Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group *G*, we want:

to be a continuous homomorphism of groups.

## Continuous Representations of Topological Groups

Let *G* be a topological group.

Our main objects of study are continuous representations These correspond to -modules (also called *G*-modules) *V* of finite complex dimension, with an additional condition for continuity:

- for any and (dual), the map given by is continuous.

[Note the above condition condition is independent of coordinates and does not require a choice of basis. When we replace *G* with later, we can also replace the word continuous with “analytic” to obtain a narrower class of representations, but let’s not get ahead of ourselves.]

A function of the form as above is called a **matrix coefficient**. E.g. if we fix a basis for *V* which gives us picking entry (*i*, *j*) in the matrix gives us a matrix coefficient

*Henceforth, all representations of topological groups are assumed to be continuous and finite-dimensional.*

### Constructing New Representations

These constructions are identical to the case for representations of finite groups.

- Given any complex vector space
*V*, the trivial action takes*gv*=*v*for any ; it is clearly continuous.

- Given a
*G*-module*V*, a vector space is said to be*G*–**invariant**if Clearly, the resulting is continuous.- We say that
*V*is**irreducible**if and it has no non-trivial*G*-invariant submodules. An irreducible representation*V*is often called**irrep**for short.

- We say that

- If is
*G*-invariant, then*V*/*W*is a*G*-module; clearly it is continuous.

- If
*V*is a*G*-module, the dual is also continuous, where*G*acts via

- Given
*G*-modules*V*,*W*, the action of*G*on and are continuous as well.

- Finally we have naturally. Using the above constructions, we get a continuous
*G*-representation for this space via:

How do we know that the above constructions are all continuous?

Here’s one example. To show that is a continuous representation, pick a basis (resp. ) of *V* (resp. *W*) so that form a basis of The matrix for is obtained from the matrices of and by multiplying entries; thus the map taking *g* to a fixed entry of the matrix for is continuous.

Finally, a linear map of *G*-modules is said to be *G*–**equivariant** if

Note that this is equivalent to saying *G* acts trivially on

## Compact Groups

From now till the end of the next article, *G* is assumed to be compact and Hausdorff; this case will be instrumental to the general theory even though is not compact. Here are some common examples of compact topological groups in representation theory:

These groups are compact because they are closed and bounded subsets of E.g. the condition for can be written out as a sequence of quadratic equations in and for Thus the set is closed; it is bounded since the condition gives:

so each

## Haar Measure

To proceed, we will borrow a result from harmonic analysis:

Theorem. If G is a compact Hausdorff group, then there is a measure m on G (called theHaar measure) satisfying:

- m(G) = 1;
- m is “left-invariant”, i.e. for any continuous , we have:
The Haar measure is unique and in fact right-invariant, i.e. We will implicitly assume the Haar measure is used when integrating over G.

**Note**

The Haar measure is usually defined for locally compact Hausdorff groups in general, in which case we have to modify the statements a little. In particular, left-invariant measures are not necessarily right-invariant.

## Semisimplicity of Continuous Representations

The following result is crucial.

Theorem. Let V be a G-module. If is a G-invariant subspace, then we can find a G-invariant subspace such that

**Proof**

Let be any projection map onto *W*, so and . Now define:

[Explicitly, one may take an isomorphism so each is a tuple of complex numbers; the above integral can be carried out component-wise, as *N* integrals of continuous functions.]

We then have:

- The image of is in
*W*, since and for

- If then and

- : indeed if then by the first property and so by the second.

- The map is
*G*-equivariant: for we have Indeed by left-invariance we have: Replacing*v*with*yv*gives us the desired outcome.

Hence, is projection onto *W* and its kernel *W’* satisfies Since is *G*-equivariant, *W’* is a *G*-invariant subspace. ♦

Thus, every continuous representation of *G* can be written as a direct sum of irreps. Next, we have:

Schur’s Lemma. If are irreps of G, then:

The proof is identical to that of the usual Schur’s lemma.

**Proof**

Suppose and is *G*-equivariant. Then and are subrepresentations, so and . By considering various cases we have either

- , in which case
*f*is an isomorphism or - , in which case
*f*= 0.

It remains to show that when .

It suffices to show that any isomorphism is a scalar: let be an eigenvector of *f *with eigenvalue so Then is *G*-equivariant and its kernel is non-zero; hence ♦