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 $L \cup L'$?
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1$\begingroup$ Looks like homework... $\endgroup$– Alain ValetteCommented Oct 23, 2011 at 15:34
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$\begingroup$ Of course I can construct myself the stabilizer, I was just wondering if it has (i.e. someone sees) different presentation than the trivial one. $\endgroup$– HicEtNuncCommented Oct 23, 2011 at 15:57
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$\begingroup$ If the lines are in general position you might as well take them to be the lines $z=0$ and $y=0$, in which case you can describe the group in terms of the matrix elements. $\endgroup$– Will SawinCommented Oct 23, 2011 at 18:37
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$\begingroup$ Thank you for your comment. In fact that's what I did, but it doesn't seem very enlightening, I expected to spot two copies of SL2 (or a subset of SL2 x SL2) but I don't really manage... $\endgroup$– HicEtNuncCommented Oct 23, 2011 at 19:41
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1 Answer
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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.
answered Oct 23, 2011 at 20:53