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Higgs bundles for real groups and the Hitchin-Kostant-Rallis section

Authors: Oscar García-Prada, Ana Peón-Nieto and S. Ramanan
Journal: Trans. Amer. Math. Soc. 370 (2018), 2907-2953
MSC (2010): Primary 14H60; Secondary 53C07, 58D29
Published electronically: November 14, 2017
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Abstract: We consider the moduli space of polystable $ L$-twisted $ G$-Higgs bundles over a compact Riemann surface $ X$, where $ G$ is a real reductive Lie group and $ L$ is a holomorphic line bundle over $ X$. Evaluating the Higgs field on a basis of the ring of polynomial invariants of the isotropy representation defines the Hitchin map. This is a map to an affine space whose dimension is determined by $ L$ and the degrees of the polynomials in the basis. In this paper, we construct a section of this map and identify the connected components of the moduli space containing the image. This section factors through the moduli space for $ G_{\mathrm {split}}$, a split real subgroup of $ G$. Our results generalize those by Hitchin, who considered the case when $ L$ is the canonical line bundle of $ X$ and $ G$ is complex. In this case, the image of the section is related to the Hitchin-Teichmüller components of the moduli space of representations of the fundamental group of $ X$ in $ G_{\mathrm {split}}$, a split real form of $ G$. The construction involves the notion of a maximal split subgroup of a real reductive Lie group and builds on results by Kostant and Rallis.

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Additional Information

Oscar García-Prada
Affiliation: Instituto de Ciencias Matemáticas, CSIC-UAM-UCM-UC3M, Nicolás Cabrera 13, 28049 Madrid, Spain

Ana Peón-Nieto
Affiliation: Ruprecht-Karls-Universität, Heidelberg Mathematisches Institut, Im Neuenheimer Fel, 69120 Heidelberg, Germany

S. Ramanan
Affiliation: Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India

Received by editor(s): February 8, 2017
Published electronically: November 14, 2017
Additional Notes: The second author was supported by an FPU grant from the Ministerio de Educación
Article copyright: © Copyright 2017 American Mathematical Society

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