Langlands duality for Hitchin systems

Ron Donagi; Tony Pantev

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Langlands duality for Hitchin systems

Ron Donagi·Tony Pantev

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Abstract

We show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In particular, the general fiber of the connected component $\Higgs_0$ of the Hitchin system for $G$ is an abelian variety which is dual to the corresponding fiber of the connected component of the Hitchin system for $\lan{G}$. The non-neutral connected components $\Higgs_α$ form torsors over $\Higgs_0$. We show that their duals are gerbes over $\Higgs_0$ which are induced by the gerbe of $G$-Higgs bundles $\gHiggs$. More generally, we establish a duality between the gerbe $\gHiggs$ of $G$-Higgs bundles and the gerbe $\lan{\gHiggs}$ of $\lan{G}$-Higgs bundles, which incorporates all the previous dualities. All these results extend immediately to an arbirtary connected complex reductive group $\mathbb{G}$.

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BibTeX RIS

Ron Donagi, Tony Pantev. 2011-12-22. Langlands duality for Hitchin systems. https://arxiv.org/abs/math/0604617

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