no. 3
Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants
[Fibrés paraboliques, produits de classes de conjugaison, et cohomologie quantique]
Teleman, Constantin1 ; Woodward, Christopher2
1 University of Cambridge, DPMMS, CMS, Wilberforce Road, Cambridge CB3 0WB (Grande-Bretagne)
2 Rutgers University, Mathematics, Hill Center, 110 Frelinghuysen Road, Piscataway NJ 08854-8019 (USA)
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 : 10.5802/aif.1957
Classification : 14L30, 14N35, 05E99
Keywords: conjugacy classes, parabolic bundles, quantum cohomology
Mot clés : classes de conjugaison, fibrés paraboliques, cohomologie quantique

Teleman, Constantin 1 ; Woodward, Christopher 2

1 University of Cambridge, DPMMS, CMS, Wilberforce Road, Cambridge CB3 0WB (Grande-Bretagne)
2 Rutgers University, Mathematics, Hill Center, 110 Frelinghuysen Road, Piscataway NJ 08854-8019 (USA) 
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.1957/}
}
TY  - JOUR
AU  - Teleman, Constantin
AU  - Woodward, Christopher
TI  - Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 713
EP  - 748
VL  - 53
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1957/
DO  - 10.5802/aif.1957
LA  - en
ID  - AIF_2003__53_3_713_0
ER  - 
%0 Journal Article
%A Teleman, Constantin
%A Woodward, Christopher
%T Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants
%J Annales de l'Institut Fourier
%D 2003
%P 713-748
%V 53
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1957/
%R 10.5802/aif.1957
%G en
%F AIF_2003__53_3_713_0
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. https://aif.centre-mersenne.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 | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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

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

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

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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

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

[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

[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 | DOI | MR | Zbl

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

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

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

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

Cité par Sources :