Abstract
In this article, we study the local invariants associated to the Hamiltonian action of a compact torus. Our main results are wall-crossing formulas between invariants attached to adjacent connected components of regular values of the moment map.
Mathematics Subject Classification (2010): 58J20, 53D20, 53D50
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References
1 M.F.ATIYAH,Ellipticoperatorsandcompactgroups,Springer,1974.LecturenotesinMathematics, 401.
M.F. ATIYAH, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14, 1982, pp. 1–15.
M.F. ATIYAH and G.B. SEGAL, The index of elliptic operators II, Ann. Math. 87, 1968, pp. 531–545.
M.F. ATIYAH and I.M. SINGER, The index of elliptic operators I, Ann. Math. 87, 1968, pp. 484–530.
M.F. ATIYAH and I.M. SINGER, The index of elliptic operators III, Ann. Math. 87, 1968, pp. 546–604.
M.F. ATIYAH and I.M. SINGER, The index of elliptic operators IV, Ann. Math. 93, 1971, pp. 139–141.
N. BERLINE,E. GETZLER,and M. VERGNE, Heat kernels and Dirac operators, Grundlehren, vol. 298, Springer, Berlin, 1991.
N.BERLINE andM.VERGNE,TheCherncharacterofatransversallyellipticsymbolandthe equivariant index, Invent. Math. 124, 1996, pp. 11–49.
N. BERLINE and M. VERGNE, L’indice equivariant des op´erateurs transversalement ellip ´tiques, Invent. Math. 124, 1996, pp. 51–101.
S. BILLEY,V. GUILLEMIN,and E. RASSART, A vector partition function for the multiplicities of slk (C), J. Algebra 278, 2004, pp. 251–293.
A. BOYSAL and M. VERGNE, Paradan’s wall crossing formula for partition functions and Khovanski–Pukhlikov differential operator, arXiv:math, CO/0803.2810.
M. BRION and C. PROCESI, Action d’un tore dans une vari´et´e projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), pp. 509–539, Progr. Math., 92, Birkh¨auser Boston, Boston, MA, 1990.
W. DAHMEN and C.A. MICHELLI, The number of solutions to linear Diophantine equations and multivariate splines, Trans. Amer. Math. Soc. 308, 1988, pp. 509–532.
C.DECONCINI,C.PROCESI,andM.VERGNE,PartitionfunctionandgeneralizedDahmen– Micchelli spaces, arXiv:math, CO/0805.2907.
J.J.DUISTERMAAT andG.J.HECKMAN,Onthevariationinthecohomologyinthesymplectic form of the reduced phase space, Invent. Math. 69, 1982, pp. 259–268; addendum, ibid. 72, 1983; pp. 153–158.
J.J. DUISTERMAAT, The heat Lefschetz fixed point formula for the Spin c -Dirac operator, Progress in Nonlinear Differential Equations and Their Applications, vol. 18, Birkh¨auser, Boston, 1996.
V. GUILLEMIN,E. LERMAN,and S. STERNBERG, Symplectic fibrations and multiplicity diagrams. Cambridge University Press 1996.
V. GUILLEMIN and S. STERNBERG, Convexity properties of the moment mapping, Invent. Math. 67, 1982, pp. 491–513.
V. GUILLEMIN and S. STERNBERG,Geometricquantizationandmultiplicitiesofgrouprepresentations, Invent. Math. 67, 1982, pp. 515–538.
V.GUILLEMIN andS.STERNBERG,Birationalequivalenceinthesymplecticcategory, Invent. Math. 97, 1989, pp. 485–522.
V. GUILLEMIN and S. STERNBERG, Supersymmetry and equivariant de Rham theory. With an appendix containing two reprints by Henri Cartan. Mathematics Past and Present. Springer-Verlag, Berlin, 1999.
G. J. HECKMAN, Projections of orbits and asymptotic behavior of multiplicities for compact Lie groups, PhD thesis, University of Leiden, 1980.
T. KAWASAKI, The Riemann–Roch theorem for complex V-manifolds, Osaka J. Math. 16, 1979, pp. 151–157.
F. KIRWAN, Cohomology of quotients in symplectic and algebraic geometry, Princeton Univ. Press, Princeton, 1984.
B. KOSTANT, On convexity, the Weyl group and the Iwasawa decomposition, Annales Scientifiques de l’E. N. S. 6, 1973, pp. 413–455.
S. KUMAR and M. VERGNE, Equivariant cohomology with generalized coefficients, Ast´erisque, 215, 1993, pp. 109–204.
E. MEINRENKEN,On Riemann–Roch formulas for multiplicities, J. Amer. Math. Soc. 9, 1996, pp. 373–389.
E.MEINRENKEN,SymplecticsurgeryandtheSpinc-Diracoperator, Adv. in Math. 134, 1998, pp. 240–277.
E.MEINRENKEN andS.SJAMAAR,Singularreductionandquantization, Topology 38, 1999, pp. 699–762.
P.-E. PARADAN, Formules de localisation en cohomologie ´equivariante, Compositio Mathematica 117, 1999, pp. 243–293.
P.-E.PARADAN,Themomentmapandequivariantcohomologywithgeneralizedcoefficient, Topology 39, 2000, pp. 401–444.
P.-E. PARADAN, Localization of the Riemann–Roch character, J. Funct. Anal. 187, 2001, pp. 442–509.
P.-E. PARADAN,NotesurlesformulesdesautdeGuillemin–Kalkman, C. R. Acad. Sci. Paris, Ser. I 339, 2004, pp. 467–472.
R. SJAMAAR, Symplectic reduction and Riemann–Roch formulas for multiplicities, Bull. Amer. Math. Soc. 33, 1996, pp. 327–338.
B. STURMFELS, On vector partition functions, J. Combinatorial Theory 71, 1995, pp. 302–309.
A. SZENES and M. VERGNE, Residue formulae for vector partitions and Euler–Maclaurin sums, Adv. in Applied Math. 30, 2003, pp. 295–342.
M. VERGNE, Multiplicity formula for geometric quantization, Part I, Part II, and Part III, Duke Math. Journal 82, 1996, pp. 143–179, pp. 181–194, pp. 637–652.
M.VERGNE,Quantificationg´eom´etriqueetr´eductionsymplectique, S´eminaire Bourbaki 888, 2001.
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Dedicated to Hans Duistermaat on the occasion of his 65th birthday
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Paradan, PE. (2011). Wall-crossing formulas in Hamiltonian geometry. In: Kolk, J., van den Ban, E. (eds) Geometric Aspects of Analysis and Mechanics. Progress in Mathematics, vol 292. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8244-6_11
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DOI: https://doi.org/10.1007/978-0-8176-8244-6_11
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