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. (more…)