The Generalized Flag Variety

Suppose that we wish to generalize the facts on the previous page from $
\DeclareMathOperator{\Fl}{F\ell}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SP}{SP}\GL_n$ to arbitrary semisimple linear algebraic groups. What does it mean to have a flag if you don’t have a space to put them in? For groups such as the symplectic group $\SP_{2n}$, one can perhaps imagine flags living in a symplectic vector space… but it is still not clear what the definition should be for an arbitrary group.

Therefore, we first need to come up with alternative ways of stating the definition in the classical case, that only depend on the structure of $\GL_n$.

Two Alternative Definitions

There are two other ways of defining the flag manifold that are somewhat less explicit but more generalizable. The group $\GL_n(\mathbb{C})$ acts on the set of flags by left multiplication on its sequence of vectors. Under this action, the stabilizer of the standard flag defined by the standard basis $\mathbf{e}_1,\mathbf{e}_2,\ldots,\mathbf{e}_n$ is the subgroup $B$ consisting of all invertible upper-triangular matrices. Notice that $\GL_n$ acts transitively on flags via change-of-basis matrices, and so the stabilizer of any arbitrary flag is simply a conjugation $gBg^{-1}$ of $B$. We can therefore define the flag variety as the quotient $\GL_n/B$, and define its variety structure accordingly.

Alternatively, we can associate to each coset $gB$ in $\GL_n/B$ the subgroup $gBg^{-1}$, and define the flag variety as the set $\mathcal{B}$ of all subgroups conjugate to $B$. Since $B$ is its own normalizer in $G$, these sets $\mathcal{B}$ and $\GL_n/B$ are in one-to-one correspondence.

Borel Subgroups

The subgroup $B$ of invertible upper-triangular matrices is an example of a Borel subgroup of $\GL_n$, that is, a maximal connected solvable subgroup.

It is connected because it is the product of the torus $(\mathbb{C}^\ast)^n$ and $\binom{n}{2}$ copies of $\mathbb{C}$. We can also show that it is solvable, meaning that its derived series of commutators $$\begin{eqnarray*}B_0&:=&B, \\ B_1&:=&[B_0,B_0], \\ B_2&:=&[B_1,B_1], \\ \vdots \end{eqnarray*}$$ terminates. Indeed, $[B,B]$ is the set of all matrices of the form $bcb^{-1}c^{-1}$ for $b$ and $c$ in $B$. Writing $b=(d_1+n_1)$ and $c=(d_1+n_2)$ where $d_1$ and $d_2$ are diagonal matrices and $n_1$ and $n_2$ strictly upper-triangular, it is not hard to show that $bcb^{-1}c^{-1}$ has all $1$’s on the diagonal. By a similar argument, one can show that the elements of $B_2$ have $1$’s on the diagonal and $0$’s on the off-diagonal, and $B_3$ has two off-diagonal rows of $0$’s, and so on. Thus the derived series is eventually the trivial group.

In fact, a well-known theorem of Lie and Kolchin states that all solvable subgroups of $\GL_n$ consist of upper triangular matrices in some basis. This implies that $B$ is maximal as well among solvable subgroups. Therefore $B$ is a Borel.

Notice that, by the Lie-Kolchin theorem, it follows that all the Borel subgroups in $\GL_n$ are of the form $gBg^{-1}$ (and all such groups are Borels). That is:

All Borel subgroups are conjugate.

It turns out that this is true for any semisimple linear algebraic group $G$, and additionally, any Borel is its own normalizer. By an argument identical to that in the previous section, it follows that the groups $G/B$ are independent of the choice of borel $B$ (up to isomorphism) and are also isomorphic to the set $\mathcal{B}$ of all Borel subgroups of $G$ as well. Therefore we can think of $\mathcal{B}$ as an algebraic variety by inheriting the structure from $G/B$ for any Borel subgroup $B$.

Finally, we define the flag variety of a linear algebraic group $G$ to be $G/B$ where $B$ is a borel subgroup. This is isomorphic to the space $\mathcal{B}$ of all Borel subgroups of $G$.

Intersection Cohomology

Is it possible that the cohomology ring of a general flag variety $G/B$ is as nice as it is in the classical case? At least when $G$ is a Lie group, we are in luck: it is isomorphic to a graded representation of the associated Weyl group. Define the reflection representation of a Weyl group $W$ to be the (non-irreducible) representation $V$ formed by the action of $W$ on the weight lattice by reflections. For instance, in the case of $S_n$, the action is on $\mathbb{C}^n$ by permuting the coordinates.

Then the cohomology ring $H^\ast(G/B)$ is isomorphic to $S(V)/I$ where $S(V)$ is the symmetric algebra of $V$ and $I$ is the ideal generated by the positive-degree invariants. Beautiful!

To be continued…

Next time, we’ll look at subvarieties of the Flag variety, called Springer fibers, whose cohomology rings give rise to the Hall-Littlewood polynomials in the classical case, and in general parameterize the irreducible representations of arbitrary Weyl groups.

Next: The Springer Correspondence, Part II