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

**1. Introduction**

Now we want to consider a more general question. Let be an (affine) algebraic group over , acting on the finite-dimensional vector space . We’d like to ask what the invariant polynomials on are, or in other words what is . 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 is reductive.

What is a reductive group? For our purposes, a reductive group over is an algebraic group such that the category of (algebraic) finite-dimensional representations is semisimple. In other words, the analog of Maschke’s theorem is true for . The “classical groups” (the general linear, special linear, orthogonal, and symplectic groups) are all reductive. There is a geometric definition (which works in characteristic 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 over contains a maximal compact Lie subgroup (which is *not* algebraic, e.g. the unitary group in ), and the category of algebraic representations of is equivalent (in the natural way) to the category of continuous representations of . Since the category of continuous representations is always semisimple (by the same averaging idea as in Maschke’s theorem, with a Haar measure on ), is clearly semisimple. But this is .

Anyway, here’s what we wish to prove:

Theorem 1Let be a reductive group over acting on the finite-dimensional vector space . Then the algebra of invariant polynomials on is finitely generated.

**2. The Reynolds operator**

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

from to , 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 . Since, in a semisimple category, the decomposition into isotypic components is unique and natural, we necessarily have:

Proposition 2For a reductive group over , such a natural projection exists.

Now we will want to use the Reynolds operator on -algebras on which acts locally finitely. That is, we want a map . We can do this, because of the local finiteness and because of the naturality of . We will need the following identity, called the Reynolds identity:

To prove this, we simply use the naturality of , and the fact that multiplication by is a -endomorphism of .

**3. Proof of the theorem**

Finally, we can prove the result. Let be a representation of the reductive group , as before, and let be the algebra of all polynomials on . We want to show that is a finitely generated algebra.

To do this, we shall use the fact that is naturally a *graded* subalgebra of . We are going to show that the “irrelevant” ideal 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 is a finitely generated algebra over .

However, we know that is a finitely generated ideal in , because is noetherian; we can thus find generators for this ideal, which we may assume live in itself. The claim is that the do not only generate the ideal in , but even the ideal in .

To do this, let live in positive degrees. We know that, since is certainly in , we can write

where the are not necessarily -invariant. But now we apply the Reynolds operator, which fixes , and use the Reynolds identity:

where the necessarily live in . This shows that the irrelevant ideal in is generated by the , proving the theorem.

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

**4. Geometric quotients**

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

Proposition 3Let be an affine variety with a -action (with reductive). Then is a finitely generated -algebra.

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

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

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

and consequently the latter ring is finitely generated over , as we saw earlier that the former ring is.

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