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Symplectic convexity theorems and applications to the structure theory of semisimple Lie groups
Otto, Michael

2004, Doctor of Philosophy, Ohio State University, Mathematics.

Atiyah’s well known convexity theorem states that for a Hamiltonian torus action 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 such that 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 involutions 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.

Robert Stanton (Advisor)

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Otto, M. (2004). Symplectic convexity theorems and applications to the structure theory of semisimple Lie groups. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Otto, Michael. "Symplectic convexity theorems and applications to the structure theory of semisimple Lie groups." Electronic Thesis or Dissertation. Ohio State University, 2004. OhioLINK Electronic Theses and Dissertations Center. 26 Sep 2017.

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Otto, Michael "Symplectic convexity theorems and applications to the structure theory of semisimple Lie groups." Electronic Thesis or Dissertation. Ohio State University, 2004. https://etd.ohiolink.edu/

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