Can every parabolic subgroup be conjugated to its opposite by an element of the Weyl group? 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. 
 A: If I understood your question correct, then the answer is no. I will assume for simplicity that you are talking about parabolic subgroups of complex simple Lie groups. Then your question translates to the corresponding question about closed subsystems of root systems. Recall that the standard parabolic subgroups bijectively correspond to the closed subsystems $R$ of the root system $\Phi$ such that $R$ contains the set of all positive roots $\Phi^+$. (Here `closed' means $\alpha\in R$, $\beta\in R$ implies $\alpha+\beta\in R$.) Your question is equivalent to the following one: given a closed subsystem $R$ containing all positive roots $\Phi^+$, is there an element $w$ of Weyl group such that $w.R=-R$?
If the Weyl group contains $-id$, then the question is yes. However, for the root systems of types $A_n$ ($n\ge2$), $D_{2n+1}$ and $E_6$ this is not true: $-id$ does not belong to the Weyl group. Therefore for these groups there may exist parabolic subgroups which are not conjugate to their opposites via the action of Weyl group. And indeed, looking at $A_2$, we see that the two standard maximal parabolics are not conjugate to their opposites via Weyl group: there is no element in Weyl group which transforms the root subsystem $R_1=\Phi^+\cup\{-\alpha_1\}$ to $-R_1$, and the same is true for $R_2=\Phi^+\cup\{-\alpha_2\}$.
A: I believe this is false, and that a counterexample occurs in $\mathrm{SL}(3)$ already. 
Let $P$ be the block upper triangular subgroup with a $2\times2$ block in the top left corner. If I understand your terminology correctly, it's "opposite" is the block lower triangular group with the same block. 
If a Weyl group element were to conjugate one to the other, it would have to map the root corresponding to the (1,2)-entry to that of the (2,1)-entry, and also (1,3) to (3,1). That is impossible for a single element of $S_3$. 
A: 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.
