Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL\_3$ to $L\cup L'$. Can you tell in what way does $SL\_3$ act on $L \cup L'$? What is the stabilizer of $L \cup L'$?



A weird action of SL_3 on a pair of lines
One way to describe this, that fits into various larger patterns, is as a minimal parabolic intersected with its conjugate by a simple root-reflection, and with that reflection adjoined.


In coordinates: take lines $x$-axis and $y$-axis. The upper-triangular matrices $P$ form a standard minimal parabolic. The positive simple roots are $diag(a\_1,a\_2,a\_3)\rightarrow a\_1/a\_2$ and $a\_2/a\_3$. The corresponding reflections are
$$
\sigma\_1=\pmatrix{
0 & 1 & 0 \cr
1 & 0 & 0 \cr
0 & 0 & 1
}
\hskip30pt
\sigma\_2=\pmatrix{
1 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 1 & 0
}
$$
The stabilizer of the union of the two lines is the group generated by $P\cap \sigma\_1P\sigma\_1$ and $\sigma\_1$, the latter interchanging the two lines, the former subgroup stabilizing both lines individually.


Perhaps this is not entirely satisfying, but it is a structural description.


