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.

`$B$`

is fixed and the rank is`$r$`

, there are`$2^r$`

nonconjugatestandardparabolics containing`$B$`

. While`$B$`

itself is minimal in this collection, it's usual to call those with a single negative root "minimal"; similarly "maximal" includes "proper" in this context. $\endgroup$ – Jim Humphreys Apr 21 '11 at 13:30