## Characters

Definition. Thecharacterof a continuous G-module V is defined as:This is a continuous map since it is an example of a matrix coefficient.

Clearly for any . The following are quite easy to show:

The last equality, that is the complex conjugate of , follows from the following:

Lemma 1. For , the linear map has all eigenvalues on the unit circle T.

**Proof**

Suppose the eigenvalues are with the right multiplicity. Then has trace But is continuous; since *G* is compact the image is bounded. It remains to show: if is bounded for all integers *m* we must have all .

Suppose ; replacing *g* by its inverse, we assume and If , increases much faster than the other and we get a contradiction. On the other hand, if , write for where *R*>1 and are real. Consider the continuous homomorphism

Its image lies in the torus group which is compact. Hence, there are arbitrarily large *m* such that (see exercise). So the real part of exceeds and increases much faster than the remaining ♦

**Exercise**

Prove that if is a continuous homomorphism to a compact Hausdorff group, then is infinite for any open neighbourhood *U* of *e*.

Lemma 2. For , the linear map is diagonalizable.

**Proof**

Write in block Jordan form. Suppose the matrix is not diagonal, say entry (1, 2) is 1 with corresponding eigenvalue . The matrix for has entry (1, 2) equal to which is unbounded since However, *G* is compact so taking entry (1, 2) of the representation has bounded image, which is a contradiction. ♦

## Orthogonality of Characters

The space of *G*-invariant elements of a *G*-module *V* is denoted by:

We have:

Proposition. The dimension of is:

**Proof**

Take via

Left-invariance then implies that the image of π lies in Also, if then Putting these two facts together gives us so π is the projection map onto Hence:

as desired. ♦

In particular, replacing *V* by , note that:

So the dimension of this space is the integral of:

In other words:

which we denote by Schur’s lemma then gives us:

Orthonormality of Irreducible Characters.If V, W are irreps of G, then

## Abelian Case

Suppose now *G* is compact Hausdorff and abelian.

Theorem. Each irreducible representation V of G is 1-dimensional.

**Proof**

First we show that every acts as a scalar. By lemma 2 above, *V* decomposes as a direct sum of eigenspaces for Each is *G*-invariant since:

Since *V* is irreducible, we have so *g* acts as a scalar.

Since every element of *G* acts as a scalar, every vector subspace is *G*-invariant; so dim(*V*) = 1. ♦

Hence by lemma 1, every irrep of *G* is a continuous homomorphism , where is the unit circle. The following special case is of interest.

Definition. Then-torusis the topological group

[In fact, any compact Hausdorff abelian *G* must be of the form (finite group) but we won’t be needing this fact.]

For *i* = 1,…,*n*, let be the homomorphism taken by projecting to the *i*-th coordinate. More generally, for any , we define:

Thus it takes to

Theorem. Any irrep of is of the form for some

**Proof**

Suppose *n*=1: for a continuous write *T* as and compose The subgroup lies in the kernel; since the kernel is a closed subgroup it is either or for some *m*>0. So is of the form for some

For *n*>1 consider the homomorphism for *i *= 1,…,*n*, by taking where the *i*-th component is *v*. By the previous case, is of the form for some Since for all we get:

♦

Corollary. A character of is a finite linear combination of whose coefficients are positive integers.

Replacing with variables , this gives us a polynomial in , also known as a **Laurent polynomial** in Hence we have a correspondence: