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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Symplectic branching laws and Hermitian symmetric spaces
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by Benjamin Schwarz and Henrik Seppänen PDF
Trans. Amer. Math. Soc. 365 (2013), 6595-6623 Request permission

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
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6595-6623
  • MSC (2010): Primary 22E46; Secondary 53D20, 17C50, 32M15, 32L05
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05929-6
  • MathSciNet review: 3105764