Symplectic branching laws and Hermitian symmetric spaces
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- by Benjamin Schwarz and Henrik Seppänen PDF
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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.References
- Wolfgang Bertram, The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, vol. 1754, Springer-Verlag, Berlin, 2000. MR 1809879, DOI 10.1007/b76884
- J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431, DOI 10.1007/978-3-642-56936-4
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
- Harish-Chandra, Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math. 78 (1956), 564–628. MR 82056, DOI 10.2307/2372674
- Kaveh, K., Crystal bases and Newton-Okounkov bodies, preprint, arxiv.org/abs/1101.1687v1
- Frances Kirwan, Convexity properties of the moment mapping. III, Invent. Math. 77 (1984), no. 3, 547–552. MR 759257, DOI 10.1007/BF01388838
- Toshiyuki Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, Representation theory and automorphic forms, Progr. Math., vol. 255, Birkhäuser Boston, Boston, MA, 2008, pp. 45–109. MR 2369496, DOI 10.1007/978-0-8176-4646-2_{3}
- Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835 (English, with English and French summaries). MR 2571958, DOI 10.24033/asens.2109
- Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721
- O. Loos, Bounded symmetric domains and Jordan pairs, Lecture Notes, University of Irvine (1977)
- Ottmar Loos, Homogeneous algebraic varieties defined by Jordan pairs, Monatsh. Math. 86 (1978/79), no. 2, 107–129. MR 516835, DOI 10.1007/BF01320204
- Ottmar Loos, Diagonalization in Jordan pairs, J. Algebra 143 (1991), no. 1, 252–268. MR 1128659, DOI 10.1016/0021-8693(91)90264-9
- Ottmar Loos, Decomposition of projective spaces defined by unit-regular Jordan pairs, Comm. Algebra 22 (1994), no. 10, 3925–3964. MR 1280101, DOI 10.1080/00927879408825058
- Andrei Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405–411. MR 1400312, DOI 10.1007/s002220050081
- Andreĭ Okounkov, Multiplicities and Newton polytopes, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 231–244. MR 1618759, DOI 10.1090/trans2/181/07
- Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR 591460
- Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
- Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87–129. MR 1314032, DOI 10.2307/2118628
- Harald Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), no. 1, 1–25 (1986). MR 821311, DOI 10.2307/2374466
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