α∈∆ g[α]. Given such a complex structure on g/t we can extend it complex linearly on C C ∼ L C g /t = α∈∆ gα, see e.g. Chapter 1.2, which we denote by abuse of notation again as J.
C Lemma 2.2. The complex structure J leaves the root spaces gα invariant. Proof. Since J is T -invariant the t-invariance of J follows by differentiating, i.e. for all C ξ ∈ gα and τ ∈ t we get ad τ Jξ = J (ad τ ξ). Then the equality [τ, Jξ] = α(τ)Jξ follows. We get the same equations with τ ∈ tC by complex linearity. This shows by the definition C C of the root spaces that Jξ lies in gα for all ξ ∈ gα.
Corollary 2.3. The complex structure J leaves the space g[α] invariant.
C C C Proof. The complex structure J leaves the space g[α] = gα ⊕ g−α invariant. Since J ∼ L originates from a complex structure on g/t = α∈∆ g[α] the claim follows.
C Now consider the complex eigenspaces of J on g[α]. Using the notation of Chapter 1.2 we get the following decomposition:
C C C i −i g[α] = gα ⊕ g−α = g[α] ⊕ g[α].
C We want to show that these two decompositions of g[α] are actually the same. For all ξ in i −i g[α] the element ξ − iJξ is in g[α] and σ (ξ − iJξ) = σ(ξ) + iσ(Jξ) = ξ + iJξ ∈ g[α], i.e. σ i −i exchanges g[α] and g[α]. i −i Since J is T -invariant, the subspaces g[α] and g[α] are one-dimensional and irreducible in C C i C C −i i gα ⊕ g−α. Hence g[α] ∈ gα, g−α and g[α] = σ g[α] . So J is given by multiplication with C C i or −i on gα and by multiplication by the complex conjugate on g−α. Conversely we can construct an invariant complex structure by the above method. Hence we have shown the following theorem.
Theorem 2.4. There exist 2|∆|/2 many T -invariant, almost complex structures on O = G/T .
27 2.1.2 The case T ⊂ H ⊂ K
Now let H = GX0 be a compact centralizer of a given point X ∈ t, i.e. X ∈ t\S ker α 0 0 α∈∆p using Lemma 1.53. Thus we are in the case where the inclusions T ⊂ H ⊂ K hold. The corresponding Lie algebra h = gX0 lies in k and we can decompose g into h ⊕ m, where m is orthogonal to h with respect to the Killing form. Since T lies in H, it naturally acts on h C C and on its orthogonal complement m. Let ∆h := ∆(h , t ) be the set of roots which belong to hC with respect to the Cartan subalgebra tC. Hence we get the following decompositions.
C C M C h = t ⊕ gα,
α∈∆h M h = t ⊕ g[α],
α∈∆h
C M C m = gα,
α∈∆\∆h M m = g[α].
α∈∆\∆h
C The equation [X0, ξ] = α(X0)ξ = 0 for ξ ∈ gα implies that α ∈ ∆h if and only if α ∈ ∆ and α vanishes on X0. Note that we are in the case of H = T if and only if ∆h = ∅. Now we are looking for H-invariant almost complex structures J which are T -invariant on m ∼= g/h. Analogously to the case where H = T in Chapter 2.1.1 the complex structures are given by multiplication by i or −i on the root spaces in mC. Since the H-invariance gives us more obstructions, the multiplications by i or −i can not be chosen freely anymore for every ±α ∈ ∆ \ ∆h. Keeping this in mind we get the following Theorem 2.5 ([BH, Proposition 13.4]). Let O ∼= G/H be equipped with a homogeneous almost complex structure J. Furthermore let m be decomposed into irreducible subspaces Lm nk with respect to H, i.e. m = k=1 nk. H Then the subspaces nk are all different as representation of H with EndR(nk) = C. Fur- thermore there exist 2m different invariant almost complex structures on O.
Proof. The subspaces nj are all invariant under T hence they are direct sums of subspaces g[α] where the corresponding roots ±α are the weights of the restricted representation. Since any two roots are different it follows that ni nj as representations for all i, j = 1, . . . , m. Using Schur’s lemma, every H-equivariant map leaves nj invariant for all j. Keeping the assumption that there exists a H-invariant almost complex structure J, which means that there is a H-equivariant map on nj without real eigenvalues, the algebra of endomorphisms of nj is not equal to R. The only other options using Schur’s Lemma are C and H. In both C cases we get the eigenspace decomposition of nj with respect to J and the eigenvalues i and −i. C i −i nj = nj ⊕ nj for all j i −i C Where nj and nj are complex irreducible subspaces of nj . This follows from the irre- i −i ducibility of nj (see Chapter 1.2). If α is a weight of nj, then −α is a weight of nj . Hence
28 i −i the subspaces nj, nj are not isomorphic as complex representations. Using Corollary 1.31 we get the following equation. H ∼ EndR(nj) = C.
On each nj for j = 1, . . . , m there exist exactly two different invariant complex structures.
2.2 Integrable invariant almost complex structures In this chapter we want to analyze the integrability (see Chapter 1.3) of invariant almost complex structures on G/H and its relationship to the roots of gC with respect to Cartan subalgebra tC where t is a compact Cartan subalgebra in g. For similar results the reader is referred to [BFR, Chapter 3] although the approach is slightly different. Using Theorem 1.34 we want to construct explicitly Lie subalgebras q in gC which induce an integrable invariant almost complex structure on G/H. We will still use the following decompositions.
C C M C h = t ⊕ gα,
α∈∆h M h = t ⊕ g[α],
α∈∆h
C M C m = gα,
α∈∆\∆h M m = g[α].
α∈∆\∆h
+ − Now we decompose the roots of ∆ \ ∆h into two distinct subsets ∆J and ∆J where for + − + − every α in ∆J the root −α lies in ∆J , i.e. ∆ \ ∆h = ∆J ∪ ∆J . With this decomposition we are able to define the following subspaces of gC.
+ C M C q = h ⊕ gα + α∈∆J C M C M C = t ⊕ gα ⊕ gα
α∈∆h + α∈∆J and
− C M C q = h ⊕ gα − α∈∆J C M C M C = t ⊕ gα ⊕ gα.
α∈∆h − α∈∆J
Almost all conditions of Theorem 1.34 are fulfilled. The equations q+ ∩ q− = hC, σ(q+) = q− and q+ + q− = gC hold. The only property that is missing is the fact that q+
29 and q− are Lie subalgebras of gC. Therefore the following definition plays an important role.
Definition 2.6. A subset ∆0 ⊂ ∆ is closed, if for two roots α, β ∈ ∆0 such that α + β is a root the sum α + β also lies in ∆0.
+ Examples of closed subsets of ∆ are ∆, ∆h, ∆k and any system ∆ of positive roots in C C C ∆. The set of roots ∆h and ∆k are closed because h and k are Lie subalgebras of g .
C + − Proposition 2.7. Let ∆h be the set of roots given by h and let ∆ = ∆h ∪ ∆J ∪ ∆J be the disjoint decomposition introduced above. Then the following properties are equivalent.
+ i) The set ∆h ∪ ∆J is closed.
− ii) The set ∆h ∪ ∆J is closed.
+ iii) The set ∆J is closed.
+ + If one of these properties holds then we have ∆h + ∆J ⊂ ∆J . Proof. For the first equivalency of i) and ii) we will first show the ”only if” part. − Since ∆h is already closed, we only have to consider the following case. Let α ∈ ∆h ∪ ∆J − − + and β ∈ ∆J with α + β∈ / ∆h ∪ ∆J , i.e. α + β ∈ ∆J . According to the construction −α + + lies in ∆h ∪ ∆J . Under the assumption that ∆h ∪ ∆J is closed, the root α + β − α = β lies + − in ∆h ∪ ∆J which contradicts β ∈ ∆J . This shows the first equivalency since the ”if” part follows analogously. + Now we show the eqivalency of the first and the third property. Let ∆h ∪ ∆J be closed + + and ∆J be not closed. Then there exist roots α, β ∈ ∆J with α + β ∈ ∆h. Now −α lies − − in ∆J and using the already proven equivalency we get β = α + β − α ∈ ∆h ∪ ∆J which + contradicts β ∈ ∆J . + + + Conversely let ∆J be closed and ∆h ∪ ∆J not. Then there exist roots α ∈ ∆h and β ∈ ∆J − + + with α + β ∈ ∆J . Then −α − β lies in ∆J and using the closedness of ∆J we get + −α = −α − β + β ∈ ∆J . This contradicts the assumption ±α ∈ ∆h. + To show the last assumption let α ∈ ∆h, β ∈ ∆J and α + β be a root. If α + β lies in ∆h then because of the closedness of ∆h also does β which contradicts the assumption. Theorem 2.8. The vector subspace
+ C M C C M C M C q = h ⊕ gα = t ⊕ gα ⊕ gα
+ α∈∆h + α∈∆J α∈∆J
C + is a Lie subalgebra of g if and only if the set ∆h ∪ ∆J is closed in ∆.
C C C Proof. Since the root spaces fulfill the equation [gα, gβ ] ⊂ gα+β for all α, β ∈ ∆ the proof follows.
30 + Remark 2.9. Theorem 2.8 shows that every closed subset ∆h∪∆J induces a Lie subalgebra + − − + − q . Thus the set ∆h ∪ ∆J induces a Lie subalgebra q such that q and q fulfill the conditions in Theorem 1.34. Conversely two Lie subalgebras q+ and q− fulfilling the conditions can be decomposed into C C + − root spaces since t ⊂ h . With Theorem 2.8 we get closed subsets ∆h ∪ ∆J and ∆h ∪ ∆J . Theorem 2.10 ([BH, Theorem 4.9]). Let ∆0 be a closed subset of ∆ such that it contains exactly one of the roots ±α for all α ∈ ∆. Then the set ∆0 is a set of positive roots. Theorem 2.11 ([BH, Theorem 4.10]). Let ∆0 be a closed subset of ∆ such that it contains at least one of the roots of ±α for all α ∈ ∆. Then the set ∆0 contains a set of positive roots. For the next theorem we need to refine the definition of a Weyl chamber. It is known (cf. [Hu2, Chapter 10]) that the sets positive roots are in one-to-one correspondence with Weyl chambers in t. Weyl chambers are defined as the connected components of the open dense subsets in t obtained by removing all hyperplanes {X ∈ t|α(X) = 0} given by each root α ∈ ∆+. + Similarly, one can show that closed positive subsets ∆J are in one-to-one correspondence with Weyl chambers in ˆt, where ˆt := {X ∈ t|α(X) = 0 for all α ∈ ∆h}. Again the Weyl chambers are defined as the connected components of the open dense subset in ˆt obtained ˆ + by removing all hyperplanes {X ∈ t|α(X) = 0} given by each root α ∈ ∆J . + Theorem 2.12. The set of closed subsets ∆J of ∆ \ ∆h, where for all α ∈ ∆ \ ∆h the set + ˆ ∆J contains exactly one root of ±α, is bijective to the set of Weyl chambers in t. + + Proof. Let ∆J be closed. Using Theorem 2.11 the set ∆h ∪ ∆J contains a set of positive + + + roots. Let ∆k with k = 1, . . . , r be all possible sets of positive roots in ∆h ∪ ∆J . Since ∆k is a positive root system there exists a corresponding Weyl chamber Ck in t. Now define Tr ˆ the set CJ = k=1 Ck and we will show that CJ is a Weyl chamber in t. For this let β be + + a root in ∆J , i.e. β ∈ ∆k for all k. The root β is positive on CJ since it is positive on Ck for all k = 1, . . . , r. If α is a root in ∆h then it vanishes on CJ since there exist positive + + + sets of roots which contain {α} ∪ ∆J or {−α} ∪ ∆J . E.g. let ∆h be a positive set of roots + + − + in ∆h then using Proposition 2.7 and Theorem 2.10 the sets ∆h ∪ ∆J and ∆h ∪ ∆J are positive sets of roots. Hence CJ is the corresponding Weyl chamber in ˆt to ∆J . Conversely let CJ be a Weyl chamber in ˆt. Since some roots vanish on CJ the space CJ must lie on the boundary of Weyl chambers in t. Let Ck, k = 1, . . . , r be the Weyl chambers + in t with CJ ⊂ Ck. Every Weyl chamber Ck in t is associated to a set of positive roots ∆k . + Tr + ˆ + Denote the set ∆J = k=1 ∆k . By definition of t and its Weyl chambers the set ∆J has + 1 the property |∆J | = 2 |∆ \ ∆h| and it contains only one of the roots ±α ∈ ∆ \ ∆h. Now + + Tr + + we have to show that the set ∆J is closed. Let α, β ∈ ∆J = k=1 ∆k with α + β ∈ ∆k + + for all k = 1, . . . , r since ∆k is closed. This shows α + β ∈ ∆J . Until now we have seen that the existence of an integrable almost complex structure on a homogeneous space is equivalent to several other conditions. Combining the previous theorems we have proven the following.
31 Theorem 2.13. The following sets are in a one-to-one correspondence.
1. The set of all integrable invariant almost complex structures J on G/H.
2. The set of all Lie subalgebras q+ ⊂ gC where q+ and its complex conjugate q− span gC and have hC as their intersection.
+ + 3. The set of all closed subsets ∆J ⊂ ∆ \ ∆h where for all α ∈ ∆ \ ∆h the set ∆J contains exactly one root of ±α.
4. The set of all Weyl chambers in ˆt.
Proof. Theorem 1.34 shows the equivalence of 1. and 2. We have seen the equivalency of 2. and 3. in Theorem 2.8 and Remark 2.9. The last equivalency of 3. and 4. is proven in Theorem 2.12. For more details on the correspondence between 1., 3. and 4. the reader is referred to [BFR, Chapter 3].
Example 2.14. In the special cases of a compact group or a simple hermitian group then the last theorem simplifies and is described in the following, cf. [BH, Proposition 13.8].
· In the general compact case, i.e. G = K is a compact Lie group and H = T holds, we consider the manifold K/T . Thus the invariant complex structures are parametrized by the Weyl chambers in t. In the notion of the last theorem we have ˆt = t since ∆h = ∅. Moreover the number of Weyl chambers is given by the order of the Weyl group of K since it acts simply transitive on the set of Weyl chambers. On the other hand the set of Weyl chambers in t are in a one-to-one correspondence to the sets of + + positive roots ∆ (= ∆J ) and the Borel Lie algebras containing t. · In the simple hermitian case there are exactly two invariant complex structures on G/K. In the notion of the last theorem we have ˆt = z(k) which decomposes into two half rays since ∆h = ∆k and the center z(k) of k is one-dimensional. Furthermore + + there exist only two sets of positive noncompact roots ∆p (= ∆J ) and two (parabolic) Lie subalgebras satisfying the last theorem.
32 2.3 Parabolics q− and Q−
− C L C C L C L C C + − Let q = h ⊕ gα = t ⊕ gα ⊕ gα be a Lie subalgebra of g , i.e. ∆J resp. ∆J − − α∈∆h α∈∆J α∈∆J is closed and contains exactly one of the roots ±α ∈ ∆ \ ∆h. In this chapter we will study the Lie subalgebra q− and its corresponding analytic Lie subgroup Q− in more detail. In fact we will see that q− is indeed a parabolic subalgebra of gC.
− L C − C − Proposition 2.15. The subspace n := gα is an ideal in q = h ⊕ n . − α∈∆J Proof. We have to show the relation [n−, q−] ⊂ n−. Since the computation of the Lie bracket breaks down into taking sums of roots the proof follows from Proposition 2.7. In − particular the proof follows from the definition of root spaces, the closedness of ∆J and − − ∆h + ∆J ⊂ ∆J .
− C − Proposition 2.16. In the case H = T , i.e. ∆h = ∅, the subspace q = t ⊕ n is a Borel subalgebra, i.e. maximal solvable.
− h − − i − − Proof. We have to show that the derived series q(n+1) := q(n), q(n) with q(0) = q ter- − minates in the zero subalgebra. Because of Theorem 2.10 the set ∆J is a set of positive roots and defines a maximal solvable Lie subalgebra in gC. This subalgebra is equal to q−. − − − C − C − The first step in the derived series gives q(1) = [q , q ] t ⊕ n , t ⊕ n which lies in the ideal n− by the definition of root spaces. After a finite time of taking the brackets of these ( 0 falls α + β∈ / ∆− subspaces the solvability follows, since gC, gC = J . α β C − gα+β falls α + β ∈ ∆J The Lie algebra q− is maximal because of its dimension. Every Borel subalgebra has the C |∆| − dimension dim t + 2 , which is exactly the dimension of q .
Corollary 2.17. In the more general case of Proposition 2.16 where ∆h 6= ∅ holds, the subspace q− is a parabolic subalgebra, i.e. contains a Borel subalgebra.
− Proof. With Theorem 2.11 we see that ∆h ∪ ∆J contains a set of positive roots such that the constructed Lie subalgebra q− contains a maximally solvable Lie subalgebra, i.e. q− is parabolic.
Now we have seen that the subspace q− is a parabolic Lie subalgebra of gC. Using this notion we see that hC is the reductive Levi subalgebra and the subspace n− is the nilpotent radical of q− (see [OV, Chapter 6.1.3]). The following proposition states that every parabolic subgroup of GC is connected, hence it is generated by a parabolic Lie subalgebra q−.
Proposition 2.18 ([OV, Theorem 1.5., Chapter 6.1.3]). Any parabolic subgroup Q− of a connected complex Lie group G is connected and coincides with its normalizer, and G/Q− is a simply connected projective algebraic variety.
33 Remark 2.19. Using [Hu, Chapter 30.2] we get the Levi decomposition Q− = HC n N − and the fact that HC × N − → Q− is an isomorphism. Where hC (resp. n−) is the Lie algebra of HC (resp. N −). Furthermore the map exp : n− → N − is a diffeomorphism (see [Kn, Corollary 1.126, Theorem 1.127]).
Lemma 2.20. The parabolic subgroups Q+ and Q− intersect exactly in HC.
Proof. The subgroup HC is per definition both in Q+ and in Q−. Now we have to show the other inclusion. Using Remark 2.19 we get
{e} = N − ∩ N + = N − ∩ Q+
− + + − C + C − since n ∩ n = 0. Let q ∈ Q ∩ Q = (H n N ) ∩ (H n N ) hence q = l1n1 = l2n2 + − C − −1 + where n1 ∈ N , n2 ∈ N and li ∈ H . Now N 3 n2 = l2 l1n1 ∈ Q follows, so n2 = e − + because of {e} = N ∩ Q . Analogously n1 = e and l1 = l2 holds. Hence we have shown that Q+ ∩ Q− ⊂ HC
Corollary 2.21. Using the previous notations we get the identity G ∩ Q− = H.
Proof. The group H is a subgroup of both G and Q− hence the inclusion from the right hand side to the left hand side is clear. Now G ∩ Q− equals the fixed point set Fixσ Q− of Q− with respect to the involution σ. Now we claim that Fixσ Q− = Fixσ Q+ ∩ Q− holds. This equation holds because Q+ and Q− are connected parabolic subgroups and hence σ exchanges Q+ and Q− as it does on the Lie algebra level with q+ and q−. Now every fixed point of Q− lies also in Q+. Using Lemma 2.20 we get the following chain of equations.
G ∩ Q− = Fixσ Q− = Fixσ Q+ ∩ Q− = Fixσ HC = H.
34 2.4 The K¨ahler structure
X0 In order for a coadjoint orbit O := OX0 to be K¨ahlerianthe centralizer G of X0 must be compact since the centralizer lies in the isometry group of the induced Riemannian metric. Therefore we will discuss in this section the compatibility of the symplectic form ω and the almost complex structure J, the induced form g (·, ·) := ω (·,J·) and wether g is a Riemannian metric on O under the assumption of GX0 being compact. Using the notation of Chapter 1.5 the base point X0 is an element of comp g. Hence it is compact. Lemma 1.40 shows that we can assume that the base point lies in k. Furthermore we can assume that the base point X0 lies in t (cf. [Kn, Theorem 4.36]). In Chapter 2.2 we have seen how closed subsets of roots, parabolic Lie subalgebras of gC and integrable, G-invariant, almost complex structure are related. Using the notation of − Chapter 2.2 and given a closed subset ∆J we consider the Lie subalgebra
− C M C q = h ⊕ gα. − α∈∆J Then we can define a complex structure on the orthogonal complement m of h in the P C C following way. Let ξ ∈ m with ξ = + ξα + ξ−α where ξα ∈ g , ξ−α ∈ g . Now define α∈∆J α −α X J(ξ) = iξα − iξ−α. (7) + α∈∆J Having Chapter 1.3.2 in mind the map J vanishes on h and J 2 = − Id. Since σ(ξ) = ξ and Lemma 1.45 hold we get σ(ξα) = ξ−α and σ(iξα) = −iξ−α. Hence J restricts to a map on m. The following theorem shows that the natural symplectic structure on the coadjoint orbit is compatible with the complex structure we discussed in the previous chapters. Theorem 2.22. The almost complex structure J defined above is compatible with the canonical symplectic structure ω introduced in Theorem 2.1, i.e. ω(JX,JY ) = ω(X,Y ) for all X,Y ∈ X(O). Proof. Since both ω and J are invariant under the G-action we can reduce the proof to the base point X0. Hence it suffices to show the following equation. ! ∼ ωX0 (Jξ, Jη) = hX0, [Jξ, Jη]i = hX0, [ξ, η]i = ωX0 (ξ, η) for all ξ, η ∈ m = g/h. P In order to show this we do the following calculations first. Let ξ = α∈∆+ ξα + ξ−α, η = P J + ηβ + η−β ∈ m then β∈∆J " # P P [Jξ, Jη] = iξα − iξ−α, iηβ − iη−β α∈∆+ β∈∆+ P J J = − [ξα, ηβ] − [ξ−α, η−β] + [ξ−α, ηβ] + [ξα, η−β] . + α,β∈∆J
35 In the same way we get P [ξ, η] = [ξα, ηβ] + [ξ−α, η−β] + [ξ−α, ηβ] + [ξα, η−β] . + α,β∈∆J
Now we consider the contraction hX0, ·i of the Killing form with the base point X0. Since the root space decomposition of gC is orthogonal with respect to the Killing form we are only interested in the case where the Lie brackets [ξα, ηβ] lies in t. This is exactly the case when α = −β (see [He, Chapter III, Theorem 4.2]). Inserting α = β in the previous + equations because ∆J contains only one of the roots of ±α ∈ ∆ \ ∆h we get the following equation.
ωX (Jξ, Jη) = hX0, [Jξ, Jη]i 0 P = hX0, − [ξα, ηβ] − [ξ−α, η−β] + [ξ−α, ηβ] + [ξα, η−β]i α,β∈∆+ P J = hX0, − [ξα, ηα] − [ξ−α, η−α] + [ξ−α, ηα] + [ξα, η−α]i α∈∆+ | {z } | {z } J ∈/t ∈/t P = hX0, [ξα, ηα] + [ξ−α, η−α] + [ξ−α, ηα] + [ξα, η−α]i α∈∆+ PJ = hX0, [ξα, ηβ] + [ξ−α, η−β] + [ξ−α, ηβ] + [ξα, η−β]i + α,β∈∆J = hX0, [ξ, η]i
= ωX0 (ξ, η).
Remark 2.23. With similar computations as in the proof of Theorem 2.22 we can check that J commutes with ad ξ on m for all ξ ∈ ∆h. More explicitely
J [ξ, η] = [ξ, Jη]
holds for all η ∈ m. Furthermore we see once again that the almost complex structure J is integrable in terms of (3) in Chapter 1.3.2.
Given the canonical symplectic form ω and a compatible integrable almost complex structure J we can define a nondegenerate, symmetric, bilinear form g by
g(X,Y ) = ω(X,JY )
for all X,Y ∈ X(O). Nondegenerency and bilinearity follow because ω has the same properties and J is a linear isomorphism. Symmetry follows by the antisymmetry of ω, the compatibility of ω and J and the property J 2 = − Id. For more detail on the topic of K¨ahlermanifolds and compatible triples the reader is referred to [Sil, Chapters 12, 13] or [Be, Chapter 2.4]. The following theorem shows under which circumstances the form g is also positive definite.
36 Theorem 2.24. The induced bilinear form g on OX0 is positive definite, hence a Rie- + mannian metric, if and only if −iα(X0) < 0 for α ∈ ∆k ∩ ∆J and −iα(X0) > 0 for + α ∈ ∆p ∩ ∆J . P Proof. Let ξ = + ξα + ξ−α. First we compute the following identity using the same α∈∆J ideas as in the proof of Theorem 2.22. " # P P [ξ, Jξ] = ξα + ξ−α, iξβ − iξ−β α∈∆+ β∈∆+ P J J = i ([ξα, ξβ] − [ξα, ξ−β] + [ξ−α, ξβ] − [ξ−α, ξ−β]) . + α,β∈∆J Then we get the following continued equation:
gX (ξ, ξ) = ωX (ξ, Jξ) = hX0, [ξ, Jξ]i 0 0 P = hX0, i ([ξα, ξβ] − [ξα, ξ−β] + [ξ−α, ξβ] − [ξ−α, ξ−β])i α,β∈∆+ P J = hX0, i (− [ξα, ξ−α] + [ξ−α, ξα])i α∈∆+ P J = −2ihX0, [ξα, ξ−α]i α∈∆+ PJ = −2ihX0, [ξα, σ(ξα)]i α∈∆+ PJ = −2ih[X0, ξα] , σ(ξα)i α∈∆+ PJ = −2ihα(X0)ξα, σ(ξα)i α∈∆+ PJ = −2iα(X0)hξα, σ(ξα)i. + α∈∆J
Using Lemma 1.52 we know that hξα, σ(ξα)i < 0 if α is a compact root and hξα, σ(ξα)i > 0 if α is noncompact. So the positivity of the bilinear form depends on the position of X0 ∈ t. Now we conclude that the bilinear form is positive definite if and only if −iα(X0) < 0 for + + α ∈ ∆k ∩ ∆J and −iα(X0) > 0 for α ∈ ∆p ∩ ∆J .
Remark 2.25. The set ∆h ⊂ ∆k does not contain any noncompact roots hence the inter- + + section ∆p ∩ ∆J is in fact a set of positive noncompact roots ∆p in the notion of Chapter + 1.6.1. Using Proposition 2.7 with h = k we get that the closedness of ∆p is equivalent to ˜ + − + the existence of a k-adapted positive root system. The set of roots ∆J = ∆k ∩ ∆J ∪ ∆p is not necessarily closed. If it is not closed it does not define a Weyl chamber in ˆt and we lose the property of positivity. If it is closed it defines another Weyl chamber in ˆt. Hence we can determine if a coadjoint orbit is a K¨ahlermanifold by looking at the Weyl + ˜ + chamber in which X0 lies and at the closedness of ∆J and ∆J .
37 + − + Corollary 2.26. Let ∆ = ∆h ∪ ∆J ∪ ∆J be a disjoint decomposition and let ∆J be closed. ˜ + + + Then the set ∆J is closed if and only if ∆p = ∆p ∩ ∆J is closed.
˜ + + + Proof. Let ∆J be a closed set of roots. Now the closedness of ∆p = ∆p ∩ ∆J follows because of ˜ + + − + + + ∆J ∩ ∆J = ∆k ∩ ∆J ∪ ∆p ∩ ∆J ∩ ∆J = ∆p ∩ ∆J and the intersection of closed sets is closed. + + − Conversely let ∆p = ∆p ∩ ∆J be closed. Then ∆k ∩ ∆J is also closed as the intersection of closed sets. Now we have to show that the union of these two sets is closed. Using + Proposition 2.7 the set ∆k ∪ ∆p ∩ ∆J is closed. The claim follows with the last property of Proposition 2.7.
Now we have equipped the coadjoint orbit OX0 with an integrable almost complex + structure J which is given by a set of roots ∆J and compatible with the canonical sym- plectic structure ω such that the induced Riemannian metric g is positive definite. In other
words OX0 is also a K¨ahler manifold. Moreover we get the following theorem.
Theorem 2.27. Let X0 ∈ t. There exists a K¨ahlerstructure on a (co-)adjoint orbit + (OX0 , ω) if and only if there exists a positive, Wk-invariant set of noncompact roots ∆p 0 + + such that X0 ∈ Cmax(∆p ), i.e. −iα(X0) > 0 for all α ∈ ∆p .
Proof. In Theorem 2.24 we have seen where the point X0 has to be with respect to the + complex structure given by the set of roots ∆J . This shows the ”only if” part. Conversely + + if X0 ∈ Cmax(∆p ) where ∆p is a positive, Wk-invariant set of noncompact roots then the roots ∆h which vanish on X0 correspond to a compact Lie subgroup H of G. Moreover denote by C a Weyl chamber in Cmax which contains X0 and corresponds to the adapted + + + ˜ + positive root system ∆ = ∆k ∪∆p . Using Corollary 2.26 we see that ∆ is also a positive set of roots which corresponds to another Weyl chamber Ce in Cmax. Now the set of roots ˜ + ∆ \ ∆h satisfies the requirements of Theorem 2.24 with respect to the point X0 and the respective adjoint orbit OX0 is K¨ahler. Remark 2.28. In this chapter we concentrated on the complex structure J given by the + set of roots ∆J and where the base point X0 ∈ t is located with respect to this roots. In other words X0 has to be an element of a Weyl chamber which corresponds to an adapted ˜ + ˜ + ˜ + positive set of roots ∆ with ∆J ⊂ ∆ . The chronological order is described in the following diagram. + + ˜ + ˜ + J → ∆J ⊂ ∆ → ∆J ⊂ ∆ → C 3 X0. In Chapter 3 the point of view is different and the notation might change slightly. There + the point X0 lies in a Weyl chamber which corresponds to a positive set of roots ∆ and ˜ + the complex structure J is given by the set ∆J . In terms of a diagram the order changes in the following way.
+ + ˜ + ˜ + X0 ∈ C → ∆J ⊂ ∆ → ∆J ⊂ ∆ → J.
38 Remark 2.29. In the compact case, the existence of a positive definite metric in Theorem 2.24 is equivalent to the existence of a positive system of roots since the set of noncompact roots ∆p is empty. Thus the maximal cone Cmax is equal to the whole Cartan subalgebra t and with Theorem 2.27 every compact coadjoint orbit is a K¨ahlerianmanifold. In the noncompact case we discuss holomorphic coadjoint orbits. Let G be a real semisimple connected noncompact Lie group with finite center then there exist holomorphic coadjoint orbits if G/K is a Hermitian symmetric space, where K is a maximal compact subgroup of G. Furthermore holomorphic coadjoint orbits are a generalization of Hermitian symmetric spaces G/K.
2.5 Open embedding ∼ The goal of this section is to show that the coadjoint orbits OX0 = G/H which are K¨ahler manifolds can be embedded as open subsets into a homogeneous space GC/Q− where GC is the complexification of G in the sense of [Ho] and Q− a parabolic subgroup given by the − integrable almost complex structure J on OX0 , i.e. by the induced Lie subalgebra q . In the special case of H = K, i.e. in the Hermitian case, a proof of such an embedding is given in [He, Proposition 7.14].
Lemma 2.30. The map
φ : g/h → gC/q−, ξ + h 7→ ξ + q−
is a complex linear isomorphism.
Proof. Looking at the decompositions
C − L C g = q ⊕ gα + α∈∆J and L g = h ⊕ g[α] + α∈∆J = h ⊕ m the spaces g/h and gC/q− have the same dimension and the map φ is well defined because the inclusion h ⊂ q− holds. Since we can identify the space g/h with the subspace m, P complex linearity follows with the next calculation. Let ξ ∈ m with ξ = ξα + ξ−α and + α∈∆J C ξα ∈ gα then
39 ! P φ(J(ξ)) = φ iξα − iξ−α + α∈∆J P − = iξα − iξ−α + q + α∈∆J P − = iξα + q + α∈∆J P − = i ξα + q + α∈∆J P − = i ξα + ξ−α + q + α∈∆J = iφ(ξ). Here we have used the complex structure introduced in Chapter 2.4 equation (7). Injectivity and hence bijectivity follows since ! − C L C L C g ∩ q = g ∩ t ⊕ gα ⊕ gα − α∈∆h α∈∆ L J = t ⊕ g[α] α∈∆h = h holds. In particular the inverse is given by the map η + q− 7→ η + σ(η) + h. This shows the lemma. Corollary 2.31. The homogeneous spaces G/H and GC/Q− have the same dimension. In C − + particular dimR G/H = dimR G /Q = 2 ∆J . The previous Lemma 2.30 has shown that the tangent spaces of the two homogeneous spaces are isomorphic to each other. The next theorem will use this to construct a G- invariant embedding. Theorem 2.32. The map
Φ: G/H → GC/Q−, gH 7→ gQ− is a G-equivariant open holomorphic embedding. Proof. The set Q− is a closed Lie subgroup of GC such that the quotient GC/Q− is a complex manifold. Furthermore the quotient map GC → GC/Q− is holomorphic and the inclusion map G → GC is differentiable.
GC / GC/Q− O :
? G
Hence the map G → GC/Q− is well defined and differentiable. Since H ⊂ Q− the map is also invariant under the H action such that the induced map Φ : G/H → GC/Q−, gH 7→
40 gQ− is well defined and differentiable. The map Φ is G-equivariant by construction because G acts on itself and on its complexification GC via left multiplication. The group G acts on the orbit O ∼= G/H transitively, so the map Φ has constant rank. Using Lemma 2.30 we observe that the map Φ has full constant rank since φ = Φ∗(eH) is an isomorphism. Furthermore it is holomorphic and a local diffeomorphism. So the image of the map Φ is an open G-orbit in GC/Q−. Injectivity is equivalent to the equation G ∩ Q− = H and follows by Corollary 2.21. So we have shown that Φ is a diffeomorphism and then biholomorphic onto its image.
41 3 Coadjoint orbihedra
From now on if not stated otherwise we will consider a connected, compatible and quasi- hermitian Lie group G. Since its Lie algebra g is quasihermitian it contains a θ-invariant compact Cartan subalgebra t. In this chapter we will study its K¨ahleriancoadjoint orbits and their projection onto the compact Cartan subalgebra t. We will generalize the result of [BGH] which are the following. First we show that the faces of the convex hull Ob of the coadjoint orbit are exposed and then we bring the faces of Ob in a one-to-one correspondence with the faces of its projection. In addition we will prove that these faces are also in one-to-one correspondence to the faces of the convex hull of the compact homogeneous space (see section 2.5) in which O is embedded into.
3.1 Momentum map In this section we consider momentum maps and components of this maps in particular. We will discuss under which circumstances components of the momentum map has connected fibers. For a more detailed description on the topic of momentum maps the reader is referred to [Be, Chapter 4] or [Sil, Parts VIII, X].
Definition 3.1. Let (M, ω) be a symplectic G-manifold, where G acts by symplectomor- phisms. A map µ : M → g∗ is called a momentum map for the group G if for all Y ∈ g we have Y dµ = ιY M ω. Here Y M denotes the fundamental vector field on M associated to Y .
In the setting of coadjoint orbits the momentum map µ : O ,→ g∗ is given as the inclusion. The momentum map with respect to the torus action is then given by restriction to the torus. Identifying the Lie algebra g with its dual g∗ the momentum map is the inclusion with respect to the G-action. The momentum map to the torus is then the projection along [t, g].
3.1.1 Connected Fibers After a K-equivariant symplectomorphism the momentum map µ on O ∼= K · X × p, with respect to the adjoint K-actions on O and K · X × p, has the following form for Y ∈ t, ξ ∈ K · X and all v ∈ p (see [De]):
Y 1 µK·X×p(ξ, v) = hξ, Y i + 2 Ωp (v, [Y, v]) , where the bracket h·, ·i denotes the Killing form and Ωp is a K-invariant symplectic form on p. The first summand on the right hand side is the momentum map on the compact
42 coadjoint orbit K · X. In order to bring the second summand, which is a quadratic form in v, into a diagonal form with respect to a chosen basis we fix an element Z0 ∈ z(k) such + that every root in ∆p is positive on Z0. Furthermore we use that θ|p = − Id, v ∈ p and X X v = λαeα + θ(λαeα) = λαeα + λαe−α = λα,1(eα + e−α) + λα,2i(eα − e−α), + + α∈∆p α∈∆p
C where eα ∈ gα and {eα + e−α, i(eα − e−α)} is an orthogonal basis of g[α] with respect to the Killing form. Then we get the following continued equation.
Y 1 µK·X×p(ξ, v) = hξ, Y i + 2 Ωp (v, [Y, v]) 1 = hξ, Y i − 2 hv, θ ([Z0, [Y, v]])i 1 = hξ, Y i + 2 hv, [Z0, [Y, v]]i 1 = hξ, Y i + 2 h[v, Z0] , [Y, v]i 1 = hξ, Y i − 2 h[Z0, v] , [Y, v]i * + 1 P P = hξ, Y i − 2 α(Z0)(λαeα − λαe−α), α(Y )(λαeα − λαe−α) + + α∈∆p α∈∆p 1 P = hξ, Y i − 2 α(Z0)α(Y ) λαeα − λαe−α, λαeα − λαe−α + α∈∆p 1 P = hξ, Y i − 2 (−iα(Z0))(−iα(Y )) i(λαeα − λαe−α), i(λαeα − λαe−α) + α∈∆p 1 P 2 = hξ, Y i − 2 (−iα(Z0))(−iα(Y )) λα,2 heα + e−α, eα + e−αi + α∈∆p 2 + λα,1 hi(eα − e−α), i(eα − e−α)i .
Remark 3.2. The momentum map on K · X × p is the sum of the momentum map on the compact orbit K · X and a quadratic form. Notice that the quadratic form is already in diagonal form and that its signature (n+, n−, n0) depends on the position of Y in t with respect to the noncompact roots. Furthermore the last equation shows that the numbers n+, n− and n0 are all even.
n Lemma 3.3. Let Q be a quadratic form on R with signature (n+, n−, n0) where n+ and n− are not 1. Then every fiber of Q is connected.
Proof. We may assume that n0 = 0 since a cylinder of a connected base set is also con- nected. Moreover positive scalars do not change the connectedness of the fibers such that 2 2 2 2 we may assume that Q is of the form x1 + ... + xn+ − (y1 + ... + yn− ). So the point (x, y) ∈ Rn+ × Rn− lies in Q−1(c) if and only if
2 2 2 2 x1 + ... + xn+ − (y1 + ... + yn− ) = c
43 holds. If (n+, n−) equals (n, 0) or (0, n) then all fibers are either spheres, points or the empty set hence connected. −1 For arbitrary (n+, n−) if c = 0 then for every (x, y) ∈ Q (0) the line R · (x, y) lies also in Q−1(0) hence the fiber is connected. Now let n+ > 1 and c > 0. For every fixed y, e.g. y = 0, we get a connected n+ −1 sphere in R since n+ > 1. Let (x, 0), (˜x, y˜) ∈ Q (c) withy ˜ 6= 0. The curve γ(t) = √ √ ky˜k x˜ y˜ x˜ √ c cosh t kx˜k , sinh t ky˜k connects γ(0) = c kx˜k , 0 and γ arcsinh c = (˜x, y˜). Hence there is a curve connecting (x, 0) and (˜x, y˜) which shows that the fiber is connected. The case c < 0 is analogous to the previous case with n− > 1.
−1 Remark 3.4. If n+ = 1 or n− = 1 there is at least a fiber Q (c) which is not connected since a coordinate hyperplane disconnects two components of the fiber.
n Proposition 3.5. Let Q be a quadratic form on R with signature (n+, n−, n0) where n+ 6= 1 and n− 6= 1, M a compact coadjoint orbit with a function f : M → R, which is a component of the momentum map. Then every fiber of f + Q is connected in M × Rn. Proof. Since f is a component of the momentum map on M there exists a source and a sink, i.e. there is an open subset of points, for which the gradient flow Φ : R×M → M flows into the critical submanifold with minimal critical value as t → ∞ and into the critical submanifold with maximal critical value as t → −∞. After a monotone reparametrization [−∞, ∞] → [0, 1] and fixing a point X ∈ M, which is not critical with respect to f, the map ΦX : [0, 1] → M, ΦX (t) := Φ(t, X) connects the source and the sink and crosses every fiber of f. Furthermore, being a component of the momentum map, every fiber of f is connected −1 hence for every two points m1, m2 in M we can connect m1 with ΦX (t1) ∈ f (f(m1)) in −1 −1 −1 the fiber f (f(m1)) and m2 with ΦX (t2) ∈ f (f(m2)) in the fiber f (f(m2)). Thus using the map ΦX we can construct a continuous curve c connecting the two points such that the function f ◦c is monotone. Note that in general f ◦c is constant in a neighborhood of t = 0 and t = 1 and strictly monotone otherwise. A sketch of this situation is given in Figure 6.
44 Figure 6: Sketch of the connecting curve c
Now consider the continuous curve γ(t) = (c(t),X(t),Y (t)). Here X(t) is a curve in Rn+ ( pf ◦ c(t), 0,..., 0 if f ◦ c(t) > 0 and Y (t) a curve in Rn− which are given by X(t) = 0 else ( p−f ◦ c(t), 0,..., 0 if f ◦ c(t) < 0 and Y (t) = . It follows that the curve γ lies in 0 else the fiber (f + Q)−1(0), i.e. f ◦ c(t) + kX(t)k2 − kY (t)k2 = 0. We are only considering the zero fiber since we may translate f with a constant. Now for any two points (Z,X,Y ), (Z0,X0,Y 0) ∈ (f +Q)−1(0) the previous Lemma 3.3 shows that we find curves connecting the point (Z,X,Y ) with (Z,X(0),Y (0)) and (Z0,X0,Y 0) with (Z0,X(1),Y (1)). Hence the fiber is connected.
Corollary 3.6. The fibers of the momentum map component µY on K ·X ×p are connected for all Y ∈ k.
45 3.2 Face structure In this chapter the momentum map will be used to describe the coadjoint orbit. More precisely we will give a description of the critical set of a component function µY of the momentum map. This will lead to the study of its maxima and then the induced exposed faces of Ob. Ultimately we will verify that any face of Ob is an exposed face.
3.2.1 Faces as orbihedra Let U be a compact Lie group and let G ⊂ U C be a semisimple compatible connected subgroup with real quasihermitian Lie algebra g. We fix a θ-invariant compact Cartan + subalgebra t and a set of positive noncompact roots ∆p . Hence we fix Cmax and Cmin in t. Definition 3.7.
1. An adjoint orbit OX in g is called admissible if there exists an adapted positive root system such that X ∈ Cmax.
2. An adjoint orbit OX in g is called strictly admissible if there exists a closed convex invariant subset C ⊂ g with lin C = {0} and X ∈ relint(C).
Furthermore [Ne, Lemma VIII.1.27] shows that if an orbit is strictly admissible then it is also admissible. Most of the time we will consider a strictly admissible coadjoint orbit. Then we will use the last definition where C is a pointed generating invariant cone in a hermitian Lie algebra. Hence the orbit OX is strictly admissible if and only if X lies in the 0 interior of the maximal cone Cmax. The existence of such cones follow from Proposition 1.71 and Theorem 1.74.
Example 3.8. In the case of Sp(2, R) we have seen in Examples 1.63 and 1.73 and Figure 5 that the cone Wmax is an elliptic cone in sp(2, R). The orbits in Wmax which are given by connected components of the two-sheeted hyperboloids are exactly the strictly admissible orbits in sp(2, R). In view of the following definition and lemma one can see in Figure 5 0 that the convex hull of elliptic orbits, i.e. the orbits in Wmax, has the orbit as the set of extreme points. The other orbits do not have a similar property. There are the nilpotent orbits and the one-sheeted hyperboloids. The former type of orbits and their convex hull are not closed. In this case the convex hull is Wmax \{0} and there are no extreme points. The convex hull of the latter type of orbit is the whole space sp(2, R) hence there do not exist any extreme points or faces. These circumstances reveal one main reason that distinguishes the noncompact with the compact setting. In the compact case every orbit has in this sense the properties of an elliptic orbit. The noncompact orbits on the other hand diversify into different classes of orbits with each class having distinct properties.
46 Definition 3.9. An orbihedron Ob of G in a reductive quasihermitian Lie algebra g is the convex envelope Ob := conv O of the adjoint orbit O with the following properties. It divides into a compact orbit O1 in the compact part and a strictly admissible orbit ∼ O2 in the hermitian part with O = O1 + O2 = O1 × O2 with respect to the ideal sum decomposition of g.
Lemma 3.10. We have exp Ob = ext Ob = O and ext F = F ∩ O for any face F of Ob.
Proof. For the first assertion we recall that conv O = conv O1 + conv O2 (see [Schn, Theo- rem 1.1.2]). Hence it suffices to show the assertion for the compact orbit and the strictly admissible orbit using Lemma 1.11. The convex envelope of the compact orbit is an or- bitope and hence the assumption follows from [BGH, Lemma 16, Theorem 38]. For the strictly admissible orbit we refer to [Ne, Proposition VIII.1.30]. The second assertion follows from Lemma 1.4.
In this work we are among other things interested if any face of Ob is exposed. The previous lemma shows this for the faces that are points. We start to consider the exposed faces of an orbihedron Ob. Let Y be a nonzero vector in g. Since the momentum map µ is just the inclusion O ,→ g the function µY in the adjoint setting is given by µY (X) := hX,Y i , where h·, ·i denotes the Killing form on g. Define the set of maximal value on O with respect to µY by n o Max(Y ) := X ∈ O µY (X) = max µY . O In the case of noncompact Lie groups it is not clear for which Y ∈ g the set Max(Y ) is nonempty since the orbit might be unbounded. The main result is the following.
Theorem 3.11. Let Max(Y ) 6= ∅. Then Y ∈ G · Cmax ⊂ Wmax and Max(Y ) is a connected GY -orbit. In particular it is a (GY )0- orbit.
Later we will see that if Y is an element in G · Cmax then the set Max(Y ) is nonempty. If the elements X and Y lie in the same torus then we will also be able to make statements about their relative positions with respect to their respective Weyl chambers. In order to prove Theorem 3.11 we will first do some preparatory work. Let g be a real semisimple Lie algebra with Cartan decomposition g = k ⊕ p and t ⊂ k a compact Cartan subalgebra of g. We denote by W := Wk = W (k, t) = W (K,T ) = W (G, T ) the Weyl group of t in G. Furthermore if C is a Weyl chamber with respect to an adapted + + + positive root system ∆ = ∆k ∪ ∆p we denote by C− the Weyl chamber associated to ˜ + − + ∆ = ∆k ∪ ∆p recalling the notation at the end of Chapter 2.4.
47 Lemma 3.12. If X,Y ∈ Cmax then there exists a Weyl chamber C such that X ∈ C and Y ∈ C− if and only if (−iα(X))(−iα(Y )) ≥ 0 for all α ∈ ∆p and (−iα(X))(−iα(Y )) ≤ 0 for all α ∈ ∆k. Proof. Let ∆+ be the set of positive roots associated to the Weyl chamber C. Since + + + X ∈ Cmax the set of positive roots decomposes into ∆ = ∆k ∪ ∆p . The ”only if” part follows from the definition of C− and the fact that Y ∈ C− holds. To prove the ”if” part we choose a set of positive roots ∆+ such that −iα(X) ≥ 0 for every α ∈ ∆+. Since X is + + + + contained in Cmax, the set ∆ decomposes into ∆ = ∆k ∪ ∆p . We proceed by induction + + on n := #{α ∈ ∆k | − iα(Y ) > 0}. If n = 0 then the Weyl chamber C given by ∆ satisfies X ∈ C and Y ∈ C−. Otherwise if n is greater than zero then by the assumption + there exists a simple compact root α0 ∈ ∆k with −iα0(Y ) > 0 and −iα0(X) = 0. If σ is the reflection associated to the simple compact root α0, then σ(C) is the Weyl chamber ˆ + + + associated to ∆ := {α ◦ σ|α ∈ ∆ } = {α0 ◦ σ}∪˙ (∆ \{α0}) (see [Hu2, Chapter 10.2, ˆ + Lemma B]). Since σ(X) = X we have −iα(X) ≥ 0 for every α ∈ ∆ while −iα0 ◦σ(Y ) < 0. ˆ + It follows that #{α ∈ ∆k | − iα(Y ) > 0} = n − 1. By the induction hypothesis there is a τ ∈ W such that X ∈ τσ(C) and Y ∈ τσ(C)−.
Lemma 3.13. Let C be a Weyl chamber in Cmax and let X ∈ C and Y ∈ C−. Let 0 0 0 0 0 X ∈ Wk · X then there exists a Weyl chamber C such that X ∈ C and Y ∈ C− if and 0 only if there exists an element w ∈ Wk such that w · X = X and w · Y = Y hold.
Proof. The ”if” part follows from the definition of a Weyl chamber and the Wk-invariance + 0 of ∆p such that C := w(C) satisfies the conditions. 0 0 0 Assume the existence of a Weyl chamber C such that X = σX ∈ C for some σ ∈ Wk 0 0 and Y ∈ C−. Let w be in Wk such that w(C) = C since all considered Weyl chambers lie −1 0 in Cmax and Wk acts transitively on the set of these Weyl chambers. The points w X = −1 −1 0 w σX and X belong to C and the Weyl orbit Wk · X. Hence w X = X (see [Hu2, 0 0 −1 Chapter 10.3, Lemma B]), i.e. X = wX. Furthermore w(C−) = C− and w Y and Y belong to C−. Thus also wY = Y holds. This concludes the proof. Proposition 3.14. Let G be a real connected semisimple Lie group with Lie algebra g, t a compact Cartan subalgebra and X ∈ comp (g) ∩ t.
Y Y 0 1. Let Y ∈ g. If Y = g · ξ ∈ G · t then the equation OX ∩ g = OX ∩ (G ) · g · t holds. Y Otherwise OX ∩ g is empty.
Y Y 2. Let Y ∈ t and Wk := {w ∈ Wk|w · Y = Y }. Then for any w ∈ Wk there is a k ∈ (KY )0 such that Ad k t = t and Ad k Z = w · Z for every Z ∈ t.
Proof. 1. Let us first assume that Y in g is not G-conjugate to an element in t. Assume Y that there is an element η ∈ OX ∩ g . Since η lies in OX there is a g ∈ G with Ad g η ∈ t. Ad g Y Ad g η Hence Ad g η ∈ OX ∩ g and Ad g Y ∈ g . Furthermore Ad g Y∈ / k otherwise it would be conjugate to an element in t. It follows that gAd g η 6⊂ k is not compact which contradicts Ad g η ∈ OX and X ∈ comp(g).
48 Before we treat the case Y = g · ξ we assume Y = ξ ∈ t after conjugation (see [Kn, Propositions 6.59, 6.61]), i.e. g = e. The inclusion from the right to the left is clear since t ⊂ gY and gY is invariant under the action of GY . We will see that t is (modulo the Y Y Y Y center) a Cartan algebra in g . Let g = z ⊕ gss be the decomposition of g into its center 0 0 Y and its semisimple part. Analogously t = z ⊕ t with t ⊂ gss. Now t0 is nilpotent, as a subalgebra of t which is abelian, and an easy calculation shows 0 Y Y that it is selfnormalizing. Hence t is a compact Cartan algebra of gss. Let ξ ∈ OX ∩ g ξ Y ξ Y ξ with ξ = ξz + ξss. Since ξ lies in OX the centralizer G is compact. Also (G ) = G ∩ G is compact. Since ξz lies in the center, the element ξss has the same compact centralizer in Y Y Y Y G as ξ. The group Gss, which is the analytic subgroup of G with Lie algebra gss, is a Y closed subgroup of G (see [Kn, Corollary 7.11]). Hence the centralizer of ξss is compact Y Y Y in Gss, i.e. ξss ∈ comp(gss). After conjugation with an element g ∈ Gss we can assume 0 Y that ξss lies in the compact Cartan algebra t ⊂ gss. Consequently ξ ∈ t after conjugation with an element of (GY )0. Now we have shown the equation. The set is also not empty Y 0 0 since t ⊂ g and OX intersects t. Using this result let Y = g · Y with Y ∈ t. Then we get the following continued equation
Y Y 0 Y 0 0 Y 0 0 −1 Y 0 OX ∩g = g· OX ∩ g = g· OX ∩ (G ) · t = OX ∩g·(G ) ·g ·g·t = OX ∩(G ) ·g·t.
Y Y 2. Now Wk = W (k , t) equals the algebraically defined Weyl group which is equal to the analytically defined Weyl group for the compact connected Lie group (KY )0 (see [Kn, Proposition 2.72, Theorem 4.54]). Hence there is a k ∈ N(KY )0 (t) with the desired property.
Lemma 3.15. Crit(µY ) = O ∩ gY .
Proof. Let X ∈ O be a critical point of µY . Then for every η ∈ g we get
d Y d 0 = µ (Ad exp tη X) = hAd exp tη X, Y i = had ηX, Y i = hη, ad X(Y )i . d t 0 d t 0 This is equivalent to ad X(Y ) = 0 since the Killing form is nondegenerate. Hence X lies in gY . Conversely if X lies in O ∩ gY then ad X(Y ) = 0 holds and with the above equation X is a critical point of µY .
Proposition 3.16. Let G be a real connected semisimple Lie group with Lie algebra g, t a θ-invariant compact Cartan subalgebra, X ∈ comp (g) ∩ t and Y ∈ g · t for some g ∈ G. Then Y Y 0 Y 0 Crit(µ ) = (G ) · g · Wk · X = (G ) · g · NG(t) · X. Proof. Lemma 3.15 shows the equality of the critical points of µY with the set O ∩ gY . Together with Proposition 3.14 we get Crit(µY ) = O ∩ gY Y 0 = OX ∩ (G ) · g · t.
49 Using the G-invariance of OX and the fact that the Weyl group Wk is of the form Wk = NK (t)/ZK (t) = NG(t)/ZG(t) (see [Kn, 7.92-93, p. 489]) we then get Y Y 0 Crit(µ ) = (G ) · g · (OX ∩ t) Y 0 = (G ) · g · Wk · X Y 0 = (G ) · g · NK (t) · X Y 0 = (G ) · g · NG(t) · X.
Y From now on we will only consider momentum map components µ : OX → R, where the element Y lies in the compact Cartan subalgebra t. We are interested in maxima and hence in critical points of µY . In order for the function µY to have critical points the element Y has to be conjugate to an element in t (see Lemma 3.15, Proposition 3.14). Moreover the proof of Proposition 3.14 showed that the critical set of µY and µAd g Y are just conjugates to each other by g. Thus we assume that Y lies in comp(g) ∩ t. Proposition 3.17. Let G be a real connected semisimple Lie group with Lie algebra g and t a compact Cartan subalgebra of g. Assume that X ∈ comp(g) ∩ t and Y ∈ t. Then Y X is a local maximum of µ : OX → R if and only if X lies in the interior of Cmax for an + adapted system ∆ and there exists a Weyl chamber C ⊂ t such that X ∈ C and Y ∈ C−.
C C P C Proof. Let ∆ be the set of roots of (g , t ) and let ξ = ξ0 + α∈∆ ξα with ξα ∈ gα. Fix a + + + + set of positive roots ∆ = ∆k ∪ ∆p such that −iα(X) ≥ 0 for every α ∈ ∆ . The positive set of roots is apriori not necessarily adapted but we will show that it has to be in order to have a local maximum. We have M g = t ⊕ g[α]. α∈∆+ ( ∼ g[α] if α(X) 6= 0 Since TX O = g · X = [g,X] and [g[α],X] = . We have 0 if α(X) = 0
M ∼ OX p ⊂ g[α] = TX O, with ξ 7→ ξ (X). −iα(X)>0 The inclusion of p follows from the fact that X lies in comp(g) and hence every noncompact O root does not vanish on X. If w ∈ TX O, choose ξ ∈ g such that w = ξ (X) = [ξ, X] and set γ(t) := Ad(exp(tξ))X. Then γ(0) = X, γ˙ (t) = [ξ, γ(t)], γ¨(0) = [ξ, [ξ, X]] and
2 2 Y d Y D µ (X)(w, w) = 2 µ (γ(t)) = hγ¨(0),Y i = h[ξ, [ξ, X]],Y i = − h[ξ, X], [ξ, Y ]i . d t 0
P C We can assume that ξ = −iα(X)>0 ξα +ξ−α with ξα ∈ gα and θ(ξα) = ξ−α. This determines ξα uniquely and X [X, ξ] = α(X)zα −iα(X)>0
50 where zα = ξα − ξ−α. Moreover the vectors zα are orthogonal to each other, izα ∈ k for α P compact and izα ∈ p for α noncompact. Similarly [Y, ξ] = α∈∆+ α(Y )zα. So D2µY (X)(w, w) = − h[ξ, X], [ξ, Y ]i * + P P = − α(X)zα, α(Y )zα −iα(X)>0 −iα(X)>0 P = − α(X)α(Y ) hzα, zαi −iα(X)>0 P = − (−iα(X))(−iα(Y )) hizα, izαi −iα(X)>0 P = − (−iα(X))(−iα(Y )) hizα, izαi α∈∆+ | {z } | {z } p >0 ≥0 P − (−iα(X))(−iα(Y )) hizα, izαi . −iα(X)>0 | {z } | {z } + >0 ≤0 α∈∆k
( + + < 0 if α ∈ ∆p If there is an α in ∆ such that (−iα(X))(−iα(Y )) + , then X is not a > 0 if α ∈ ∆k local maximum. Otherwise the Hessian is negative semidefinite where the Hessian D2µY (X)(w, w) equals 2 Y zero if and only if zα 6= 0 implies α(Y ) = 0. This means that the kernel of D µ (X) is Y Y g · X = TX Crit(µ ). So the Hessian is degenerate only along the critical submanifold and is negative definite in the transverse direction. It follows that X is a local maximum. Furthermore if the Hessian is negative definite then every root in ∆˜ + must be positive on + Y . Corollary 2.26 shows that ∆p is a closed set of roots and Proposition 2.7 in the case of + + ˜ + h = k shows that ∆p is invariant under the Weyl group Wk hence ∆ and ∆ are adapted positive sets of roots. Moreover X and Y lie in a maximal cone Cmax. Summing up we have shown that X is a local maximum point of µY if and only if (−iα(X))(−iα(Y )) ≥ 0 + + for α ∈ ∆p and (−iα(X))(−iα(Y )) ≤ 0 for α ∈ ∆k . By Lemma 3.12 this is equivalent to the condition that there is a Weyl chamber C such that X ∈ C and Y ∈ C−.
Theorem (3.11). Let Max(Y ) 6= ∅. Then Y ∈ G·Cmax ⊂ Wmax and Max(Y ) is a connected GY -orbit. In particular it is a (GY )0- orbit. Proof of Theorem 3.11. We start assuming that G is semisimple. The case where G is reductive and might have a center will be dealt with later in this proof. Let E be the set of all local maxima of µY . Since the function µY is GY -invariant, the sets E and Max(Y ) are GY -invariant. Using the assumption that the set Max(Y ) is nonempty there exists an element X ∈ Max(Y ) ⊂ E. Hence we get X ∈ O ∩ gY and Y ∈ G · t by Proposition 3.14. So X and Y commute and lie in some compact torus in the compact subalgebra gX . Then this torus can be extended to a compact Cartan subalgebra t in g. If Z ∈ E, then by Proposition 3.16 there are g ∈ (GY )0 andw ˜ ∈ W such that Z = g · w˜ · X. Since Z ∈ E, alsow ˜ · X ∈ E. By Proposition 3.17 there are Weyl chambers C,C0 such that 0 0 X ∈ C, w˜ · X ∈ C and Y ∈ C− ∩ C−. By Lemma 3.13 there is a w ∈ W such that w · X =w ˜ · X and w · Y = Y . By Proposition 3.14 there is a k ∈ (KY )0 such that
51 w · X = k · X. It follows that Z = g · k · X ∈ (GY )0 · X. So E ⊂ (GY )0 · X. Since (GY )0 · X ⊂ Max(Y ) ⊂ E we conclude that E = Max(Y ) = (GY )0 · X. In particular Max(Y ) is connected and Max(Y ) = GY · X since it is GY -invariant. If G is not semisimple with a compact Cartan subalgebra t, then split g into z ⊕ [g, g] 0 with z = z(g) ⊂ t. Since G is connected, the equation G = (ZG) · Gss holds (see [Kn, Corollary 7.11]). If O = G · X split X = X0 + X1 with X0 ∈ z and X1 ∈ gss. Then O = X0 + O1 where O1 = Gss · X1. Similarly split Y = Y0 + Y1 with Y0 ∈ z and Y1 ∈ gss. Then Max(Y ) = X0 + Max(Y1). By Lemma 1.36 and Lemma 1.69 the subgroup Gss is a semisimple compatible quasihermitian subgroup with a compact Cartan subalgebra. Y1 Y1 0 Therefore we know that Max(Y1) is a connected orbit of both Gss and (Gss ) . Since Y Y1 Y G = Z(G) · Gss , we conclude that Max(Y ) is a connected orbit of G . Therefore it is also an orbit of (GY )0.
Remark 3.18. The proof of Theorem 3.11 shows that the set E of all local maxima of µY is connected, equal to Max(Y ) and a GY -orbit. Hence every local maximum is also a global maximum.
Corollary 3.19. Let Max(Y ) 6= ∅ and let FY (Ob) be the exposed face of Ob defined by Y . Y Y Y 0 Then ext FY (Ob) = Max(Y ),FY (Ob) ⊂ g and ext FY (Ob) is both a G - and a (G ) -orbit. Proof. We have
ext FY (Ob) = O ∩ FY (Ob) = {X ∈ O| hX,Y i = max hZ,Y i} Z∈Ob = {X ∈ O| hX,Y i = max hZ,Y i} = Max(Y ), Z∈O where we use Lemma 1.4 and the third equation follows from the linearity of the product and the fact that Ob is the convex hull of O. Since Crit(µY ) = O ∩ gY , we see that Y Y 0 Y FY (Ob) ⊂ g . By Theorem 3.11 ext FY (Ob) = Max(Y ) is an orbit of (G ) and G .
Remark 3.20. Note that ext FY (Ob) is a again an orbihedron with respect to the group Y Y G with Y ∈ G · Cmax. First of all the group G is indeed compatible and quasihermitian. That it is quasihermitian follows from the fact that G is compatible and quasihermitian Y and the Lemmas 1.68 and 1.67. Furthermore G is compatible with respect to intg0 U if Ad g0 Y ∈ Cmax ⊂ t and using Lemma 1.36 c). The convex hull of ext FY (Ob) is then an orbihedron, i.e. the orbit decomposes into a compact and a hermitian orbit, because after a conjugation we may assume that X lies in ext FY (Ob) ⊂ OX and since −iα(X) is positive for all noncompact roots then it is also positive for all noncompact roots which vanish on Y .
52 Proposition 3.21. Let F be a nonempty face of Ob. Then there is an abelian subalgebra s ⊂ t0, where t0 is conjugate to the compact Cartan subalgebra t, such that F is an s 0 orbihedron of the compatible and quasihermitian subgroup (G ) , i.e. F ⊂ zg(s) and ext F is an orbit of (Gs)0. If F is proper, then s 6= {0}.
Proof. Fix a maximal chain of faces F = F0 ( F1 ( ··· ( Fk = Ob, such that for any i there is no face strictly contained between Fi−1 and Fi. This is possible by Lemma 1.8. We will prove the result by induction on k. If k = 0, then F = Ob, so it is enough to set s = {0}. Let k > 0 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 length k. By the inductive hypothesis the theorem holds for F1, so there is a nontrivial abelian subalgebra s1 s1 0 s1 such that F1 ⊂ g and ext F1 is an orbit of (G ) . In other words F1 is an orbihedron of Gs1 , which is a compatible and quasihermitian subgroup by the induction hypothesis. Since every proper face is contained in an exposed face and F is a maximal face of F1, it is exposed. Hence F = FY (F1) where the element Y lies up to conjugation in the center s1 of g , a torus of the compact part or in Cmax in the hermitian part with respect to the ideal sum deomposition of reductive quasihermitian cf. Remark 1.70. Set s = s1 ⊕ RY . 0 The subalgebra s1 lies in some compact maximal torus t1. So does Y after conjugation with some element in Gs1 . Hence s is conjugate to a subalgebra of t. By Corollary 3.19 the inclusion F ⊂ (gs1 )Y = gs holds and ext F is an orbit of ((Gs1 )Y )0 = (Gs)0. With Remark 3.20 we see that Gs is again compatible and quasihermitian. Thus the inductive step is completed. If s = {0} then (Gs)0 = G, ext F = O and F = Ob. So for proper faces s 6= {0}.
3.2.2 All faces are exposed
In this section we will prove that every face of an orbihedron Ob is exposed. Figure 1 in Section 1.1 shows that not every face of a convex set is exposed. The fact that all faces of an orbitope, i.e. a compact orbihedron, are exposed has been proven in [BGH]. Let G ⊂ U C be a connected compatible subgroup with quasihermitian Lie algebra g and O an admissible orbit in g. In general dim Ob might be less than dim g and there might be some normal subgroup of G that acts trivially on O. We will decompose G in order to deal with this degeneracy. Since the group G is compatible we get that it is also reductive and its Lie algebra g decomposes into its center and its semisimple part, hence g = z⊕gss. Note that every simple ideal of gss is either compact or hermitian since g is quasihermitian (see k k P + P [Ne, p. 241]). Let Aff(O) := λixi k ∈ N , λi ∈ R, xi ∈ O, λi = 1 be the affine i=1 i=1 hull of the orbit O. We can write Aff(O) = x0 + g1 where g1 ⊂ g is a linear subspace and x0 ∈ g.
53 Proposition 3.22. The linear subspace g1 is a semisimple quasihermitian ideal of g and its complement g0 is a reductive ideal. If G1,G0 are the corresponding connected subgroups C of G, then G1 is compatible with U and G = G0 · G1 holds. If x ∈ O, then x = x0 + x1 for some x0 ∈ z(g), x1 ∈ g1 and O = x0 + G1 · x1. Proof. Since G fixes O, it fixes its affine hull Aff(O). Furthermore for any g ∈ G and ˜ ˜ x0 + ξ ∈ Aff(O) = x0 + g1 the element g · (x0 + ξ) = x0 + ξ + g · ξ lies in x0 + g1. Since ξ ˜ and ξ + g · ξ are in g1 the vector g · ξ lies in g1. This shows that the linear subspace g1 is invariant under the action of G and then an ideal in g. The orbit O is admissible and intersects a θ-invariant compact Cartan subalgebra. Then the orbit O is invariant under the involution θ and, with similar computations as for the invariance under G, the affine hull Aff(O) and the linear subspace g1 are also θ-invariant. Let x0 = xz + xss be the decomposition with respect to the decomposition of g = z(g) ⊕ gss. The group G does not act on the center hence the ideal g1 is an ideal in gss. This shows that g1 is a semisimple ideal in gss and since every simple factor is either compact or hermitian it is also quasihermitian. If we impose x0 to be orthogonal to g1 with respect to the Killing form then we show that this choice is unique and it must be an element in the center. Assume that Aff(O) = x0 + g1 =x ˜0 + g1, where x0, x˜0 ⊥ g1. The difference x0 − x˜0 is an element in g1 and must be orthogonal to any element in g1 with respect to the Killing form. The Lie algebra g1 is semisimple hence its Killing form is nondegenerate which shows that x0 − x˜0 = 0, i.e. x0 =x ˜0. The uniqueness of x0 and the G-invariance of Aff(O) show that x0 is also invariant under G and hence must be an element in the center z(g). Now the complement g0 of g1 is also an ideal and invariant under θ hence reductive. Furthermore g0 and g1 are commuting ideals which span g such that the corresponding connected analytic subgroups G0 and G1 are normal and commute with each other and fulfill G = G0 · G1. Pick x ∈ O. We can split x = x0 + x1 where x0 ∈ z(g) ⊂ g0 is as above and x1 ∈ g1. It follows that
O = G · x0 + G · x1 = x0 + G1 · x1.
Here we used that G0 acts trivially on g1 and G acts trivially on z. Furthermore G1 is normal and closed ([Kn, Proposition VII.7.9]). We want to show that it is also compatible. The group Gss with the Lie algebra gss = g1 ⊕ [g0, g0] is compatible due to Lemma 1.36 b). Hence G1 = ZGss ([g0, g0]) is compatible with Lemma 1.36 c).
Remark 3.23. The decomposition of g can be further refined by setting g2 := [g0, g0] and g3 := z(g) = z(g0). They are both θ-invariant ideals of g, orthogonal to each other and to g1 with respect to the Killing form. Furthermore the element x0 lies in g3, g2 is semisimple and quasihermitian and the decomposition
g = g1 ⊕ g2 ⊕ g3 holds. Similarly to the last proof the decomposition also holds on the group level, i.e. G = G1 · G2 · G3.
54 Lemma 3.24. With the same assumptions as in the previous proposition we may refine the decomposition such that g1 = g1,c ⊕ g1,h, where g1,c is a compact and g1,h a hermitian ideal in g1. So now we have the following decomposition of g.
g = g1,c ⊕ g1,h ⊕ g2 ⊕ g3
Hence we have O = x0 +G1,c ·xc +G1,h ·xh, where G1,c and G1,h are the respective connected analytic Lie subgroups associated to g1,c and g1,h and x1 = xc + xh. In other words the admissible orbit O is the sum of a compact orbit and a strictly admissible orbit translated by an element of the center.
Proof. In the previous proposition we have seen that g1 is a semisimple quasihermitian k L ideal in g. Hence we may decompose it into its simple ideals g1 = g1,j, where g1,j are j=1 either compact or hermitian ideals in g1. Knowing this we define the ideals
k k M M g1,c := g1,j and g1,h := g1,j. j=1 j=1 g1,j compact g1,j hermitian
Now the respective connected analytic subgroups G1,c and G1,h are normal in G1 and commute with each other. Hence with the decomposition x1 := xc + xh with xc ∈ g1,c and xh ∈ g1,h we get G1 · x1 = G1 · xc + G1 · xh = G1,c · xc + G1,h · xh. Furthermore the first summand is a compact orbit of the compact group G1,c and the second summand is a strictly admissible orbit since O was an admissible orbit with
0 < −iα(x) = −iα(x0) − iα(xc) − iα(xh) = −iα(xh)
+ 0 for all α ∈ ∆p , where xh ∈ Cmax lies in a pointed generating invariant cone in g1,h.
Remark 3.25. We may use the last Proposition 3.22 not only on the orbihedron Ob but also on the set of faces. So if F is a face with an abelian subalgebra s (see Proposition s 3.21) such that F is an orbihedron of G then one can define gF as the linear subspace s 0 s with Aff(F ) = x0 + gF . The decompositions are then given by g = gF ⊕ gF and G = 0 GF ·GF . One may remark that gF is semisimple and quasihermitian with Proposition 3.22. Furthermore gF defined by Aff(F ) = x0 + gF (and hence also GF ) is uniquely determined by the face F . The rest is not uniquely defined since s is not unique for F . Analogously 0 the compact Cartan subalgebra t = tF ⊕ tF decomposes into subalgebras with respect to 0 0 F , where tF is a compact Cartan subalgebra in gF and tF in gF (up to the center). Let t ⊂ g be a compact Cartan subalgebra and π : g → t denote the orthogonal projection. Define the set P := π(O). Let us first consider O to be an admissible orbit in a quasihermitian Lie algebra g. Then we have the following theorem.
55 0 Theorem 3.26 ([Ne, Theorem VIII.1.36, Remark VIII.1.20]). If X ∈ Cmax then
P := π(OX ) = conv(Wk · X) + Cmin.
Hence P is a convex polyhedron and ext P = O ∩ t is a Wk-orbit. This theorem will be illustrated in the following figures and examples. Especially the polyhedron P will be dealt with in the lower dimensional symplectic and unitary case.
∼ Figure 7: The orbit O and its projection in the Sp(2, R) = SU(1, 1)-case
Example 3.27. In Figure 7 we see the only example of rank 1 in the noncompact setting up to isomorphisms. Recall that Figure 5 shows all types of orbits in this case. Here it shows the orbit O of Sp(2, R) and its projection onto P by the map µ = π. The orbit O we consider is a connected component of a two-sheeted hyperboloid in g. The projection onto the one-dimensional torus decomposes in the sense of the last theorem into a point and a half ray.
56 (a) The degenerate Sp(4, R)-case (b) The general Sp(4, R)-case Figure 8: The projection in the Sp(4, R)-case
Example 3.28. Examples of the polyhedron P in the cases of Sp(4, R) and SU(2, 1) can be seen in Figure 8 and 9. The convex hull of the Weyl group orbit in the degenerate case is a point and in the general case a line segment. The cones in these cases are described in Example 1.59 and 1.60 and can be seen in Figure 4. These are all classical examples in the simple noncompact setting of rank 2 up to the isomorphisms discussed in Theorem 1.77.
(a) The degenerate SU(2, 1)-case (b) The general SU(2, 1)-case Figure 9: The projection in the SU(2, 1)-case
Theorem 3.26 is a generalization of the following theorem of Kostant [Ko].
Theorem 3.29 (Kostant). Let G be a compact Lie group and x ∈ t ∩ O. Then P = conv(W · x). In particular, P is a convex polytope and ext P = O ∩ t is a W -orbit.
∗ Corollary 3.30. Lemma 1.21 and Theorem 3.26 show that every Y ∈ G·Cmax = G·(Cmin) defines a supporting hyperplane to P and hence to Ob. Together with Theorem 3.11 we get that an element Y ∈ g defines a supporting hyperplane to Ob if and only if Y ∈ G · Cmax.
57 Lemma 3.31 ([Ne, Propositions VIII.1.25, VIII.1.30]). Let O be a strictly admissible orbit. Then
i) the convex hull Ob is closed and exp(Ob) = ext(Ob) = O holds.
ii) P = π(O) = π(Ob) = Ob ∩ t holds.
0 Lemma 3.32 ([Ne, Lemma VIII.3.17, Proposition VIII.3.20]). Let Wmax be an invariant convex and maximal elliptic cone. Then the following assertions hold.
0 i) Let Ce ⊂ Wmax be an invariant convex subset either closed or open. Then π(Ce) = C := Ce ∩ t is convex with Cmin + C ⊂ C.
0 ii) Let C ⊂ Cmax be a Wk-invariant closed convex subset of t with Cmin ⊂ rec C which is 0 contained in C0 + Cmin, where C0 ⊂ Cmax is compact. Then the set G · C is convex 0 and contained in Wmax with π(G · C) = C.
Let F be a face of an orbihedron Ob. From now on we denote by σ the projection of the orbit ext F in O onto the torus t which turns out to be a face of the polyhedron P . In the following sections it should be clear from the context if σ denotes a face or an antiholomorphic involution.
Proposition 3.33. Let F be a nonempty face of an orbihedron Ob and s be a subalgebra such that F is an orbihedron of (Gs)0 (see Proposition 3.21). Let t be a compact Cartan subalgebra containing s. Set σ := π(ext F ). Then σ = π(F ) = F ∩ t and σ is a nonempty face of P . If F is proper, then σ is proper. Furthermore F is an orbihedron of (Gσ⊥ )0, where σ⊥ ⊂ t denotes the orthogonal to the tangent space of σ. Moreover ext F is an orbit of Gσ⊥ and F = Gσ⊥ · σ.
Proof. Using Proposition 3.21 we choose a subalgebra s such that F is an orbihedron of (Gs)0. Then the set ext F is an (Gs)0-orbit and t ⊂ gs holds. On this orbit we may apply Proposition 3.22 and Remark 3.23. We get a semisimple quasihermitian compatible s 0 s s normal subgroup G1 of (G ) , a decomposition g = g1 ⊕ g2 ⊕ g3 where g3 = z(g ) and their s 0 s 0 respective connected analytic subgroups G1,G2,G3 = Z(G ) such that (G ) = G1 ·G2 ·G3. According to Lemma 3.24 we may refine this decomposition such that g1 = g1,c ⊕ g1,h with their respective subgroups G1,c and G1,h in G1. It follows that t = t1,c ⊕ t1,h ⊕ t2 ⊕ g3, where t1,· := t ∩ g1,· and t2 = t ∩ g2 are compact Cartan subalgebras in g1,· resp. g2. s Moreover the orbit ext F = G · x = x0 + G1,c · xc + G1,h · xh decomposes into an orbit of a compact group and an orbit of an hermitian group where x = x0 + xc + xh ∈ ext F ∩ t and x0 ∈ g3, xc ∈ g1,c, xh ∈ g1,h. Since taking the convex hull is compatible with the addition (cf. [Schn, Theorem 1.1.2]) we may discuss the two orbits seperately. The restriction of the projection π to g1,· is the orthogonal projection g1,· → t1,· and the affine hull of σ is x0 + t1. By Theorem 3.26 and Theorem 3.29 we get the continued equation
58 σ = π(ext F ) = π(x0) + π(G1,c · xc) + π(G1,h · xh) | {z } | {z } =:σc =:σh = x0 + conv(G1,c · xc ∩ t1,c) + conv(G1,h · xh ∩ t1,h) + Cmin
= conv(Wk1 · x) + Cmin, where Wk1 denotes the Weyl group W (k1, t1) and Cmin the minimal cone in the hermitian part of g1. By Lemma 3.31 and a similar result in the compact case (see [BGH, Lemma 32]) we get
σ = π(x0) + π(G1,c · xc) + π(G1,h · xh)
= π(x0) + π(G\1,c · xc) + π(G\1,h · xh) = π(ext[F ) = π(F ), and similarly
σ = π(x0) + π(G1,c · xc) + π(G1,h · xh)
= x0 ∩ g3 + G\1,c · xc ∩ t1,c + G\1,h · xh ∩ t1,h
= x0 ∩ t + G\1,c · xc ∩ t + G\1,h · xh ∩ t = F ∩ t. Now we have shown that σ = π(F ) = F ∩ t holds and the fact that σ is a face of P follows directly from Lemma 1.9, while σ = π(F ) 6= ∅ since F 6= ∅. ⊥ The affine hull of σ is x0 + t1 hence the the orthogonal complement in t is σ = t2 ⊕ g3. The ideals g1, g2 are semisimple, centralize s ⊂ t and g1, g2, g3 are ideals in g. Thus ⊥ σ⊥ s σ⊥ σ⊥ s σ⊥ s s ⊂ g3 ⊂ σ , G ⊂ G and G = G1 ·G3. So G1 ⊂ G ⊂ G and G1 ·x ⊂ G ·x ⊂ G ·x. s σ⊥ Since G1 · x = G · x = ext F we get that ext F is an orbit of G . The connectedness σ⊥ 0 ⊥ of ext F implies that it is also an orbit of (G ) . Since σ = t2 ⊕ g3 holds we get σ⊥ x0 + g1 ⊂ g3 ⊕ g1 = g for the affine hull of F . This shows that the face F is an orbihedron of (Gσ⊥ )0. Furthermore Gσ⊥ is a compatible subgroup, since this holds after conjugation with Lemma 1.36 c) and σ⊥ ⊂ t. The ideal gσ⊥ is quasihermitian as the sum of a quasihermitian ideal g1 and and the center g3. σ⊥ σ⊥ s We have to prove that F = G ·σ. The equality G ·σ = G ·σ holds, because G2 acts s σ⊥ s trivially on g1 ⊕ g3 ⊃ g1 + x0 ⊃ σ. Since F is G -invariant, σ ⊂ F and G ⊂ G , we get Gσ⊥ · σ ⊂ F . On the other hand ext F ⊂ Gs · σ. We will show that the set Gs · σ is convex s hence F ⊂ G ·σ. The projection σ of the face F decomposes into σ = x0 +σc +σh. Both σc and σh are invariant under their respective Weyl groups and additionally σh is of the form of Theorem 3.26. Then Lemma 3.32 (with G = G1,h) and [Gi, Lemma 7] (in the compact setting, i.e. with G = G1,c) show that G1,c · σc,G1,h · σh are convex with π(G1,c · σc) = σc and π(G1,h · σh) = σh. Since the respective groups act trivially on the other parts we get s s that x0 +G1,c ·σc +G1,h ·σh = G1 ·σ is convex. Also G ·σ is convex because x0 ∈ g3 = z(g ) s σ⊥ and G2,G3 act trivially on σ. Therefore we get F = G · σ = G · σ. Finally we have to show that σ is proper when F is proper. Assume first that the affine hull of Ob is g. Then the affine hull of P is t. If F is proper, then s 6= {0}, so t1 ( t and σ ( P .
59 Corollary 3.34. Let F1,F2 be proper faces of Ob, and let s1, s2 ⊂ g be subalgebras such that si Fi is a (G )-orbihedron. Assume that there exists a maximally Cartan subalgebra t ⊂ g containing both s1 and s2. If F1 ∩ t = F2 ∩ t, then F1 = F2.
σ⊥ Proof. If σ := Fi ∩ t then F1 = G · σ = F2.
Theorem 3.35. All proper faces of Ob are exposed. Proof. Let F be a proper face of Ob and choose a subalgebra s such that F is an (Gs)0- orbihedron and a maximal compact Cartan subalgebra t containing s. By Proposition 3.33 σ := F ∩t is a proper face of P := π(O). Due to the fact that all proper faces of a polyhedron are exposed (cf. Corollary 1.23), there exists a vector Y ∈ t such that σ = FY (P ), where FY (P ) is the intersection of P and the hyperplane HY defined by Y . This hyperplane is an orthogonal complement to RY since the Killing form is nondegenerate on RY ⊂ t. Furthermore Lemma 3.31 shows that P = π(O) = π(Ob) and ext(Ob) = O hold hence we have max hY, xi = max hY, xi = max hY, xi. x∈P x∈O x∈Ob 0 Let F := FY (Ob) be the exposed face of Ob defined by Y . Now we want to show that F = F 0. If x ∈ F then π(x) ∈ σ, so
hY, xi = hY, π(x)i = max hY, xi = max hY, xi x∈P x∈Ob
0 holds because Y ⊥ x − π(x), π(x) ∈ σ and σ = FY (P ). Hence F ⊂ F . We also have
0 σ = FY (P ) = HY ∩ P = HY ∩ Ob ∩ t = FY (Ob) ∩ t = F ∩ t.
So we have two faces F,F 0 with F ∩ t = F 0 ∩ t = σ. Set s0 = RY ⊂ t. By Corollary 3.19 the 0 s0 0 0 face F is an orbihedron of (G ) . Applying Corollary 3.34 we get F = F = FY (Ob).
3.3 Correspondences of faces We will conclude this thesis with several correspondences of the set of faces. At the end of this chapter we will see that the faces of Ob are not only in correlation with the set of faces of its projection P but also with the set of faces of an orbitope. The first correspondence between the set of faces of Ob and P generalizes the results of [BGH] from the compact to the noncompact setting. On the other hand the second correspondence between the set of faces of the orbihedron Ob and a certain orbitope draws a connection between the noncompact and the compact setting.
3.3.1 The correspondence of the faces of the polyhedron and the orbihedron In this section we consider only strictly admissible orbits and their convex hulls. In other words we just consider the orbits of a hermitian Lie group hence in the quasihermitian ideal decomposition of g there exist no compact ideals. The compact case is already well understood (cf. [BGH, BGH2]) and we have already
60 seen that an orbihedron Ob decomposes into a compact orbit and an orbit of an hermitian Lie group with respect to the decomposition of the group G. Furthermore even the Weyl group W (k, t) decomposes into a product of Weyl groups respectively. Hence the general quasihermitian case follows from the combination of the compact and the hermitian case. We fix a compact Cartan subalgebra t in a hermitian Lie algebra g. Denote by F(Ob) the set of proper faces of Ob and by F(P ) the set of proper faces of the polyhedron P . If F is a face of Ob and g ∈ G, then g · F is still a face of Ob, i.e. the group G acts on F(Ob). Similarly the Weyl group Wk = W (k, t) acts on the set of faces F(P ). The goal is to show that these two sets of faces are in bijection up to their respective actions, i.e. ∼ F(Ob)/G = F(P )/Wk.
Lemma 3.36. For every face F of Ob there is a g0 ∈ G such that g0 · F is exposed with respect to some Y ∈ Cmax ⊂ t. The face g0 · F is unique up to an element in NG(t).
Proof. By Theorem 3.35 there is a Y ∈ g with F = FY (Ob). Choose g0 ∈ G such that
Ad g0Y ∈ Cmax ⊂ t using Corollary 3.30. Then g0 · F = FAd g0Y (Ob) is exposed with respect to Ad g0Y which lies in t.
To prove the second statement it is enough to show that if F = FY (Ob) = Fg0·Y (Ob) with −1 Y ∈ t and Ad g0Y ∈ t, then there is a g ∈ NG(t) with g0 · F = g · F . Since Y ∈ t ∩ Ad g0 t, −1 Y Y both t and Ad g0 t are compact Cartan subalgebras of g (modulo the center of g ). Y −1 Hence there is a ge ∈ G such that Ad g0 t = Ad get. Therefore w := g0 · ge ∈ NG(t) and g · F = g · F (O) = g · F (O) = g g · F = w · F . 0 0 Y b 0 Ad gYe b 0e ⊥ We consider the following mapϕ ˜ : F(P ) → F(Ob), σ 7→ Gσ · σ. We will prove that 0 0 σ ∼Wk σ if and only ifϕ ˜(σ) ∼G ϕ˜(σ ). Then the mapϕ ˜ induces an injective map
ϕ : F(P )/Wk → F(Ob)/G
[σ] 7→ [Gσ⊥ · σ] on the sets of equivalence classes. Then we show that ϕ is also surjective which proves the main theorem of this subsection. First of allϕ ˜ is a well defined map since for a face σ of P it is given as σ = FY (P ) for some Y ∈ Cmax. Furthermore with Theorem 3.11 the set F := FY (Ob) is a face of Ob. Then Proposition 3.33 shows that F is given by Gσ⊥ · σ. Theorem 3.37. The map ϕ is bijective. Therefore the sets of equivalence classes of faces F(P )/Wk and F(Ob)/G are in bijective correspondence.
0 0 σ0⊥ σ⊥ −1 0 Proof. If σ = w · σ for any σ, σ ∈ F(P ), w ∈ Wk, then G = wG w and soϕ ˜(σ ) = w · ϕ˜(σ). This shows that the induced map ϕ is well-defined. Vice-versa assume g · F1 = F2 for some g ∈ G with Fi =ϕ ˜(σi). Hence Fi are orbihedra of abelian subalgebras si with −1 s2 = gs1g . Using the decomposition and notation of Remark 3.25 we get that GF2 = −1 gGF1 g .
61 t −1 g ⊥ ⊥ −1 g0 Also we have that σi = π(Fi) = Fi ∩ which implies σ2, gσ1g ⊂ F2 and σ2 , gσ1 g ⊂ F2 . For the compact Cartan subalgebras we get
−1 tF2 , gtF1 g ⊂ gF2
and 0 0 −1 0 tF2 , gtF1 g ⊂ gF2 . Note that σ⊥ = t0 holds and t is just the linear tangent space of σ . These subalge- i Fi Fi i bras have to be conjugate with respect to their respective subgroups (as compact Cartan 0 subalgebras up to the center), i.e. there exist elements g1 ∈ GF2 and g2 ∈ GF2 such t −1 −1 t t0 −1 −1 t0 0 that g1g F1 g g1 = F2 and g2g F1 g g2 = F2 . Defineg ˜ := g1 · g2. Since GF2 and GF2 −1 −1 commute (see Remark 3.25) the same equations hold forg ˜ e.g.gg ˜ tF1 g g˜ = tF2 and t0 −1 −1 t0 t −1 −1 t gg˜ F1 g g˜ = F2 . Hence we havegg ˜ g g˜ = , i.e w :=g ˜ · g is an element in the nor- t 0 s2 malizer NG( ). Furthermoreg ˜ ∈ GF2 · GF2 = G holds andg ˜ leaves the face F2, which is a s2 G -orbihedron, invariant. This implies the equationgF ˜ 2 = F2 and the continued equality w·F1 =ggF ˜ 1 =gF ˜ 2 = F2 follows. Finally we have wσ1 = w(F1 ∩t) = wF1 ∩t = F2 ∩t = σ2. This shows the injectivity of the map ϕ. To show surjectivity let F be a face of Ob. Then F = FY (Ob) for some Y ∈ g. Theorem 3.11 shows that Y is conjugate to some element in Cmax. Hence there exists an element −1 −1 g ∈ G such that gY g ∈ Cmax ⊂ t and g tg is (up to the center) a compact Cartan subalgebra of gY (up to the center). By Proposition 3.33 we get that gF = Gσ⊥ · σ holds where σ = gF ∩ t. This shows the surjectivity of ϕ. The following examples and figures will portray the strong connection which has now been proven between the set of faces of the orbihedron Ob and its polyhedron P .
(a) The compact face (b) The noncompact face (c) The extreme point Figure 10: The general Sp(4, R)-case and the faces of the polyhedron P up to equivalency
Example 3.38. In the previous Example 3.27 and Figure 7 one can see that in the case of Sp(2, R) the only proper face of the polyhedron and the orbihedron up to equivalency is an extreme point. In order to portray the last theorem we want to discuss the case of Sp(4, R). The different
62 proper faces of the polyhedron P up to equivalency can be seen in Figure 10. The vector Y ∈ Cmax gives rise to a hyperplane which supports the given face. On the other hand Figure 11 illustrates the orbit O or rather its convex hull and its different faces which correspond to the faces of P . In this generic case, i.e. the case where the isotropy is a two-dimensional compact torus, the orbit O is eight-dimensional. The compact face in Figure 10a corresponds to the disk in Figure 11 which is a face of Ob given by a two-dimensional orbit of SU(2). The noncompact face in Figure 10b corresponds to the hyperbola in Figure 11 which is a face of Ob given by a two-dimensional orbit of Sp(2, R). The hyperbola describes in fact the hyperboloid in the case of Sp(2, R) which projects onto a half ray we have already seen in Example 3.27. The extreme point in Figure 10c corresponds to any point on the orbit in Figure 11.
Figure 11: Sketch of the general Sp(4, R)-orbihedron and its faces
3.3.2 Complex geometry related to the faces In the compact case the set of faces corresponds to the set of closed orbits of parabolic subgroups of GC inside O (see [BGH, Theorem 64, Lemma 66] or [BGH2, Proposition 3.9]). In the noncompact setting the complexified group GC does not act on the G-orbit O. This can be seen for example in the case of Sp(2, R) in Example 3.27 and Figure 7. In this case the orbit O can be embedded into the projective space P1(C), where the complexification Sp(2, C) = Sp(2, R)C acts transitively. Hence there is no chance to get a similar result as in the compact setting. Since we can embed the orbit openly into a complex generalized flag manifold Z (see section 2.5) one can hope that there exists a correspondence between the set of faces of Ob and the closed orbits of parabolic subgroups of GC on the flag manifold Z.
63 0 ˜ Lemma 3.39. Let X ∈ Cmax. Then the element X := −τ(X) lies in Cmax and has the same centralizer as X, where τ is the reflection with respect to the hyperplane Hk orthogonal ˜ to the one-dimensional center of k. Furthermore w · X = w^· X for all w ∈ Wk. Proof. The reflection τ leaves the centralizer of X invariant since X has compact centralizer and every compact root lies in Hk. Hence every hyperplane corresponding to a compact root is orthogonal to Hk. So if some compact root vanishes on X then it also does on ˜ ˜ X. To show that X is an element in Cmax note that both the multiplication by −1 and the reflection τ exchange the noncompact positive roots with the noncompact negative roots. For the last assertion we decompose w into its simple reflections by the hyperplanes given by the compact roots. Since Hk is orthogonal to these hyperplanes each simple reflection commutes with τ and the multiplication by −1. Hence w · X˜ = w^· X holds for all w ∈ Wk.
+ Remark 3.40. Note that if the roots of ∆h vanish on X and the roots ∆J are positive on ˜ ˜ + X then the same holds for X with respect to ∆h and ∆J where we have used the notation ˜ of Remark 2.25. Hence the map −τ gives us for any point X in Cmax the correct point X with respect to the complex structure J. In other words if X lies in a Weyl chamber C ˜ then X lies in the Weyl chamber C−. Furthermore in the special case where all compact roots vanish on X then the orbit is a hermitian symmetric space and the points X and X˜ coincide. There are also cases where X˜ is an element of the Weyl group orbit of X but this is not in general. This might happen for example in the case where the torus is two-dimensional. Then the one-dimensional center z(k) is itself a hyperplane in the torus t. Particularly the map τ is exactly the reflection given be the hyperplane z(k) hence −τ is the only nontrivial element in the Weyl group Wk.
For the remainder of this subsection we denote the coadjoint G-orbit through X as OG ˜ C and the coadjoint U-orbit through X as OU where U is the compact real form of G . We ∼ have already seen (cf. Chapter 2.5) that the coadjoint orbit OG = G/H can be embedded holomorphically and G-equivariantly into GC/Q− ∼= U/H. Here
˜ QX− := Q− := {g ∈ GC| lim exp(itX˜)g exp(−itX˜) exists in GC} t→∞ is the parabolic subgroup of GC we discussed in this context. Its Lie algebra q− is given ˜ ˜ ˜ by the centralizer zgC (X) of X and the sum of eigenspaces with negative eigenvalues on X, − ˜ L C i.e. q = zgC (X) ⊕ ˜ − g . α∈∆J α One may look at the projections of the two coadjoint orbits onto the compact Cartan subalgebra t. We will denote the projection of OG resp. OU as PG and PU respectively. As ˜ mentioned earlier PU is a polytope and the convex hull of the Weyl group orbit W · X and PG is a polyhedron and the sum of the convex hull of Wk · X and the minimal cone Cmin. Remark 3.41. Geometrically we describe the momentum image of a generalized flag manifold Z ∼= GC/Q− and compare it with the momentum image of an open G-orbit
64 X ∼= G/H with a compact centralizer. Note that X and Z are both K¨ahlermanifolds with respect to the same complex structure. The symplectic structures are quite different since one is defined by G-equivariance and the other one by U-equivariance. Hence their respective momentum maps are different and the momentum images are µU (Z) = OU and µG(X) = OG. Surprisingly we will prove in this section that the face structure of ObG and ObU have a strong connection. In other words the maximal sets of the momentum Y Y components µG and µU for some Y in Cmax are related to each other. In section 2.5 we have seen that we may embed G/H holomorphically into GC/Q−. The following results in this section will show that this property is inherited by the submanifolds of OG which describe the faces. Hence every submanifold in OG which gives rise to a face in ObG is an open submanifold in a compact submanifold of OU which gives rise to a face of ObU . Lemma 3.42. The map ψ : F(PG)/Wk → F(ObU )/U,
[σ] = [FY (PG)] 7→ [FY (ObU )] is well defined. ˜ Proof. First we will show that the map ψ : F(PG) → F(ObU ), σ = FY (PG) 7→ FY (ObU ) is well defined and induces the map ψ on the set of equivalence classes. Let σ be a face of PG through the point X which is given by σ = FY (PG). Then a face Y of ObG is given by the convex hull of the orbit G · X. We may choose a Weyl chamber C ˜ with X ∈ C sucht that Y and X are elements of the Weyl chamber C− (see Proposition 3.17 and Remark 3.40). Now the element Y defines a parabolic subgroup QY− . The Levi part is just the com- plexification of the centralizer of Y . Since X˜ and Y lie in the same Weyl chamber the respective parabolic subgroups share a borel subgroup which fixes the point X˜. Hence Y C Y− ˜ the (G ) -orbit and the Q -orbit through X coincide and are compact in OU . The only Y− closed Q -orbit gives rise to the face F of ObU with F = FY (ObU ) (see [BGH2, Proposition 3.9]). The next step is to show that the map ψ˜ does not depend on the choice of Y . Let Y1 Y2 Y1,Y2 ∈ C− ⊂ t with σ = FY1 (PG) = FY2 (PG). Then we have that G · X = G · X holds. Similar to the calculations in Chapter 2.5 we get a G-equivariant embedding of OG into C − ∼ Y1 ˜ Y2 ˜ G /Q = OU . Using this map we also get the equality G · X = G · X. Computing the dimensions of these orbits using the roots that vanish on Y1 resp. Y2 one sees that GY1 · X˜ = GY2 · X˜ is open in the respective orbits of the parabolic subgroups QY1− · X˜ and QY1− · X˜. Since QY1− and QY2− share the same Borel subgroup B, the orbits QY1− · X˜ and QY1− · X˜ decompose into B-orbits and have the same B-fixpoint X˜ ∈ GY1 · X˜ = GY2 · X˜. Thus we get that all B-orbits of QY1− · X˜ lie in QY2− · X˜ and vice versa. This shows the equality QY1− · X˜ = QY1− · X˜ and that the map ψ˜ does not depend on the choice of Y but only of σ. Thus the map ψ˜ is well defined.
65 To show that the map ψ is well defined consider an element w ∈ Wk and the face w · σ w·Y− Y− −1 ˜ Y− ˜ with w · σ = Fw·Y (PG). Consequently Q · w^· X = wQ w w · X = w · Q · X. This shows that the induced faces in ObU are equivalent with respect to the U-action.
Remark 3.43. Starting with a G-orbit OG = G·X and its orbihedron ObG we have already ˜ seen that we may embed the orbit holomorphically into an U-orbit OU = U ·X. The convex hull ObU of OU is then an orbitope. Consider a face F of the orbihedron ObG which is given by F = FY (ObG). In the previous sections we have already seen that the extreme points of F are given by an orbit of the Y centralizer G . Now we are interested in faces of the orbitope ObU which arise naturally ˜ with respect to the face F of ObG. The previous lemma describes ψ resp. ψ which gives us such a map. In particular the induced face in ObU is the convex hull of a closed orbit given by the parabolic subgroup QY− . The situation we discussed in this remark is described in the following diagram.
˜ ObG ⊃ OG = G · X / U · X = OU ⊂ ObU O O S S ? ? Y Y− ˜ \Y− ˜ FY (ObG) = F ⊃ ext F = G · X / Q · X ⊂ Q · X Lemma 3.44. Let G be a complex Lie group which acts on a complex manifold Z and σ an antiholomorphic involution on G. In addition let G1,G2 be σ-invariant complex subgroups of G which are equal up to their ineffectivity (denoted by N1 resp. N2) on Z. Then G1 and G2 also coincide in ineffective elements g where σ(g) acts effectively.
Proof. Assume that there is an element g ∈ N1 such that σ(g) acts on Z effectively. Hence σ(g) is an element of both G1 and G2. Since G2 is invariant under the involution σ we have that g ∈ N2. Using the same argument vice-versa we have proven the lemma.
Lemma 3.42 shows that the map ψ : F(PG)/Wk → F(ObU )/U is well defined. In order to show the injectivity of this map the following theorem plays a key role. Theorem 3.45 ([O, § 15, Theorem 2]). Let M ∼= G/P be a homogeneous flag manifold of a connected complex semisimple Lie group G, acting on M effectively. Then M decomposes into irreducible factors ∼ M = M1 × · · · × Mr,
where Mi are homogeneous flag manifolds Gi/Pi and Gi are simple with G = G1 × · · · × Gr and P = P1 × · · · × Pr. This decomposition gives rise to an isomorphism 0 ∼ 0 0 (Bih(M)) = (Bih(M1)) × · · · × (Bih(Mr)) . If G is simple then the identity component (Bih(M))0 of the biholomorphic transformations on M is simple too, and (Bih(M))0 = G except in the following cases:
1. G = P Sp(2n, C), (Bih(M))0 = PSL(2n, C).
66 0 2. G = G2(C), (Bih(M)) = PSO7(C).
3. G = SO(2n − 1, C), (Bih(M))0 = PSO(2n, C).
Theorem 3.46. There is a one to one correspondence between the set of faces of ObG and ObU with respect to the corresponding actions of G and U, i.e. ∼ ∼ ∼ F(ObG)/G = F(PG)/Wk = F(PU )/W = F(ObU )/U.
In other words the map ψ : F(PG)/Wk → F(ObU )/U,
[σ] = [FY (PG)] 7→ [FY (ObU )] is bijective.
Proof. We already have seen the correspondence of the faces between the orbits and their projections. Now we will draw a connection between the faces of ObG and ObU . First we will show the injectivity of the map ψ. Let Y1,Y2 ∈ Cmax ⊂ t with
closed QY1− · X˜ = QY2− · X˜ =: M ⊂ Z := GC/Q−. ˜ The elements Y1 and Y2 can be chosen such that Y1,Y2, X lie in the same Weyl chamber. Thus both parabolic subgroups QY1− and QY2− share the same Borel subgroup B which fixes the point X˜. Furthermore M = LY1 · X˜ = LY2 · X˜ holds, where LYj denotes the Y Levi part of the parabolic subgroup Q j − . The manifold M may not be irreducible, but the product decomposition of M into irreducible factors induces a product decomposition Yi of the Levi subgroups modulo the ineffectivtity (cf. Theorem 3.45) with L ⊂ Bih(Mi). Since these factors act transitively and using the classification of Onishchik in Theorem Y1 Y2 3.45 we get that the factors are contained in each other, i.e. Lj ⊂ Lj or vice versa. The group components correspond to connected pieces of the Dynkin diagram of GC hence for Y1 Y2 example the diagram describing Lj must be contained in the diagram describing Lj or vice versa. Looking at the Dynkin diagrams of 1.-3. of Theorem 3.45 we see that this only Y1 Y2 happens if and only if Lj = Lj . Now the parabolic subgroups are the same modulo the ineffectivity of their action on M. Moreover on the level of Lie algebras we get that the Y1− Y2− C ˜ Levi parts of q and q only differ in the root spaces gα where α vanishes on X but only in one of Y1 or Y2. These root spaces give rise to the ineffectivity in the respective Levi subgroups, which are invariant under the involution σ since it just exchanges the root C C spaces gα with g−α. (Even if they are not invariant we may use Lemma 3.44 on the factors of LYi .) Furthermore any root space to a root α that does not vanish on X˜ vanishes on Y1 if and only if it vanishes on Y2. If they do vanish, these roots give rise to the factors that act effectively on M. Since these parts are the same for Y1 and Y2 the real forms GY1 of LY1 and GY2 of LY2 act the same way on QY1− · X˜ = QY2− · X˜. Hence we get the Y1 ˜ Y1 ˜ Y1 Y1 equalities G · X = G · X, G · X = G · X and then FY1 (PG) = FY2 (PG). This shows the injectivity of ψ.
67 The surjectivity of ψ can be seen when we use the correspondence of the faces of ObU and PU . Then it is clear that the map [FY (PG)] 7→ [FY (PU )] is surjective since Y ∈ Cmax and it suffices to use the elements in one Weyl chamber to describe any face of PU up to equivalency. Remark 3.47. To illustrate the difficulty with the injectivity of ψ in the previous proof ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ C − C we consider the following groups. Let G = SL4(C) and Q = ∈ G 0 0 ∗ ∗ 0 0 0 ∗ be a parabolic subgroup. Furthermore define ∗ ∗ ∗ ∗ Y ∗ ∗ ∗ ∗ Q 1− = ∈ GC 0 0 ∗ ∗ 0 0 ∗ ∗ and ∗ ∗ ∗ ∗ Y 0 ∗ ∗ ∗ Q 2− = ∈ GC . 0 0 ∗ ∗ 0 0 ∗ ∗ Here the two Levi factors are the same modulo ineffectivity both being isomorphic to SL2(C). But their ineffectivities are different. With the last theorem we have finally proven the last of our goals in this work. Not only we have verified that every proper face of an orbihedron ObG is exposed and corresponds to a proper face of the polyhedron P but we were able to find a natural orbitope ObU such that their face structure coincide up to an isomorphism. In order to visualize the last theorem we will now consider the low dimensional orbits in the symplectic and unitary case. There we will describe the faces of the polyhedron PG and the polytope PU rather than the faces of their orbihedra since the dimensions of the orbihedron ObG and the orbitope ObU are too high. Example 3.48. The following Figures 12-14 illustrate the last theorem und portrays the polyhedron PG resp. the polytope PU . One will see the strong connection between their structure of faces. Figure 12 presents the polyhedra where G = Sp(2, R) and U = Sp(2). Note that we have already seen that the polyhedron PG is a half ray in Figure 7. As we have remarked at the beginning of this section the G-orbit can be embedded into an U-orbit, which in this case is the projective space P1C. The momentum image of the U-orbit is then a line segment given in Figure 12. In this example we see that the only proper face up to equivalency of PG and PU is an extreme point. Hence this is the simplest example of the isomorphy of Theorem 3.46. Due to the exceptional isomorphisms this also covers the case of G = SU(1, 1) and U = SU(2).
68 Figure 12: One dimensional projections PG and PU in the Sp(2, R)-/Sp(2)-case
The higher dimensional examples, i.e. G = SU(2, 1) and U = SU(3), of the unitary group are given in Figure 13a and 13b. In the degenerate case the G-orbit can be embedded into the U-orbit which is the two-dimensional projective space P2C. The polytope PU is then an equilateral triangle which is given on the right-hand side in Figure 13a. Except from the extreme point the only other proper face up to equivalency of the polyhedron is the extreme ray. Analogously to Figure 12 the extreme ray corresponds to the line segment in PU which is also the only proper face of PU except the extreme point. In the nondegenerate case the G-orbit can be embedded into the U-orbit which is a Grass- 3 mannian G2 of the two-dimensional planes in C . In Figure 13b one can see the polyhedra in this case. The correspondence of the faces is similar to the degenerate case. We get one more face of PG resp. PU up to equivalency which is a line segment. This face corresponds to a compact orbit isomorphic to P1C. Hence the map P1C → P1C is not only an open embedding but rather an isomorphism. The higher dimensional examples of the symplectic group are given in Figure 13c and 13d in the case of a two-dimensional torus and in Figure 14 in the case of a three- dimensional torus. The degenerate and nondegenerate cases where the torus is two-dimensional (see Figure 13c and 13d) are analogous to the unitary cases of SU(2, 1) and SU(3).
69 (a) The degenerate SU(2, 1)-/SU(3)-case (b) The general SU(2, 1)-/SU(3)-case
(c) The degenerate Sp(4, R)-/Sp(4)-case (d) The general Sp(4, R)-/Sp(4)-case
Figure 13: Two-dimensional projections PG and PU
The proper faces up to equivalency in the degenerate resp. nondegenerate symplectic case are exactly of the same type as in the degenerate resp. nondegenerate unitary case. The faces are an extreme point and an extreme ray (and in the nondegenerate case another line segment) for the polyhedron PG and an extreme point and one line segment (respec- tively two line segments in the nondegenerate case) for the polytope PU . Figure 14 shows the polyhedra in the symplectic case of a three-dimensional torus. The most degenerate case can be seen in Figure 14a. The orbit is then a simple hermitian symmetric space and has dimension dim Sp(6, R) − dim K = 21 − dim U(3) = 21 − 9 = 12. Figure 14b and 14c show the next degenerate cases. The isotropy group is then five- dimensional hence the orbit has dimension 21 − 5 = 16. Finally Figure 14d shows the nondegenerate case. Here the isotropy group equals a torus and is three-dimenional hence the orbit is of dimension 18. Note that the base of the polyhedron PG is given by the projections of a coadjoint K-orbit. In the lower dimensional setting (see Figure 12, 13) this was either an extreme point or a line segment. In this case (see Figure 14) the base is a polytope given by the group SU(3) and is either a point, an equilateral triangle or a hexagon. Furthermore we remark that every face of the polyhedra resp. polytopes is given as a lower dimensional polyhedron or polytope in a previous case or a product of such.
70 (a) First degenerate Sp(6, R)-/Sp(6)-case (b) Second degenerate Sp(6, R)-/Sp(6)-case
(c) Third degenerate Sp(6, R)-/Sp(6)-case (d) The general Sp(6, R)-/Sp(6)-case
Figure 14: Three-dimensional projections PG and PU
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