Hecke modifications, wonderful compactifications and moduli of principal bundles

Wong, Michael Lennox

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 309-367.

In this paper we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of “universal” Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the “wonderful,” or De Concini–Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.

Classification : 14D20, 32G08

[1] C. De Concini and C. Procesi, Complete symmetric varieties, In: “Invariant Theory (Proceedings, Montecatini 1982)”, 996 Lecture Notes in Math., Springer, 1982, 1–44. | MR | Zbl

[2] J. Duistermaat and J. Kolk, “Lie Groups”, Springer, 2000. | MR | Zbl

[3] S. Evens and B. F. Jones, On the wonderful compactification, arXiv:0801.0456v1 [math.AG] 3 Jan 2008, 2008.

[4] G. Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 41–68. | EuDML | MR | Zbl

[5] E. Frenkel and D. Ben-Zvi, “Vertex Algebras and Algebraic Curves”, Second Edition, American Mathematical Society, 2004. | MR | Zbl

[6] G. Harder, Halbeinfache Gruppenschemata über Dedekindringen, Invent. Math. 4 (1967), 165–191. | EuDML | MR | Zbl

[7] G. Harder and D. Kazhdan, Automorphic forms on $G{L}_2$ over function fields, In: “Proc. Sympos. Pure Math.”, 33, part 2 (1979), 357–379. | MR | Zbl

[8] J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory”, Springer, New York, NY, 1972. | MR | Zbl

[9] J. Hurtubise, On the geometry of isomonodromic deformations, J. Geom. Phys. 58 (2008), 1394–1406. | MR | Zbl

[10] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. I.H.E.S. 25 (1965), 5–48. | EuDML | Numdam | MR | Zbl

[11] V. G. Kac, “Infinite Dimensional Lie algebras”, Cambridge University Press, 1990. | MR | Zbl

[12] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric langlands program, arXiv:hep-th/0604151, 2006. | MR | Zbl

[13] I. Krichever, Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, Mosc. Math. J. 2 (2002), 717–752. | MR | Zbl

[14] S. Kumar, “Kac-Moody Groups, their Flag Varieties and Representation Theory”, Birkhäuser, 2002. | MR | Zbl

[15] J. M. Lansky, Decomposition of double cosets in p-adic groups, Pacific J. Math. 197 (2001), 97–117. | MR | Zbl

[16] P. Norbury, Magnetic monopoles on manifolds with boundary, Trans. Amer. Math. Soc., S 0002-9947(2010)04934-7, arXiv:0804.3649v2 [math.DG] 14 Oct 2008, 2008. | MR | Zbl

[17] A. Pressley and G. Segal, “Loop Groups”, Oxford University Press, New York, NY, 1986. | MR | Zbl

[18] A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152. | EuDML | MR | Zbl

[19] A. Tjurin, Classification of $n$-dimensional vector bundles over an algebraic curve of arbitrary genus, Izv. Ross. Akad. Nauk Ser. Mat. 30 (1966), 1353–1366. | MR | Zbl

[20] A. Weil, Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17 (1938), 47–87. | JFM