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Higgs bundles for real groups and the Hitchin-Kostant-Rallis section
| Content Provider | Semantic Scholar |
|---|---|
| Author | García-Prada, Oscar Pe'on-Nieto, Ana Ramanan, S. |
| Copyright Year | 2015 |
| 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 at a basis of the ring of polynomial invariants of the isotropy representation, one 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. Building up on the work of Kostant-Rallis and Hitchin, in this paper, as a first step in the study of the Hitchin map, we construct a section of this map. This generalizes the section constructed by Hitchin 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-Teichm\"uller components of the moduli space of representations of the fundamental group of $X$ in $G_{\mathrm{Split}}$, a split real form of $G$. In fact, our construction is very natural in that we can start with the moduli space for $G_{\mathrm{Split}}$, instead of $G$, and construct the section for the Hitchin map for $G_{\mathrm{Split}}$ directly. The construction involves the notion of maximal split subgroup of a real reductive Lie group. |
| Starting Page | 2907 |
| Ending Page | 2953 |
| Page Count | 47 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/tran/7363 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1511.02611v2.pdf |
| Alternate Webpage(s) | https://www.mathi.uni-heidelberg.de/~apeonnieto/Preprints/HKRweb.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/tran%2F7363 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
