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Kählerian Structures of Coadjoint Orbits of Semisimple Lie Groups and Their

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Kählerian Structures of Coadjoint Orbits of Semisimple Lie Groups and Their 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 = f(λi1 ; : : : ; λin )jij 2 f1; : : : ; ng; ij 6= ik for j 6= kg ; 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 2 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; !) 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 ! holds for all ξ 2 g. Here the function µξ : X ! R with µξ(x) := hµ(x); ξi is the component X of µ along ξ and ιξX ! is the contraction of the symplectic form ! with the vector field ξ on X induced by ξ 2 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 2 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 ! 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.
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