Abstract
We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ⊂ p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.
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References
M. F. Atiyah, Convexity and commuting Hamiltonians, Bulletin of the London Mathematical Society 14 (1982), 1–15.
L. Biliotti and A. Ghigi, Satake–Furstenberg compactifications, the moment map and λ1, American Journal of Mathematics 135 (2013), 237–274.
L. Biliotti, A. Ghigi and P. Heinzner, Coadjoint orbitopes, Osaka Journal of Mathematics 51 (2014), 935–968.
L. Biliotti, A. Ghigi and P. Heinzner, Polar orbitopes, Communications in Analysis and Geometry 21 (2013), 579–606.
L. Biliotti, A. Ghigi and P. Heinzner, A remark on the gradient map, Documenta Mathematica 19 (2014), 1017–1023.
J. Dadok, Polar coordinates induced by actions of compact Lie groups, Transactions of the American Mathematical Society 288 (1985), 125–137.
V. M. Gichev, Polar representations of compact groups and convex hulls of their orbits, Differential Geometry and its Applications 28 (2010), 608–614.
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Inventiones Mathematicae 67 (1982), 491–513.
G. Heckman, Projection of orbits and asymptotic behaviour of multiplicities of compact Lie groups, PhD thesis, University of Leiden, 1980.
P. Heinzner and A. Huckleberry, Kählerian potentials and convexity properties of the moment map, Inventiones Mathematicae 126 (1996), 65–84.
P. Heinzner and H. Stötzel, Semistable points with respect to real forms, Mathematische Annalen 338 (2007), 1–9.
P. Heinzner and P. Schützdeller, Convexity properties of gradient maps, Advances in Mathematics 225 (2010), 1119–1133.
P. Heinzner and G. W. Schwarz, Cartan decomposition of the moment map, Mathematische Annalen 337 (2007), 197–232.
P. Heinzner, G. W. Schwarz and H. Stötzel, Stratifications with respect to actions of real reductive groups, Compositio Mathematica 144 (2008), 163–185.
P. Heinzner, G. W. Schwarz and H. Stötzel, Stratifications with respect to actions of real reductive groups, arXiv:math/0611491v2.
P. Heinzner and H. Stötzel, Critical points of the square of the momentum map, in Global Aspects of Complex Geometry, Springer, Berlin, 2006, pp. 211–226.
A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, Vol. 140, Birkhäuser Boston, Boston, MA, 2002.
B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Annales Scientifiques de l’École Normale Supérieure 6 (1973), 413–455 (1974).
R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1993.
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The first two authors were partially supported by FIRB 2012 “Geometria differenziale e teoria geometrica delle funzioni”, by a grant of the Max-Planck Institut für Mathematik, Bonn and by GNSAGA of INdAM.
The second author was also supported by PRIN 2009 MIUR “Moduli, strutture geometriche e loro applicazioni”.
The third author was partially supported by DFG-priority program SPP 1388 (Darstellungstheorie)
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Biliotti, L., Ghigi, A. & Heinzner, P. Invariant convex sets in polar representations. Isr. J. Math. 213, 423–441 (2016). https://doi.org/10.1007/s11856-016-1325-6
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DOI: https://doi.org/10.1007/s11856-016-1325-6