A little earlier, we studied invariant theory for the general linear group ${GL(V)}$ for a finite-dimensional vector space ${V}$ over ${\mathbb{C}}$. We considered the canonical representation on ${V^{\otimes p} \otimes V^{* \otimes q}}$ and studied “invariant polynomials” on this space: that is, polynomials ${P: V^{\otimes p} \otimes V^{* \otimes q} \rightarrow \mathbb{C}}$ constant on orbits. We showed that these formed a finitely generated ${\mathbb{C}}$-algebra, and indeed gave a set of generators: these were given by pairing a factor of ${V}$ with a factor of ${V^*}$ with respect to the evaluation pairing. This is not, of course, a linear map, but it is a well-defined polynomial map of ${p}$ vector and ${q}$ covector variables.

1. Introduction

Now we want to consider a more general question. Let ${G}$ be an (affine) algebraic group over ${\mathbb{C}}$, acting on the finite-dimensional vector space ${V}$. We’d like to ask what the invariant polynomials on ${V}$ are, or in other words what is ${(\mathrm{Sym} V^*)^G}$. It was a Hilbert problem to show that this “ring of invariants” is finitely generated. The general answer turns out to be no, but we will show that it is the case when ${G}$ is reductive.

What is a reductive group? For our purposes, a reductive group over ${\mathbb{C}}$ is an algebraic group ${G}$ such that the category ${\mathrm{Rep}(G)}$ of (algebraic) finite-dimensional representations is semisimple. In other words, the analog of Maschke’s theorem is true for ${G}$. The “classical groups” (the general linear, special linear, orthogonal, and symplectic groups) are all reductive. There is a geometric definition (which works in characteristic ${p}$ too), but we will just take this semisimplicity as the definition.

The semisimplicity is quite a surprising phenomenon, because the method of proof of Maschke’s theorem—the averaging process—fails for reductive groups, which are never compact in the complex topology (as then they would not be affine varieties). However, it turns out that a reductive group ${G}$ over ${\mathbb{C}}$ contains a maximal compact Lie subgroup ${K}$ (which is not algebraic, e.g. the unitary group in ${GL_n}$), and the category of algebraic representations of ${G}$ is equivalent (in the natural way) to the category of continuous representations of ${K}$. Since the category of continuous representations is always semisimple (by the same averaging idea as in Maschke’s theorem, with a Haar measure on ${K}$), ${\mathrm{Rep}(K)}$ is clearly semisimple. But this is ${\mathrm{Rep}(G)}$.

Anyway, here’s what we wish to prove:

Theorem 1 Let ${G}$ be a reductive group over ${\mathbb{C}}$ acting on the finite-dimensional vector space ${V}$. Then the algebra of invariant polynomials on ${V}$ is finitely generated.

2. The Reynolds operator

To do this, we shall use the Reynolds operator. Given a finite-dimensional representation ${M}$ of ${G}$, there is a natural projection

$\displaystyle R_M: M \rightarrow M^G,$

from ${M}$ to ${M^G}$, the isotypic component of the trivial representation (this can be thought of as averaging). It is natural, in that it is a natural transformation from the identity functor to the functor ${M \mapsto M^G}$. Since, in a semisimple category, the decomposition into isotypic components is unique and natural, we necessarily have:

Proposition 2 For a reductive group ${G}$ over ${\mathbb{C}}$, such a natural projection exists.

Now we will want to use the Reynolds operator on ${\mathbb{C}}$-algebras ${T}$ on which ${G}$ acts locally finitely. That is, we want a map ${R_T: T \rightarrow T^G}$. We can do this, because of the local finiteness and because of the naturality of ${R}$. We will need the following identity, called the Reynolds identity:

$\displaystyle \boxed{ R(ab) = a R(b), \quad a \in T^G, b \in T.}$

To prove this, we simply use the naturality of ${R}$, and the fact that multiplication by ${a}$ is a ${G}$-endomorphism of ${T}$.

3. Proof of the theorem

Finally, we can prove the result. Let ${V}$ be a representation of the reductive group ${G}$, as before, and let ${T = \mathrm{Sym}V^*}$ be the algebra of all polynomials on ${V}$. We want to show that ${T^G}$ is a finitely generated algebra.

To do this, we shall use the fact that ${T^G}$ is naturally a graded subalgebra of ${T}$. We are going to show that the “irrelevant” ideal ${(T^G)_+ \subset T^G}$ of elements in positive degrees is finitely generated as an ideal. This is enough (by a well-known inductive argument, as in Proposition 1.14 of chapter 6 of the CRing project) to show that ${T^G}$ is a finitely generated algebra over ${\mathbb{C}}$.

However, we know that ${(T^G)_+ T}$ is a finitely generated ideal in ${T}$, because ${T}$ is noetherian; we can thus find generators ${t_1, \dots, t_k}$ for this ideal, which we may assume live in ${(T^G)_{+}}$ itself. The claim is that the ${\{t_i\}}$ do not only generate the ideal in ${T}$, but even the ideal in ${T^G}$.

To do this, let ${f \in T^G}$ live in positive degrees. We know that, since ${f}$ is certainly in ${T}$, we can write

$\displaystyle f = \sum f_i t_i, \quad f_i \in T,$

where the ${f_i}$ are not necessarily ${G}$-invariant. But now we apply the Reynolds operator, which fixes ${f}$, and use the Reynolds identity:

$\displaystyle f = \sum R(f_i) t_i,$

where the ${R(f_i)}$ necessarily live in ${T^G}$. This shows that the irrelevant ideal in ${T^G}$ is generated by the ${\left\{t_i\right\}}$, proving the theorem.

This proof works for a compact group, which need not even be a Lie group, acting continuously.

4. Geometric quotients

Let ${G}$ be a reductive group over ${\mathbb{C}}$, and let ${V}$ be an affine variety with a ${G}$-action (that is, a morphism of varieties ${G \times V \rightarrow V}$ that satisfies the usual conditions). We are interested in constructing a “quotient” of ${V}$ modulo ${G}$. When ${G}$ is a finite group, we can construct the categorical quotient by considering the fixed points ${\mathbb{C}[V]^G}$ in the coordinate ring. We can do the same in the case of a reductive group ${G}$, but now we need to see that we get a finitely generated object.

Proposition 3 Let ${V}$ be an affine variety with a ${G}$-action (with ${G}$ reductive). Then ${\mathbb{C}[V]^G}$ is a finitely generated ${\mathbb{C}}$-algebra.

The associated affine variety is called the “geometric” quotient.

To see this, we use the fact that ${V}$ imbeds as a closed ${G}$-subvariety of a finite-dimensional ${G}$-representation ${W}$ (an elementary fact about algebraic group actions). In particular, there is a surjection of ${G}$-equivariant algebras

$\displaystyle \mathrm{Sym} W^* \rightarrow \mathbb{C}[V],$

and, since both are locally finite with respect to the ${G}$-action, reductivity shows that there is a surjection

$\displaystyle (\mathrm{Sym} W^*)^G \rightarrow \mathbb{C}[V]^G,$

and consequently the latter ring is finitely generated over ${\mathbb{C}}$, as we saw earlier that the former ring is.