Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic.
Can we do the same thing if we choose a standard parabolic subgroup? Can we always find an element in the weyl group such that conjugation by this element takes it to the opposite parabolic? I am guessing this should be possible by modifying the longest element in the weyl group. Is this true? Any help is appreciated.