Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$

Authors

  • Álvaro Antón Sancho

DOI:

https://doi.org/10.7146/math.scand.a-26348

Abstract

In this work we study finite-order automorphisms of the moduli space of principal $G$-bundles coming from outer automorphisms of the structure group when $G$ is a simple complex Lie group. We do this by describing the subvarieties of fixed points for the action of that automorphisms on the moduli space of principal $G$-bundles. In particular, we prove that these fixed points are reductions of structure group to the subgroup of fixed points of the outer automorphism. Moreover, we study the way in which these fixed points fall into the stable or nonstable locus of the moduli.

References

Adams, J. F., Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.

Antón Sancho, Á., Principal Spin-bundles and triality, Rev. Colombiana Mat. 49 (2015), no. 2, 235–259. https://doi.org/10.15446/recolma.v49n2.60442

Antón Sancho, Á., The moduli space of principal Spin bundles, Rev. Un. Mat. Argentina 57 (2016), no. 2, 25–51.

Biswas, I., Gómez, T. L., and Muñoz, V., Automorphisms of moduli spaces of symplectic bundles, Internat. J. Math. 23 (2012), no. 5, 1250052, 27pp. https://doi.org/10.1142/S0129167X12500528

Biswas, I., Gómez, T. L., and Muñoz, V., Automorphisms of moduli spaces of vector bundles over a curve, Expo. Math. 31 (2013), no. 1, 73–86. https://doi.org/10.1016/j.exmath.2012.08.002

Bryant, R. L., Some remarks on $G_2$-structures, in “Proceedings of Gökova Geometry-Topology Conference 2005”, Gökova Geometry/Topology Conference (GGT), Gökova, 2006, pp. 75--109.

Cartan, É., Sur la structure des groupes de transformations finis et continus, Ph.D. thesis, Paris (Nony), 1894, in Cartan, E., Œuvres complètes. Partie I. Groupes de Lie, Gauthier-Villars, Paris, 1952.

Fulton, W. and Harris, J., Representation theory. A first course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0979-9

García-Prada, O., Gothen, P. B., and Mundet i Riera, I., Higgs bundles and surface group representations in the real symplectic group, J. Topol. 6 (2013), no. 1, 64–118. https://doi.org/10.1112/jtopol/jts030

Gómez, T. and Sols, I., Moduli space of principal sheaves over projective varieties, Ann. of Math. (2) 161 (2005), no. 2, 1037–1092. https://doi.org/10.4007/annals.2005.161.1037

Hwang, J.-M. and Ramanan, S., Hecke curves and Hitchin discriminant, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 801–817. https://doi.org/10.1016/j.ansens.2004.07.001

Kouvidakis, A. and Pantev, T., The automorphism group of the moduli space of semistable vector bundles, Math. Ann. 302 (1995), no. 2, 225–268. https://doi.org/10.1007/BF01444495

Mumford, D., Projective invariants of projective structures and applications, in “Proc. Internat. Congr. Mathematicians (Stockholm, 1962)'', Inst. Mittag-Leffler, Djursholm, 1963, pp. 526--530.

Narasimhan, M. S. and Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. https://doi.org/10.2307/1970710

Onishchik, A. L. and Vinberg, E. B. (eds.), Lie groups and Lie algebras III, Encyclopedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994.

Ramanan, S., Orthogonal and Spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 151–166. https://doi.org/10.1007/BF02837285

Ramanathan, A., Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152. https://doi.org/10.1007/BF01343949

Ramanathan, A., Moduli for principal bundles over algebraic curves. I, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 3, 301–328. https://doi.org/10.1007/BF02867438

Ramanathan, A., Moduli for principal bundles over algebraic curves. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 421–449. https://doi.org/10.1007/BF02837697

Rubio, R., Exceptional $G$-Higgs bundles, DEA thesis, University Autónoma, Madrid, 2007.

Serman, O., Moduli spaces of orthogonal and symplectic bundles over an algebraic curve, Compos. Math. 144 (2008), no. 3, 721–733. https://doi.org/10.1112/S0010437X07003247

Springer, T. A. and Veldkamp, F. D., Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/978-3-662-12622-6

Wolf, J. A. and Gray, A., Homogeneous spaces defined by Lie group automorphisms. I, J. Differential Geometry 2 (1968), 77–114.

Wolf, J. A. and Gray, A., Homogeneous spaces defined by Lie group automorphisms. II, J. Differential Geometry 2 (1968), 115–159.

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Published

2018-02-20

How to Cite

Antón Sancho, Álvaro. (2018). Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$. MATHEMATICA SCANDINAVICA, 122(1), 53–83. https://doi.org/10.7146/math.scand.a-26348

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