Suppose $P_I$ is a standard parabolic subgroup of $GL_n(k)$ (we can assume $k$ is finite, but I doubt it matters). Is $P_I$ conjugate to the opposite parabolic ${P_I}^-$ under some element of the Weyl group $W$? Is it perhaps the longest element of the Weyl subgroup $W_I$?
I know in general it's not possible to conjugate a parabolic to its opposite, but I wonder if it's at least possible for $GL_n(k)$? If it is true, a reference would be sufficient. Thanks.