Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants
[Fibrés paraboliques, produits de classes de conjugaison, et cohomologie quantique]
Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 713-748.

L'ensemble des classes de conjugaison apparaissant dans un produit de classes de conjugaison d'un groupe de Lie 1-connexe compact peut être identifié avec un polytope convexe dans une chambre pour le groupe affine de Weyl. Nous démontrons que les inégalités linéaires définissant ce polytope correspondent aux invariants de Gromov- Witten pour les variétés de drapeaux généralisées. Ceci généralise les résultats de Agnihotri, du deuxième auteur et de Belkale sur les valeurs propres d'un produit de matrices unitaires et la cohomologie quantique des grassmanniennes.

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1-connected Lie group K can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety G/P, where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for SU(n) of Agnihotri and the second author and Belkale on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

DOI : https://doi.org/10.5802/aif.1957
Classification : 14L30,  14N35,  05E99
Mots clés : classes de conjugaison, fibrés paraboliques, cohomologie quantique
@article{AIF_2003__53_3_713_0,
     author = {Teleman, Constantin and Woodward, Christopher},
     title = {Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants},
     journal = {Annales de l'Institut Fourier},
     pages = {713--748},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {3},
     year = {2003},
     doi = {10.5802/aif.1957},
     zbl = {1041.14025},
     mrnumber = {2008438},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1957/}
}
Teleman, Constantin; Woodward, Christopher. Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants. Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 713-748. doi : 10.5802/aif.1957. http://archive.numdam.org/articles/10.5802/aif.1957/

[1] S. Agnihotri; C. Woodward Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett, Volume 5-6 (1998), pp. 817-836 | MR 1671192 | Zbl 1004.14013

[2] A. Alekseev; A. Malkin; E. Meinrenken Lie group valued moment maps, J. Differential Geom, Volume 48 (1998) no. 3, pp. 445-495 | MR 1638045 | Zbl 0948.53045

[3] M. F. Atiyah; R. Bott The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London, Ser. A, Volume 308 (1982), pp. 523-615 | MR 702806 | Zbl 0509.14014

[4] V. Balaji; I. Biswas; D. S. Nagaraj Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J (2), Volume 53 (2001) no. 3, pp. 337-367 | Article | MR 1844373 | Zbl 01695159

[5] A. Beauville; Y. Laszlo Un lemme de descente, C. R. Acad. Sci. Paris, Sér. I Math, Volume 320 (1995) no. 3, pp. 335-340 | MR 1320381 | Zbl 0852.13005

[6] A. Beauville; Y. Laszlo; Ch. Sorger The Picard group of the moduli of G-bundles on a curve (Compositio Math., to appear) | MR 1626025 | Zbl 0976.14024

[7] P. Belkale Local systems on 1 - S for S a finite set, Compositio Math, Volume 129 (2001) no. 1, pp. 67-86 | Article | MR 1856023 | Zbl 1042.14031

[8] A. Berenstein; R. Sjamaar Coadjoint orbits, moment polytopes and the Hilbert-Mumford criterion, J. Amer. Math. Soc, (electronic), Volume 13 (2000) no. 2, pp. 433-466 | MR 1750957 | Zbl 0979.53092

[9] U. Bhosle; A. Ramanathan Moduli of parabolic G-bundles on curves, Math. Z, Volume 202 (1989) no. 2, pp. 161-180 | Article | MR 1013082 | Zbl 0686.14012

[10] I. Biswas Parabolic bundles as orbifold bundles, Duke Math. J, Volume 88 (1997) no. 2, pp. 305-325 | Article | MR 1455522 | Zbl 0955.14010

[11] I. Biswas A criterion for the existence of a parabolic stable bundle of rank two over the projective line, Internat. J. Math, Volume 9 (1998) no. 5, pp. 523-533 | Article | MR 1644048 | Zbl 0939.14015

[12] H. Boden Representations of orbifold groups and parabolic bundles, Comment. Math. Helvetici, Volume 66 (1991), pp. 389-447 | Article | MR 1120654 | Zbl 0758.57013

[13] G. D. Daskalopoulos The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geom, Volume 36 (1992) no. 3, pp. 699-746 | MR 1189501 | Zbl 0785.58014

[14] G. D. Daskalopoulos; R. A. Wentworth The Yang-Mills flow near the boundary of Teichmüller space, Math. Ann, Volume 318 (2000) no. 1, pp. 1-42 | Article | MR 1785574 | Zbl 0997.53046

[15] I. V. Dolgachev; Y. Hu Variation of geometric invariant theory quotients, with an appendix by Nicolas Ressayre, Inst. Hautes Études Sci. Publ. Math, Volume 87 (1998), pp. 5-56 | Numdam | MR 1659282 | Zbl 1001.14018

[16] S. K. Donaldson; P. Kronheimer The geometry of four-manifolds., Oxford Mathematical Monographs, Oxford University Press, New York, 1990 | MR 1079726 | Zbl 0820.57002

[17] V. G. Drinfeld; C. Simpson B-structures on G-bundles and local triviality, Math. Res. Lett, Volume 2 (1995) no. 6, pp. 823-829 | MR 1362973 | Zbl 0874.14043

[18] A. L. Edmonds; R. S. Kulkarni; R. E. Stong Realizability of branched coverings of surfaces, Trans. Amer. Math. Soc, Volume 282 (1984) no. 2, pp. 773-790 | Article | MR 732119 | Zbl 0603.57001

[19] M. Entov K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math, Volume 146 (2001) no. 1, pp. 93-141 | Article | MR 1859019 | Zbl 1039.53099

[20] G. Faltings Stable G-bundles and projective connections, J. Algebraic Geom, Volume 2 (1993) no. 3, pp. 507-568 | MR 1211997 | Zbl 0790.14019

[21] W. Fulton; R. Pandharipande Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995 (1997), pp. 45-96 | Zbl 0898.14018

[22] W. Fulton; C. Woodward Quantum products of Schubert classes (2001) (preprint) | MR 1837123

[23] M. Furuta; B. Steer Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math, Volume 96 (1992) no. 1, pp. 38-102 | Article | MR 1185787 | Zbl 0769.58009

[24] M. Gotô A theorem on compact semi-simple groups, J. Math. Soc. Japan, Volume 1 (1949), pp. 270-272 | Article | MR 33829 | Zbl 0041.36208

[25] A. Grothendieck Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math, Volume 79 (1957), pp. 121-138 | Article | MR 87176 | Zbl 0079.17001

[26] A. Grothendieck Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math (1961) no. 8, pp. 222 pp | Numdam | MR 163909 | Zbl 0118.36206

[27] A. Grothendieck Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math (1961) no. 11, pp. 167 pp | Numdam | MR 163910

[28] R. Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, volume 52, Springer-Verlag, Berlin-Heidelberg-New York, 1977 | MR 463157 | Zbl 0367.14001

[29] P. Heinzner; F. Kutzschebauch An equivariant version of Grauert's Oka principle, Invent. Math, Volume 119 (1995) no. 2, pp. 317-346 | Article | MR 1312503 | Zbl 0837.32004

[30] D. Huybrechts; M. Lehn Stable pairs on curves and surfaces, J. Algebraic Geom, Volume 4 (1995) no. 1, pp. 67-104 | MR 1299005 | Zbl 0839.14023

[31] L. C. Jeffrey Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann, Volume 298 (1994), pp. 667-692 | Article | MR 1268599 | Zbl 0794.53017

[32] A. A. Klyachko Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), Volume 4 (1998) no. 3, pp. 419-445 | Article | MR 1654578 | Zbl 0915.14010

[33] A. Knutson; T. Tao The honeycomb model of gl n (c) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc, Volume 12 (1999) no. 4, pp. 1055-1090 | Article | MR 1671451 | Zbl 0944.05097

[34] A. Knutson; T. Tao; C. Woodward Honeycombs II: Facets of the Littlewood-Richardson cone (to appear in Jour. Am. Math. Soc.)

[35] Y. Laszlo; C. Sorger The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4), Volume 30 (1997) no. 4, pp. 499-525 | Numdam | MR 1456243 | Zbl 0918.14004

[36] V.B. Mehta; C. S. Seshadri Moduli of vector bundles on curves with parabolic structure, Math. Ann, Volume 248 (1980), pp. 205-239 | Article | MR 575939 | Zbl 0454.14006

[37] E. Meinrenken; C. Woodward Hamiltonian loop group actions and Verlinde factorization, Journal of Differential Geometry, Volume 50 (1999), pp. 417-470 | MR 1690736 | Zbl 0949.37031

[38] J. Millson; B. Leeb Convex functions on symmetric spaces and geometric invariant theory for spaces of weighted configurations on flag manifolds (2000) (Preprint)

[39] D. Mumford The red book of varieties and schemes. Includes the Michigan Lectures (1974) on "curves and their Jacobians", with contributions by Enrico Arbarello, Lecture Notes in Mathematics, volume 1358, Springer-Verlag, Berlin, 1999 | MR 1748380 | Zbl 0945.14001

[40] P. E. Newstead Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay., Volume volume 51 (1978) | MR 546290 | Zbl 0411.14003

[41] C. Pauly Espaces de modules de fibrés paraboliques et blocs conformes, Duke Math. J, Volume 84 (1996), pp. 217-235 | MR 1394754 | Zbl 0877.14031

[42] D. Peterson Lectures on quantum cohomology of G/P.M.I.T (1997)

[43] J. R\hbox{\pet \aa}de On the Yang-Mills heat equation in two and three dimensions, J. reine angew. Math, Volume 431 (1992), pp. 123-163 | MR 1179335 | Zbl 0760.58041

[44] A. Ramanathan Moduli for principal bundles over algebraic curves, I, Proc. Indian Acad. Sci. Math. Sci, Volume 106 (1996) no. 3, pp. 301-328 | Article | MR 1420170 | Zbl 0901.14007

[45] A. Ramanathan Moduli for principal bundles over algebraic curves, II, Proc. Indian Acad. Sci. Math. Sci, Volume 106 (1996) no. 4, pp. 421-449 | Article | MR 1425616 | Zbl 0901.14008

[46] J.-P. Serre Cohomologie galoisienne, Springer-Verlag, Berlin, 1994 | MR 1324577 | Zbl 0812.12002

[47] C. S. Seshadri Fibrés vectoriels sur les courbes algébriques, Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980 (Astérisque), Volume volume 96 (1982) | Zbl 0517.14008

[48] C. T. Simpson Harmonic bundles on noncompact curves, J. Amer. Math. Soc, Volume 3 (1990) no. 3, pp. 713-770 | Article | MR 1040197 | Zbl 0713.58012

[49] P. Slodowy Two notes on a finiteness problem in the representation theory of finite groups, Algebraic groups and Lie groups (Austral. Math. Soc. Lect. Ser), Volume volume 9 ; appendix by G. Martin Cram (1997), pp. 331-348 | Zbl 0874.22011

[50] C. Teleman Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve, Invent. Math, Volume 134 (1998) no. 1, pp. 1-57 | Article | MR 1646586 | Zbl 0980.14025

[51] C. Teleman The quantization conjecture revisited, Ann. of Math. (2), Volume 152 (2000) no. 1, pp. 1-43 | Article | MR 1792291 | Zbl 0980.53102

[52] M. Thaddeus Geometric invariant theory and flips, J. Amer. Math. Soc, Volume 9 (1996) no. 3, pp. 691-723 | Article | MR 1333296 | Zbl 0874.14042

[53] A. Weil Remarks on the cohomology of groups, Ann. of Math. (2), Volume 80 (1964), pp. 149-157 | Article | MR 169956 | Zbl 0192.12802

[54] C. Woodward On D. Peterson's comparison formula for Gromov-Witten invariants of G/P (e-print, math.AG/0206073) | Zbl 1077.14085