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

Let’s look at . Topologically, is a three sphere, and the conjugacy classes are latitudinal two spheres. We’ll label the conjugacy classes by the line segment , where labels the conjugacy class of matrices with eigenvalues and . The conjugacy class is a sphere of radius proportional to , and hence area proportional to .

The characters of are indexed by positive integers, with

.

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

Here is the area of the conjugacy class and turns out to be the correct normalization factor.

So, we should expect that

.

All of this is pretty standard. So, what would you expect happens if you take to be on and on ? Seriously, see if you can guess what peculiar behavior these sums show.

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

.

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

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

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

Here I’ve zoomed into the left hand part, with . I show the old curve, and the result of adding on one more term. The value at jumps up to . Here is the next sums:

Gradually, it looks like the points near , but not at , are slowly converging to the correct value of . But the values are are not converging; they’re oscillating between two values which look like and . Somehow, that discontinuity way out at is producing failure of convergence way back at .

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

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

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Peter-Weyl says nothing about pointwise convergence for (not even) continuous functions, so it’s not (abstractly) surprising.

The divergent series I described in this very old post is, I think, equivalent to yours at 0 — up to normalization –. I used a Chebychev-polynomial version of the characters instead, which is why the coefficients look different. The character value for chi_m at 0 is m so that the series you write at x=0 is then very clearly divergent. (It converges in Cesaro mean).

In the first paragraph you meant to say “class functions”! This threw me off at first, because otherwise it’s the (normalised) matrix coefficients that give an o/n basis.

Fixed, thanks!