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ABELIANIZATION OF HIGGS BUNDLES FOR QUASI-SPLIT REAL GROUPS

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Abstract

We study the Hitchin map for Gℝ-Higgs bundles on a smooth curve, where Gℝ is a quasi-split real form of a complex reductive algebraic group G. By looking at the moduli stack of regular Gℝ-Higgs bundles, we prove that it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The corresponding points inside the stack of G-Higgs bundles are contained in the substack of points fixed by an involution induced by the Cartan involution of Gℝ. We determine this substack of fixed points and prove that stable points are in correspondence with stable Gℝ-Higgs bundles.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas, c/ Nicolás Cabrera 13–15, Campus de Cantoblanco, 28049, Madrid, Spain

    OSCAR GARCÍA-PRADA

  2. School of Mathematics, University of Birmingham, Watson Building, Edgebaston, Birmingham, B15 2TT, UK

    ANA PEÓN-NIETO

  3. Laboratoire de Mathématiques, J. A. Dieudonné, Université de Nice Sophia-Antipolis, 06108, Nice Cedex 02, France

    ANA PEÓN-NIETO

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  1. OSCAR GARCÍA-PRADA
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  2. ANA PEÓN-NIETO
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Correspondence to OSCAR GARCÍA-PRADA.

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Oscar García-Prada is partially supported by the Spanish MINECO under ICMAT Severo Ochoa project No. SEV-2015-0554, and under grant No. MTM2013-43963-P.

Ana Peón-Nieto is supported by the FPU doctoral scholarship number AP2008-291 from Ministerio de Educación.

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GARCÍA-PRADA, O., PEÓN-NIETO, A. ABELIANIZATION OF HIGGS BUNDLES FOR QUASI-SPLIT REAL GROUPS. Transformation Groups 28, 285–325 (2023). https://doi.org/10.1007/s00031-021-09658-9

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