# A Peter-Weyl “counter-example”

Let $K$ be a compact Lie group. The Peter-Weyl theorem gives a basis for functions on $K$. In particular, it tells us that the characters are an orthonormal basis for class functions on $K$.

Let’s look at $K=SU(2)$. Topologically, $K$ is a three sphere, and the conjugacy classes are latitudinal two spheres. We’ll label the conjugacy classes by the line segment $[0, \pi]$, where $\theta$ labels the conjugacy class of matrices with eigenvalues $e^{i \theta}$ and $e^{- i \theta}$. The conjugacy class $\theta$ is a sphere of radius proportional to $\sin \theta$, and hence area proportional to $\sin^2 \theta$.

The characters of $SU(2)$ are indexed by positive integers, with

$\displaystyle{\chi_n(\theta) = e^{- (n-1) i \theta} + e^{- (n-3) i \theta} + \cdots + e^{(n-1) i \theta} = \frac{\sin (n \theta)}{\sin \theta} }$.

So, if $F$ is a class function on $K$, then its inner products with the characters are given by the integrals

$\displaystyle{ c_n = \frac{2}{\pi} \int_0^{\pi} F(\theta) \chi_n(\theta) \sin^2(\theta) d \theta }.$

Here $\sin^2 \theta$ is the area of the conjugacy class $\theta$ and $2/\pi$ turns out to be the correct normalization factor.

So, we should expect that

$\displaystyle{ F(\theta) = \sum c_n \chi_n(\theta) }$.

All of this is pretty standard. So, what would you expect happens if you take $F$ to be $1$ on $[0, \pi/2]$ and $-1$ on $[\pi/2, \pi]$? Seriously, see if you can guess what peculiar behavior these sums show.

Of course, the integrals are easy to compute. For $n$ odd, $c_n$ vanishes. For $n$ even, we get

$\displaystyle{ c_{2m} = (-1)^{m-1} \frac{8 m}{(4 m^2-1) \pi} }$.

Here’s the sum of the first $20$ terms, plotted together with $F$:

Looks pretty good. We’re seeing some Gibbs phenomenon from the singularity at $\pi/2$, that’s to be at expected.

But what’s going on near $0$? That’s nowhere near the singularity: $F$ is constant near $0$. But our function is way down near $0.361943$; nothing like $1$. Things get even stranger if we add on one more term:

Here I’ve zoomed into the left hand part, with $\theta \in [0, 0.25]$. I show the old curve, and the result of adding on one more term. The value at $0$ jumps up to $1.63782$. Here is the next $10$ sums:

Gradually, it looks like the points near $0$, but not at $0$, are slowly converging to the correct value of $1$. But the values are $0$ are not converging; they’re oscillating between two values which look like $1-2/\pi$ and $1+2/\pi$. Somehow, that discontinuity way out at $\theta = \pi/2$ is producing failure of convergence way back at $\theta=0$.

So, a cute question: Why isn’t this a counter-example to the Peter-Weyl theorem?

And, a more challenging question: How nice does $F$ have to be in order not to get this sort of behavior?