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Automorphisms of multiplicity free Hamiltonian manifolds


Author: Friedrich Knop
Journal: J. Amer. Math. Soc. 24 (2011), 567-601
MSC (2010): Primary 53D20, 14L30, 14M27
DOI: https://doi.org/10.1090/S0894-0347-2010-00686-8
Published electronically: November 30, 2010
MathSciNet review: 2748401
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Abstract: Let $ M$ be a multiplicity free Hamiltonian manifold $ M$ for a connected compact Lie group $ K$ (not necessarily abelian). Let $ \mathcal{P}$ be the momentum polytope of $ M$. We calculate the automorphism of $ M$ as a sheaf over $ \mathcal{P}$ and show that all higher cohomology groups of this sheaf vanish. From this, and a recent theorem of Losev, we deduce a conjecture of Delzant: the momentum polytope and the principal isotropy group determine $ M$ up to isomorphism. Moreover, we give a criterion for when a polytope and a group are afforded by a multiplicity free manifold.


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Additional Information

Friedrich Knop
Affiliation: Department of Mathematics, Universität Erlangen, Bismarckstrasse $1\frac{1}2$, D-91054 Erlangen, Germany

DOI: https://doi.org/10.1090/S0894-0347-2010-00686-8
Received by editor(s): August 18, 2010
Received by editor(s) in revised form: October 1, 2010
Published electronically: November 30, 2010
Article copyright: © Copyright 2010 by Friedrich Knop

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