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

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Symplectic Convexity Theorems and Applications to the Structure Theory of Semisimple Lie Groups SYMPLECTIC CONVEXITY THEOREMS AND APPLICATIONS TO THE STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS DISSERTATION Presented in Partial Fulfillment of the Requirement for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Michael Otto ***** The Ohio State University 2004 Dissertation Committee: Approved by Professor T. Kerler Professor B. Kr¨otz,Co-Adviser Adviser Professor R. Stanton, Co-Adviser Department of Mathematics ABSTRACT Atiyah’s well known convexity theorem states that for a Hamiltonian torus ac- tion T × M → M on a compact connected symplectic manifold M the image Φ(M) under the associated moment map Φ : M → t∗ is convex. Duistermaat in addition considered antisymplectic involutions τ on M satisfying Φ ◦ τ = Φ. He showed that Φ(M) = Φ(Q) for Lagrangian submanifolds Q that arise as fixed point sets of such involutions. We prove a generalization of Duistermaat’s symplectic convexity theorem for invo- lutions τ which satisfy several compatibility properties with the torus action, but which are not necessarily antisymplectic. By the same method we can also extend a Duistermaat-type theorem for non-compact M with proper Φ. All the symplectic convexity theorems mentioned have applications to the structure theory of semisimple Lie groups. With the generalization of Duistermaat’s theorem we are able to extend existing symplectic proofs for Kostant’s and Neeb’s convexity theorems to all semisimple groups. In addition we develop the symplectic framework to refine a recently discovered complex convexity theorem. ii ACKNOWLEDGMENTS First and foremost, I wish to thank Bernhard Kr¨otz for his role as adviser through- out my graduate studies. With his energy and enthusiasm he provided a great working environment. I would also like to thank Bob Stanton for his support, especially during the process of writing this dissertation, and for his frequent valuable advice. iii VITA April 19, 1974 ................... Born — Braunschweig, Germany 2001 ........................ Diplom, TU Clausthal/Germany 2000–present ................... Graduate Teaching Associate, The Ohio State University PUBLICATION B. Kr¨otzand M. Otto, Vanishing properties of analytically extended matrix coeffi- cients, Journal of Lie Theory 12, 2002, 409-421. FIELDS OF STUDY Major Field: Mathematics iv TABLE OF CONTENTS Page Abstract ....................................... ii Acknowledgments .................................. iii Vita ......................................... iv Chapters: 1. Introduction . 1 2. Symplectic Geometry . 7 2.1 Background . 8 2.2 Local normal forms . 18 2.3 Symplectic convexity theorems . 28 3. Lie theory . 41 3.1 Kostant’s convexity theorems . 42 3.1.1 Kostant’s theorems . 42 3.1.2 The linear version . 44 3.1.3 The nonlinear version . 47 3.2 Neeb’s convexity theorem for semisimple symmetric spaces . 56 3.2.1 Statement of the theorem and reduction of the problem 57 3.2.2 A symplectic proof . 60 3.3 A refinement of the complex convexity theorem of Gindikin- Kr¨otz . 65 3.3.1 Notation and basic facts . 66 3.3.2 Complex groups . 69 3.3.3 The complex convexity theorem for non-complex groups 76 Appendix: Poisson Lie groups and Manin triples ............... 81 Bibliography .................................... 86 v CHAPTER 1 INTRODUCTION A complex (n × n)-matrix A is called Hermitian if A = A¯T . Here the bar sym- bolizes the matrix operation where each matrix entry is replaced with its complex conjugate. The exponent T denotes the transposed matrix. It is a well known fact that a given Hermitian matrix A is diagonalizable. This means one can find (com- plex) numbers λ1, . , λn, not necessarily distinct, and a complex matrix V with the property V −1 = V¯ T (such a V is called unitary), such that λ 1 −1 .. V AV = . . λn The λi’s are the eigenvalues of A and can easily shown to be real numbers. How do the entries of A depend on the λi’s, or in other words, how much infor- mation on the matrix A is encoded in its eigenvalues ? A partial answer to this question was provided by Schur in 1923 [21]. He showed that the range of the diagonal elements of A was restricted by the eigenvalues. Let 1 us be more specific. We denote the n-dimensional vector of diagonal elements of A by diag(A), and we write the eigenvalues in vector form (λ1, . , λn). Note that both n diag(A) and (λ1, . , λn) are points in R . For the finite set of points which are ob- tained from (λ1, . , λn) by permuting the entries we write Sn.(λ1, . , λn) (the orbit of (λ1, . , λn) under the symmetric group Sn). The convex hull conv(Sn.(λ1, . , λn)) n of the finite set Sn.(λ1, . , λn) is a compact region in R which is determined uniquely by the eigenvalues λ1, . , λn. Schur’s result asserts that the diagonal vector must lie in this region, i.e. diag(A) ∈ conv(Sn.(λ1, . , λn)). It turns out that diagonal vectors of Hermitian matrices with eigenvalues λ1, . , λn not only lie in conv(Sn.(λ1, . , λn)) but actually make up all that region, i.e. {diag(A): A Hermitian with eigenvalues λ1, . , λn} = conv(Sn.(λ1, . , λn)). This was proven by Horn in 1954 [12]. Statements on square matrices can often be extended to semisimple Lie groups. But it was not until 1973 that Kostant [14] proved such a generalization for the Schur-Horn result. Let G be a connected semisimple Lie group with Lie algebra g. The Iwasawa de- composition of G generalizes the Gram-Schmidt decomposition of a complex matrix g with determinant one into a unique product g = nak of an upper triangular ma- trix n with ones on the diagonal, a real diagonal matrix a with determinant one and a unitary matrix k with determinant one. The usual notation is G = NAK for 2 certain subgroups N, A, K of G that are unipotent, abelian, compact, respectively. On the Lie algebra level the Iwasawa decomposition states g = n ⊕ a ⊕ k, where n = Lie(N), a = Lie(A), k = Lie(K). It is a generalization of the fact that any com- plex matrix with trace zero can be uniquely written as the sum of a strictly upper triangular matrix, a real diagonal matrix with trace zero and a skew-Hermitian ma- trix. The group Sn from above becomes the Weyl group W = NK (a)/ZK (a). We denote bya ˜ : G = NAK → A, nak 7→ a, and pra : g = n + a + k → a,Xn + Xa + Xk 7→ Xa the middle projections in the Iwasawa decompositions. Then Kostant’s result is the following. Kostant’s convexity theorems. Let Y ∈ a. Then 1. pra(Ad(K).Y ) = conv(W.Y ) linear version 2. loga ˜(K exp(Y )) = conv(W.Y ) nonlinear version where conv(·) denotes the convex hull of (·). The Schur-Horn theorem is a special case of the linear version in case G = SL(n, C). Atiyah [2] took this subject into a new and unexpected direction. He related the Lie theoretic results of Kostant with symplectic geometry. He, and independently, Guillemin and Sternberg [6], discovered a symplectic convexity theorem from which Kostant’s linear result could be derived. 3 AGS-Theorem. Let (M, ω) be a compact connected symplectic manifold. Sup- pose a torus T acts on M in a Hamiltonian way with moment map Φ: M → t∗. Then Φ(M) is convex. More precisely, Φ(M) is the convex polytope spanned by the finite set Φ(Fix(M)), where Fix(M) denotes the T -fixed points of M. Subsequently, the AGS-theorem has been generalized in different directions by numerous authors. Some consider not only torus actions but more general Hamil- tonian actions. Others study non-compact symplectic manifolds. In many instances additional conditions on the moment map are required. One of the early extensions of the AGS-theorems will be of particular interest to us. It was given by Duistermaat [4] who considered certain involutions on the underlying symplectic manifold. Duistermaat’s Theorem. Let (M, ω) be a compact connected symplectic man- ifold. Suppose a torus T acts on M in a Hamiltonian fashion with moment map Φ: M → t∗. In addition, let τ : M → M be an antisymplectic involution which satisfies Φ ◦ τ = Φ. If Q denotes the set of τ-fixed points of M, then Φ(Q) = Φ(M). In particular, Φ(Q) is convex. The statement remains true if one replaces Q with any of its connected components. It turns out that antisymplecticity is a rather strong condition on the involution τ that is not always satisfied in applications. One of the main results of this thesis is 4 the following theorem which states how the conditions on τ in Duistermaat’s theorem can be weakened. Theorem A. [15] Let M be a compact connected symplectic manifold with Hamil- tonian torus action T × M → M and moment map Φ: M → t∗. Furthermore, let τ : M → M be an involutive diffeomorphism with fixed point set Q such that 1. t ◦ τ = τ ◦ t−1 for all t ∈ T . 2. Φ ◦ τ = Φ. 3. Q is a Lagrangian submanifold of M. Then Φ(Q) = Φ(M) = conv Φ(Fix(M)) = conv Φ(Fix(Q)). In particular Φ(Q) is a convex subset of t∗. Moreover, the same assertions hold if Q is replaced with any of its connected components. Note that this theorem extends the class of Lagrangian submanifolds Q in Duis- termaat’s theorem for which the image Φ(Q) is still the full convex set Φ(M). We will also consider the convexity theorem by Hilgert, Neeb and Plank [11] which deals with the situation that M is non-compact and the moment map Φ is proper. We will show that, again, the condition of Φ being antisymplectic can be weakened as in Theorem A.
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