Why is the generalized flag variety a “variety”?

Question

In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\mathfrak g$ forms a Zariski-closed subset of a Grassmannian on this semi-simple Lie algebra. A Borel subalgebra is a maximal solvable subalgebra, by general theorems, all of them have the same dimension, say $d$. Then the set $X$ can be identified with a subset of the Grassmannian $\mathrm{Gr}(d,\mathfrak g)$. My question is: why is $X$ a Zariski-closed subset? How to translate the condition “solvable” to an algebraic condition? I thought of Lie’s criterion on solvability, but it is not an equivalence condition, so it did not work.

Answer 1

I think, the statement "the set of Borel subalgebras of a semi-simple Lie algebra $\mathfrak g$ forms a Zariski-closed subset of a Grassmannian" can be understood in more than one way and then the question becomes less trivial than one thinks. Everything is over $\mathbb C$, by the way, otherwise things become even more complicated.

First interpretation: This is the one presumably intentend by Chriss-Ginzburg and others. It consists of two statements:

1. The set of Borel subgroups is in bijection with $G/B$.
2. The natural morphism $G/B\to Gr_d(\mathfrak g)$ is a closed embedding.

The first assertion uses the conjugacy of Borel subgroups and that Borel subgroups are selfnormalizing. The second uses that $G/B$ is projective and that $B$ is the normalizer of $\mathfrak b$ in $G$.

Second interpretation: The statement could also mean that the moduli space of Borel subalgebras is precisely the schematic image of $G/B$. This is more subtle and is not addressed in any of the standard text, as far as I know. One can see this as follows:

1. Step: A subalgebra $\mathfrak b\subseteq\mathfrak g$ is the Lie algebra of a Borel subgroup if and only if $\mathfrak b$ is solvable of maximal possible dimension $d=\dim B$. Here one has to prove mainly that a maximal dimensional solvable subalgebra is algebraic in the sense that it is the Lie algebra of some algebraic subgroup.

2. Step: There is a closed subscheme $X\subseteq Gr_d(\mathfrak g)$ which is a moduli scheme of $d$-dimensional solvable subalgebras. For this one has to translate "solvable" into an algebraic condition. This condition comes from that fact that the degree of solvability of any $\mathfrak b$ is bounded by say $\ell$. Then $X$ is defined by the condition that all "multicommutators" $[\ldots[[\xi_1,\xi_2],[\xi_3,\xi_4]],\ldots]$ vanish for all $\xi_1,\ldots\xi_{2^\ell}\in\mathfrak b$. This is easily seen to be a closed condition.

3. Step: Step 1 implies that $X$ is set theoretically the image of $G/B$ in $Gr_d(\mathfrak g)$.

4. Step: It remains to prove that $X$ is smooth, hence reduced and normal. For this we compute the tangent space $Z$ of $X$ in $\mathfrak b$. First the tangent space of $Gr_d(\mathfrak g)$ in $\mathfrak b$ is $H:=\mathrm{Hom}(\mathfrak b,\mathfrak g/\mathfrak b)$. It turns out that $Z$ is the space of $1$-cocycles (=derivations) in $H$. Moreover, the tangent space of $G/B$ in $\mathfrak b$ is the space of all $1$-coboundaries (=inner derivations) $C\cong\mathfrak g/\mathfrak b$. So $Z/C=H^1(\mathfrak b,\mathfrak g/\mathfrak b)$. The latter is easily seen to be $0$ (take a filtration with $1$-dimensional quotients and use that $\mathfrak b$ and $\mathfrak g/\mathfrak b$ have no weights in common). Thus $Z=C$ and therefore $\dim Z=\dim X$ and we are done.

Answer 2

A nice source on this is Procesi's "Lie Groups: an approach through invariants and representations". There he defines parabolic subgroups as precisely those for which $G/P$ is a projective variety and shows that this is equivalent to $P$ containing a Borel subgroup.

In a later section (called quadratic relations, I think) he shows that you can cut them out with only quadratic polynomials in an appropriate projectivised representation of $G$ (a result originally due to Kostant I believe). This is, in effect, a generalisation of the Plucker embedding for the Grassmannian.

Note, you say generalised flag variety in the title which refers to any quotient $G/P$ of semisimple $G$ by a parabolic subgroup but in the body of the question only refer to the quotient by a Borel $G/B$ which I would call a "full" flag variety.

Edit: said Borel when I meant Kostant