If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find some subgroups of $G_x$? I mean if we take the space $(x,0,0)$ then $S^1$ stabilizes this axis and it is a subgroup of $G_x$ and if we take the complementary plane $(0,y,z)$ (what we can say here?).
I would appreciate the solution with details.. Thanks
The action of $SU(3)$ on $\mathbb C^3$ is far from being transitive. In any linear representation $0$ is a fixed point, but in addition, maps in $SU(3)$ are norm-preserving. Linear algebra shows that the orbits of $SU(3)$ in $\mathbb C^3$ are the spheres of radius $r$ for $r\geq 0$. The stabilizers of all vectors $x\neq 0$ are isomorphic: Any transformation fixing $x$ acts as the identity on the complex line $\mathbb C\cdot x$ spanned by $x$. By unitarity, the transformation also fixes the complex orthocomplement $x^\perp\cong\mathbb C^2$. It is easy to see that mapping $A$ to its action on this orthocomplement gives rise to an isomorphism $G_x\cong SU(2)$. So you obtain a representation of the sphere $S^5$ as a homogeneous space $SU(3)/SU(2)$. The action of $SU(3)$ on $S^5$ preserves a rather rich geometric structure on $S^5$, called a Sasaki structure. This includes the round metric as well as the CR structure on $S^5$ defined by the maximal complex subspaces in tangent spaces. In addition, there is a "complex volume form" on the CR subbundle preserved by the action of $SU(3)$.