# From Rational Canonical Form to The Kostant Section

Suppose we have a $2\times 2$ matrix
$$M = \begin{pmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{pmatrix}$$
with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce a matrix with a given characteristic polynomial? This can be accomplished using the rational canonical form:
$$t^2 + at + b\mapsto \begin{pmatrix} 0 & -b\\ 1 & -a \end{pmatrix}.$$
We can calculate that the characteristic polynomial of this matrix to be $t^2 + at + b$. This map gives a bijection between quadratic monic polynomials in $F[t]$ and matrices of the above form. One way to understand this phenomenon is through algebraic groups. To explain, let’s stick with $F$ having characteristic zero, though much of what we do can be done in characteristic $p$ for $p$ sufficiently large as well using very different techniques.

# Algebraic Groups and Representations

We can look at the $2\times 2$ matrices with coefficients in $F$ as the $F$-points of the Lie algebra $\mathfrak{gl}_2$ of $\GL_2$. If we let $F[\mathfrak{gl}_2]$ be the coordinate ring of $\mathfrak{gl}_2$, then the coefficients of the characteristic polynomial, which in this case are the trace and the determinant (up to sign), generate the algebra $F[\mathfrak{gl}_2]^{\GL_2}$ of invariant functions under the action of $\GL_2$. Here, $\GL_2$ acts on $\mathfrak{gl}_2$ by the action of conjugation. The representation of an algebraic group $G$ on its Lie algebra $\mathfrak{g}$ in general is known as the adjoint representation. In the case of the adjoint representation, it is known that $k[\mathfrak{g}]^G$ is a polynomial algebra generated by $m$ invariants where $m$ is the rank of $G$. This is the dimension of the maximal torus of $G$, so that for $\GL_2$, we have verified this: we have two invariants.

The rational canonical form is an example of a Kostant-Weierstrass section or KW-section, as it is known in invariant theory: in this case, an affine subvariety $\cfr\subseteq\gfr$ such that the composition $\cfr\hookrightarrow\gfr\to \gfr//G$. Here, we have written $\gfr//G$ for the categorical quotient $\Spec(k[\gfr]^G)$, and the map $\gfr\to\gfr//G$ is induced by inclusion.

Question. Let $G$ be a connected reductive algebraic group over a field $F$ of arbitrary characteristic, and $V$ an algebraic representation of $V$. When does there exist an affine subspace $\cfr\subseteq V$ such that composition $\cfr\hookrightarrow V\to V//G$ is an isomorphism?

This question is interesting and subtle, and there are important examples of representations for which this question is believed to be true but not proven. However, in our case where $F$ has characteristic zero and the representation is the adjoint representation, a KW-section was constructed by Kostant. In this setting, it is usually called the Kostant section since he was the first one to notice this phenomenon.

# The Kostant Section

To reiterate our setting, we are looking at a connected reductive algebraic group $G$ over a field $F$ of characteristic zero acting on its Lie algebra $\gfr$. We denote the action of $g\in G$ on $x\in\gfr$ by $\Ad(g)x$.

The interesting thing about the Kostant section is that its construction actually gives many different possible sections. In order to introduce it, we must first introduce regular elements: we say that $x\in\gfr$ is regular if $\Ad(G)x$, the orbit of $x$, has maximal dimension. It is a theorem that $x$ is regular if and only if its centraliser in $\gfr$ has minimal dimension. Often this latter criterion is more useful since the centraliser in $\gfr$ is easy to compute.

Next, we introduce $\mathfrak{sl}_2$-triples. An $\mathfrak{sl}_2$-triple in $\gfr$ is a triple $(x,e,f)$ of three elements of $\gfr$ such that:

1. $[x,e]=2e$
2. $[x,f]=-2f$
3. $[e,f]=x$

The point is that the three ‘obvious’ basis elements
$$x=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}, e=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}, f= \begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}$$
of $\mathfrak{sl}_2$ also satisfy these relations (exercise), and that any such elements span a Lie algebra isomorphic to $\mathfrak{sl}_2$. Kostant’s section then is $e + \zfr_\gfr(f)$ where
$$\zfr_\gfr(f) = \{ y\in \gfr : [y,f] = 0\}$$
is the centraliser of $f$ in $\gfr$.

# An Example

Let’s apply the construction of the Kostant section to our example of $\GL_2$ acting on $\mathfrak{gl}_2$. First of course, we have to find a regular nilpotent element. Of course, in this case, we might as well just take
$$e = \begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}.$$
Of course, we need to verify that it is in fact regular:

Theorem. The element $e$ is regular in $\mathfrak{gl}_2$.
Proof. The computation
$$\begin{pmatrix} x & y\\ z & t \end{pmatrix} \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} – \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\begin{pmatrix} x & y\\ z & t \end{pmatrix}= \begin{pmatrix} -z & x-t\\ 0 & z \end{pmatrix}$$
shows that $\zfr_\gfr(e)$ is two-dimensional. This is the rank of $\GL_2$, so $e$ is indeed regular.

Next, we need to embed $e$ into an $\mathfrak{sl}_2$-triple. In this case this is easy because the basis of $\mathfrak{sl}_2$ we described above will work. In general, finding the $\mathfrak{sl}_2$-triple is a little more tricky, but usually not really difficult.

We then compute that
$$\zfr_\gfr(f) = \left\{ \begin{pmatrix} x & 0\\ z & x \end{pmatrix} : x,z\in F \right\}$$

So, the section is given by the inclusion of
$$e + \zfr_\gfr(f) = \left\{ \begin{pmatrix} x & 1\\ z & x \end{pmatrix} : x,z\in F \right\}$$
into $\mathfrak{gl_2}$. Notice that, as expected, there is exactly one matrix of this type one we specify what the trace and determinant must be. Here’s a parting question:

Question: Is it possible to get the rational canonical form from a Kostant section?