Osaka Journal of Mathematics

Coadjoint orbitopes

Leonardo Biliotti, Alessandro Ghigi, and Peter Heinzner

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We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of compact Lie groups. We show that up to conjugation the faces are completely determined by the geometry of the faces of the convex hull of Weyl group orbits. We also consider the geometry of the faces and show that they are themselves coadjoint orbitopes. From the complex geometric point of view the sets of extreme points of a face are realized as compact orbits of parabolic subgroups of the complexified group.

Article information

Osaka J. Math., Volume 51, Number 4 (2014), 935-969.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations 53D26


Biliotti, Leonardo; Ghigi, Alessandro; Heinzner, Peter. Coadjoint orbitopes. Osaka J. Math. 51 (2014), no. 4, 935--969.

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  • E. Arbarello, M. Cornalba and P.A. Griffiths: Geometry of Algebraic Curves, II, Grundlehren der Mathematischen Wissenschaften 268, Springer, Heidelberg, 2011.
  • M.F. Atiyah: Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1–15.
  • M. Audin: Torus Actions on Symplectic Manifolds, second revised edition, Progress in Mathematics 93, Birkhäuser, Basel, 2004.
  • A. Barvinok and I. Novik: A centrally symmetric version of the cyclic polytope, Discrete Comput. Geom. 39 (2008), 76–99.
  • M. Berger: Geometry, I, translated from the French by M. Cole and S. Levy, Universitext, Springer, Berlin, 1994.
  • L. Biliotti and A. Ghigi: Satake–Furstenberg compactifications, the moment map and $\lambda_{1}$, Amer. J. Math. 135 (2013), 237–274.
  • A. Borel and L. Ji: Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Boston, MA, 2006.
  • J.-P. Bourguignon, P. Li and S.-T. Yau: Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv. 69 (1994), 199–207.
  • \begingroup C. Carathéodory: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95–115. \endgroup
  • J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan: Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math. 49 (1983), 309–398.
  • V.M. Gichev: Polar representations of compact groups and convex hulls of their orbits, Differential Geom. Appl. 28 (2010), 608–614.
  • V. Guillemin and S. Sternberg: Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491–513.
  • V. Guillemin and S. Sternberg: Symplectic Techniques in Physics, second edition, Cambridge Univ. Press, Cambridge, 1990.
  • G. Heckman: Projection of orbits and asymptotic behaviour of multiplicities of compact Lie groups, PhD thesis (1980).
  • P. Heinzner, G.W. Schwarz and H. Stötzel: Stratifications with respect to actions of real reductive groups, Compos. Math. 144 (2008), 163–185.
  • A. Huckleberry: Introduction to group actions in symplectic and complex geometry; in Infinite Dimensional Kähler Manifolds (Oberwolfach, 1995), DMV Sem. 31, Birkhäuser, Basel, 2001, 1–129.
  • A.A. Kirillov: Lectures on the Orbit Method, Graduate Studies in Mathematics 64, Amer. Math. Soc., Providence, RI, 2004.
  • A.W. Knapp: Lie Groups Beyond an Introduction, second edition, Progress in Mathematics 140, Birkhäuser Boston, Boston, MA, 2002.
  • B. Kostant: Quantization and unitary representations, I. Prequantization; in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math. 170, Springer, Berlin, 1970, 87–208.
  • B. Kostant: On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974).
  • D. McDuff and D. Salamon: Introduction to Symplectic Topology, second edition, Oxford Mathematical Monographs, Oxford Univ. Press, New York, 1998.
  • C.C. Moore: Compactifications of symmetric spaces, Amer. J. Math. 86 (1964), 201–218.
  • R. Sanyal, F. Sottile and B. Sturmfels: Orbitopes, Mathematika 57 (2011), 275–314.
  • I. Satake: On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. (2) 71 (1960), 77–110.
  • R. Schneider: Convex Bodies: the Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press, Cambridge, 1993.
  • Z. Smilansky: Convex hulls of generalized moment curves, Israel J. Math. 52 (1985), 115–128. \endthebibliography*