We attempt to identify the irreducible rational representations of From the last article, we may tensor it with a suitable power of det and assume it is polynomial.

One key ingredient is the following rather ambiguous statement.

**Peter-Weyl Principle**: any irrep can be embedded inside the ring of functions on*G*.

To make sense of this statement, we need to define a group action on the latter. This article is rather general; throughout, *G* is allowed to be any topological group and all representations are assumed to be continuous and finite-dimensional.

*C*(*G*) as a Representation

Let *C*(*G*) be the ring of continuous functions

Definition. The action of on is defined as follows:One checks that .

Some examples of functions in *C*(*G*) are the *matrix coefficients*; recall that these are functions of the form for some representation , and A more economical way of writing this is for some

Proposition. Let be the set of matrix coefficients. Then is a (unital) -subalgebra of .

**Proof**

To prove , let be the trivial representation.

Closure under addition and multiplication: suppose , for *i* = 1,2; let be the matrix coefficient. Taking:

we have and ♦

## Identifying Matrix Coefficients in *C*(*G*)

Next we have:

Lemma. For any , the following are equivalent.

- The span of all over all is finite-dimensional.
- The span of all over all is finite-dimensional.
- The span of all over all is finite-dimensional.
- is a matrix coefficient.

**Proof**

That 1 ⇒ 2, 3 is obvious. Let us now prove 4 ⇒ 1. Suppose for and . Then the function takes

where is a linear map which takes . Hence the span of all lies in and (1) follows since is finite-dimensional.

Finally, we prove 3 ⇒ 4; the case of 2 ⇒ 4 is similar. Let *V* be the space spanned by all , so *V* is a finite-dimensional subspace of *C*(*G*) which is -invariant. Let

- be given by the action of on
*V*; - be the map .

Now tracing through gives:

Thus is a matrix coefficient. [Note: for 2 ⇒ 4, one needs to take the dual of *V*.] ♦

Corollary. is the sum of all finite-dimensional -subrepresentations of .

## Weak Peter-Weyl Theorem

Let *V* be a representation of *G*; the vector space End(*V*) becomes a -rep via:

Another way of seeing this is: naturally and the action of on the LHS is obtained from the RHS. Next, we obtain a linear map:

(*)

This is -equivariant:

- Indeed (
*x*,*y*) takes to so dually is taken to the function - In , (
*x*,*y*) takes to Letting it gets taken to so the following commutes:

We have so its dual is naturally isomorphic to End(*V*) as vector spaces and as *G*-reps, but not as -reps. This is because the isomorphism requires a twist of the two components, which is not -equivariant.

Now suppose *G* is *compact Hausdorff*.

Lemma. Let V, W be irreducible representations of G. Then:

- is an irreducible representation of ;
- as -reps if and only if as G-reps.

**Proof**

The character of End(*V*) is: Taking the inner product:

This proves both claims together. ♦

Taking the direct sum of (*) over all irreps *V* of *G*, we obtain:

Weak Peter-Weyl Theorem. The above map is an isomorphism.

**Proof**

To show *injectivity*: the kernel of the map is a -submodule, so it is a direct sum of -irreps On the other hand, clearly cannot map to 0, since that would mean

To show *surjectivity*: each is, by definition, of the form for some , . Thus *f *lies in the image. ♦

*We remind the reader that this theorem only works when G is compact Hausdorff.*