We introduce and describe the "regular quotient" for the Hitchin fibration for symmetric spaces and explain some basic consequences for Higgs bundles. We include an invariant theoretic approach to spectral covers in this setting for the particular space GL₂ₙ/(GLₙ× GLₙ). We also include a study of the regular centralizer group scheme for quasisplit pairs, including a Galois description of a closely related group scheme. We collect some basic consequences for Hitchin systems associated to such pairs.
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Hameister et al. (2024) studied this question.
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