Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.

We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $M_1$ and $M_2$:

• Recall that $P_1$ and $P_2$ are said to be associate if the projections $P_1 \cap P_2 \to M_1$ and $P_1 \cap P_2 \to M_2$ are surjective.
• Furthermore, they are called opposite if the intersection $P_1 \cap P_2$ is a Levi subgroup.
• I will say that $P_1$ and $P_2$ somewhat opposite if there exists a parabolic $P$ containing both, with the property that the images of $P_1$ and $P_2$ in the Levi quotient of $P$ are opposite parabolics.

Is it true that any pair of associate parabolics are connected by a sequence of pairwise somewhat opposite parabolics?

Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.

We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $M_1$ and $M_2$:

• Recall that $P_1$ and $P_2$ are said to be associate if the projections $P_1 \cap P_2 \to M_1$ and $P_1 \cap P_2 \to M_2$ are surjective.
• Furthermore, they are called opposite if the intersection $P_1 \cap P_2$ is a Levi subgroup.
• I will say that $P_1$ and $P_2$ are somewhat opposite if there exists a parabolic $P$ containing both, with the property that the images of $P_1$ and $P_2$ in the Levi quotient of $P$ are opposite parabolics.

Is it true that any pair of associate parabolics are connected by a sequence of pairwise somewhat opposite parabolics?