# An Example Using Chevalley Restriction

Here is a classic problem of geometric invariant theory: let $G$ be a reductive linear algebraic group such as $\mathrm{GL}_n$ and let $\mathfrak{g}$ be its Lie algebra. Determine the invariant functions $k[\mathfrak{g}]^G$, where $G$ acts on $\mathfrak{g}$ via the adjoint action. This problem is motivated by the search for quotients: What is the quotient $\mathfrak{g}/G$? Here, the action of $G$ on $\mathfrak{g}$ is given by the adjoint action. More explicitly, an element $g\in G$ acts via the differentiation of $\mathrm{Int}_g$, where $\mathrm{Int}_g$ is conjugation by $g$ on $G$.

For simplicity, we will stay in the realm of varieties over an algebraically closed field $k$ of characteristic zero.

First, we should ask:

What should $\mathfrak{g}/G$ even mean?

Obviously, it should be a variety. Perhaps the weakest definition is that the quotient should be any variety $Y$ with a map $\mathfrak{g}\to Y$, constant along orbits of $G$, and such that $Y$ is universal for this property. In other words, if $Z$ is a variety with a map $\mathfrak{g}\to Z$ constant along $G$-orbits, then $\mathfrak{g}\to Z$ should factor through $\mathfrak{g}\to Y$. (In other words, in the category of morphisms $\mathfrak{g}\to *$, the object $\mathfrak{g}\to Y$ is an initial object.) Certainly, we would like a quotient to satisfy this property. If such a $Y$ exists, it is unique up to unique isomorphism and is called a categorical quotient.

### Quotients

Now, where would we go looking for a quotient like this? One possibility is to consider the problem on the algebraic side. The regular functions $k[\mathfrak{g}]$ on $\mathfrak{g}$ also have a right-action of $G$. Indeed, if $g\in G$ and $\varphi\in k[\mathfrak{g}]$ then $(\varphi*g)(x) = \varphi(gx)$. Now, since the affine $k$-algebras and the affine varieties are antiequivalent, it makes sense to look at the invariants: $k[\mathfrak{g}]^G$. If this is an affine $k$-algebra, then the corresponding variety would be a reasonable candidate for a quotient.

It is a nontrivial result that if a reductive group $G$ over our algebraically closed field $k$ acts on an affine variety $X$ then $k[X]^G$ is a finitely generated $k$-algebra and the corresponding variety is in fact a categorical quotient. Here “acts” means there is a morphism of varieties $G\times X\to X$ satisfying the axionms of a group action.

### Calculations

However, this post we actually be concentrated on describing $k[\mathfrak{g}]^G$ in fairly simple terms. As an example, let us try $G = \mathrm{GL}_2$. Then $\mathfrak{g} = \mathfrak{gl}_2$, the Lie algebra of all $2\times 2$ matrices, and the adjoint action is given by conjugation. The Lie algebra has $k[x_1,x_2,x_3,x_4]$ as its coordinate ring, and is isomorphic to affine $4$-space.

Let $A = \left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix}\right)\in\mathrm{GL}_2$ and let $X = \left(\begin{smallmatrix} x_1 & x_2\\ x_3 & x_4\end{smallmatrix}\right)\in\mathfrak{gl}_2$. What does $AXA^{-1}$ look like? This is an elementary matrix calculation, and it ends up being:

$\frac{1}{ad – bc}\begin{pmatrix} adx_1 + bdx_3 – acx_2 – bcx_4 & -abx_1 – b^2x_3 + a^2x_2 + abx_4\\cdx_1 + d^2x_3 – c^2x_2 – cdx_4 & -cbx_1 – bdx_3 + acx_2 + adx_4\end{pmatrix}$

If you meditate with this matrix for a bit, you might notice that the trace of this matrix is $x_1 + x_4$, which is the trace of $X$. Thus the function $x_1 + x_4\in k[x_1,x_2,x_3,x_4] = k[\mathfrak{g}]$ is $G$-invariant. Is there another? By another of course, I mean another algebraically independent from $x_1+x_4$.

One could try and play around a bit, perhaps trying something nice and familiar like the determinant (hehe), but the computations can get a bit messy. A computer algebra system might help, but what are you supposed to do if you’re stuck in a cabin up north with only a candle and a piece of charcoal to occupy your time? You use the Chevalley restriction theorem!

### Chevalley Restriction Theorem

We keep the setting of $G$ being a reductive algebraic group over an algebraically closed field with Lie algebra $\mathfrak{g}$. Select a maximal torus $T\subseteq G$. Then its Lie algebra $\mathfrak{t}$ is a subalgebra $\mathfrak{t}\subseteq\mathfrak{g}$, and there is a restriction map $k[\mathfrak{g}]\to k[\mathfrak{t}]$, and hence an induced map

$k[\mathfrak{g}]^G\to k[\mathfrak{t}]$

We define the Weyl group $W = W(\mathfrak{t})$ of $\mathfrak{t}$ to be the group $W = N_G(\mathfrak{t})/N_G(\mathfrak{t})$. This group acts on $\mathfrak{t}$, and hence on $k[\mathfrak{t}]$, and any regular function invariant under $G$ is also invariant under $N_G(\mathfrak{t})$. Hence we obtain a map

$k[\mathfrak{g}]^G\to k[\mathfrak{t}]^W$.

There are various generalisations of next theorem, but the formulation here will be enough for our purposes.

Theorem (Chevalley Restriction). Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic zero and $T$ a maximal torus. Let $\mathfrak{g} = \mathrm{Lie}(G)$ and $\mathfrak{t} = \mathrm{Lie}(T)$. Then the morphism $k[\mathfrak{g}]^G\to k[\mathfrak{t}]^W$ induced by restriction is an isomorphism.

This result is amazing. It reduces the problem of determining $G$-invariant functions on $\mathfrak{g}$ to $W$-invariant functions on $\mathfrak{t}$. The interesting part is that the Weyl group is a finite group (hint: use rigidity of tori), so computing the $W$-invariant functions in $k[\mathfrak{t}]$ is already going to be much easier.

(To be honest, I am not aware of the most general forms of this theorem; for instance, it holds in positive characteristic $p$ as long as $p \nmid |W|$ and it holds when $k$ isn’t algebraically closed if $G$ is split, and there are other variations, so to keep it simple I excluded these, but in the future I may write a more comprehensive post about this.)

### Back to $\mathrm{GL}_2$

So what are the $\mathrm{GL}_2$-invariant functions in $k[\mathfrak{gl}_2]$? Consider the maximal torus of diagonal matrices. The first thing is to determine the Weyl group. This is a straightforward matrix computation, which I will leave to the reader. The answer is $W\cong \mathbb{Z}/2 = \{1,\sigma\}$, and the action of $\sigma$ on $\mathfrak{t}$ is given by permuting the diagonal.

The $k$-algebra representing $\mathfrak{t}$ is $k[x,y]$, and so the invariant subalgebra $k[x,y]^W$ are the symmetric polynomials. It is a standard result that all such polynomials in two variables are generated by $x + y$ and $xy$. So we have determined that $k[\mathfrak{gl}_2]^{\mathrm{GL}_2}$ is isomorphic to a polynomial ring in two variables, generated by homogeneous polynomials. Isn’t this a much easier calculation?

This in particular shows that the categorical quotient $\mathfrak{gl}_2/\mathrm{GL}_2$ is isomorphic to $\mathbb{A}^2$.