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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

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Can every parabolic subgroup be conjugated to its opposite by an element of the Weyl group?

For real semisimple Lie groups with finite center the answer is YES: any two parabolic subgroups are conjugate by an element of the Weyl group.
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