Borel fixed-point theorem

Remarks

The proof goes by induction on $\dim G$. The derived group $G'$ is connected, normal and of smaller dimension, so its fixed-point set $Y = X^{G'}$ is nonempty, and it is closed, hence complete. The group $G$ acts on $Y$, and an orbit of minimal dimension $G \cdot y$ is closed, so it is complete. The stabilizer $G_y$ contains $G'$, so it is normal and $G / G_y$ is an affine group. The orbit map gives a bijective morphism from $G / G_y$ onto the complete orbit $G \cdot y$, and one checks that $G / G_y$ is then complete as well. A connected variety that is both affine and complete is a point, so the orbit is a point and $y$ is fixed.

Example: a connected solvable closed subgroup $H$ of $\mathrm{GL}_n$ acts on the variety of full flags in $k^n$, which is complete. The theorem gives a flag fixed by $H$, and this proves the Lie–Kolchin theorem: $H$ is conjugate to a subgroup of the group $B$ of invertible upper triangular matrices.

Completeness is needed: $\mathbb{G}_m$ acts on $\mathbb{A}^1 \setminus \set{0}$ by multiplication without a fixed point. Solvability is needed: $\mathrm{SL}_2$ acts on $\mathbb{P}^1$ transitively, so without a fixed point. Connectedness is needed: the group $\Z / 2$ swaps the two points of a two-point variety.

Going deeper

For a student

The theorem is the basic tool for Borel subgroups. Applied to a Borel subgroup $B$ acting on $G / P$, it shows that $B$ is conjugate into every parabolic subgroup $P$. It gives the conjugacy of all Borel subgroups, and with it the conjugacy of maximal tori. It also gives the Lie–Kolchin theorem on triangularizing connected solvable matrix groups.