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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symplectic branching laws and Hermitian symmetric spaces


Authors: Benjamin Schwarz and Henrik Seppänen
Journal: Trans. Amer. Math. Soc. 365 (2013), 6595-6623
MSC (2010): Primary 22E46; Secondary 53D20, 17C50, 32M15, 32L05
Published electronically: May 14, 2013
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Abstract: Let $ G$ be a complex simple Lie group, and let $ U \subseteq G$ be a maximal compact subgroup. Assume that $ G$ admits a homogenous space $ X=G/Q=U/K$ which is a compact Hermitian symmetric space. Let $ \mathscr {L} \rightarrow X$ be the ample line bundle which generates the Picard group of $ X$. In this paper we study the restrictions to $ K$ of the family $ (H^0(X, \mathscr {L}^k))_{k \in \mathbb{N}}$ of irreducible $ G$-representations. We explicitly describe the moment polytopes for the moment maps $ X \rightarrow \mathfrak{k}^*$ associated to positive integer multiples of the Kostant-Kirillov symplectic form on $ X$, and we use these, together with an explicit characterization of the closed $ K^{\mathbb{C}}$-orbits on $ X$, to find the decompositions of the spaces $ H^0(X,\mathscr {L}^k)$. We also construct a natural Okounkov body for $ \mathscr {L}$ and the $ K$-action, and we identify it with the smallest of the moment polytopes above. In particular, the Okounkov body is a convex polytope. In fact, we even prove the stronger property that the semigroup defining the Okounkov body is finitely generated.


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Additional Information

Benjamin Schwarz
Affiliation: Fakultät für Elektrotechnik, Informatik und Mathematik, Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
Email: bschwarz@math.upb.de

Henrik Seppänen
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany
Email: hseppaen@uni-math.gwdg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05929-6
PII: S 0002-9947(2013)05929-6
Keywords: Branching law, holomorphic line bundle, Hermitian symmetric space, moment map, Jordan pair, Okounkov body
Received by editor(s): November 28, 2011
Received by editor(s) in revised form: May 7, 2012, and August 2, 2012
Published electronically: May 14, 2013
Additional Notes: The second author was supported by the DFG Priority Programme 1388 Representation Theory
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.