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
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:
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?