Automorphisms of multiplicity free Hamiltonian manifolds

Knop, Friedrich

doi:10.1090/S0894-0347-2010-00686-8

Automorphisms of multiplicity free Hamiltonian manifolds
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by Friedrich Knop

J. Amer. Math. Soc. 24 (2011), 567-601

DOI: https://doi.org/10.1090/S0894-0347-2010-00686-8

Published electronically: November 30, 2010

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Abstract:

Let $M$ be a multiplicity free Hamiltonian manifold $M$ for a connected compact Lie group $K$ (not necessarily abelian). Let $\mathcal {P}$ be the momentum polytope of $M$. We calculate the automorphism of $M$ as a sheaf over $\mathcal {P}$ and show that all higher cohomology groups of this sheaf vanish. From this, and a recent theorem of Losev, we deduce a conjecture of Delzant: the momentum polytope and the principal isotropy group determine $M$ up to isomorphism. Moreover, we give a criterion for when a polytope and a group are afforded by a multiplicity free manifold.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
Edward Bierstone and Pierre D. Milman, Composite differentiable functions, Ann. of Math. (2) 116 (1982), no. 3, 541–558. MR 678480, DOI 10.2307/2007022
Michel Brion, Sur l’image de l’application moment, Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986) Lecture Notes in Math., vol. 1296, Springer, Berlin, 1987, pp. 177–192 (French). MR 932055, DOI 10.1007/BFb0078526
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
Camus, R., Variétés sphériques affines lisses, Thèse de doctorat (Université J. Fourier) (2001).
Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
Thomas Delzant, Classification des actions hamiltoniennes complètement intégrables de rang deux, Ann. Global Anal. Geom. 8 (1990), no. 1, 87–112 (French). MR 1075241, DOI 10.1007/BF00055020
Victor Guillemin and Reyer Sjamaar, Convexity properties of Hamiltonian group actions, CRM Monograph Series, vol. 26, American Mathematical Society, Providence, RI, 2005. MR 2175783, DOI 10.1090/crmm/026
Victor Guillemin and Shlomo Sternberg, Multiplicity-free spaces, J. Differential Geom. 19 (1984), no. 1, 31–56. MR 739781
Patrick Iglésias, Les $\textrm {SO}(3)$-variétés symplectiques et leur classification en dimension $4$, Bull. Soc. Math. France 119 (1991), no. 3, 371–396 (French, with English summary). MR 1125672
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Yael Karshon and Eugene Lerman, The centralizer of invariant functions and division properties of the moment map, Illinois J. Math. 41 (1997), no. 3, 462–487. MR 1458185
Frances Kirwan, Convexity properties of the moment mapping. III, Invent. Math. 77 (1984), no. 3, 547–552. MR 759257, DOI 10.1007/BF01388838
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
Friedrich Knop, Weylgruppe und Momentabbildung, Invent. Math. 99 (1990), no. 1, 1–23 (German, with English summary). MR 1029388, DOI 10.1007/BF01234409
Friedrich Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), no. 1-3, 309–328. MR 1253195, DOI 10.1007/BF01231563
Friedrich Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153–174. MR 1311823, DOI 10.1090/S0894-0347-96-00179-8
—, Towards a classification of multiplicity free manifolds, Handout for the conference “Journées Hamiltoniennes” in Grenoble (Nov. 29–30, 1997), 8 pages.
—, Weyl groups of Hamiltonian manifolds, I, Preprint (1997), 33 pages, dg-ga/9712010.
Friedrich Knop, Convexity of Hamiltonian manifolds, J. Lie Theory 12 (2002), no. 2, 571–582. MR 1923787
Friedrich Knop and Bart Van Steirteghem, Classification of smooth affine spherical varieties, Transform. Groups 11 (2006), no. 3, 495–516. MR 2264463, DOI 10.1007/s00031-005-1116-3
Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
Eugene Lerman, Eckhard Meinrenken, Sue Tolman, and Chris Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998), no. 2, 245–259. MR 1489203, DOI 10.1016/S0040-9383(97)00030-X
Ivan V. Losev, Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 1105–1134 (English, with English and French summaries). MR 2543664
D. Luna, Variétés sphériques de type $A$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226 (French). MR 1896179, DOI 10.1007/s10240-001-8194-0
Andy R. Magid, Equivariant completions and tensor products, Group actions and invariant theory (Montreal, PQ, 1988) CMS Conf. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 133–136. MR 1021285
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
A. S. Miščenko and A. T. Fomenko, A generalized Liouville method for the integration of Hamiltonian systems, Funkcional. Anal. i Priložen. 12 (1978), no. 2, 46–56, 96 (Russian). MR 0516342
Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
Reyer Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), no. 1, 46–91. MR 1645052, DOI 10.1006/aima.1998.1739
Chris Woodward, The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom. 14 (1996), no. 1, 3–42. MR 1375064, DOI 10.1007/BF00128193
Chris T. Woodward, Spherical varieties and existence of invariant Kähler structures, Duke Math. J. 93 (1998), no. 2, 345–377. MR 1625995, DOI 10.1215/S0012-7094-98-09312-7