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We study topologically trivial  G -Higgs bundles over an elliptic curve  X  when the structure group  G  is a connected real form of a complex semisimple Lie groupG   ℂ    . We achieve a description of their (reduced) moduli space, the associated Hitchin fibration and the finite morphism to the moduli space ofG   ℂ    -Higgs bundles.

Higgs bundles over elliptic curves for real groups