In order to calculate the Dirac spectrum on Berger's sphere $(S^3,g_t)$, I came across irreps of SU(2) (see Hitchin p. 30).
Apperently, Hitchin restricts the Dirac operator to the eigenspaces of the Laplacian relative to the standard metric because they commute. That's clear. Now he says, that they are given by irreps of SU(2), ie. homogeneous polynomials in two complex variables. And that's the part I do not understand.
The eigenspaces of the corresponding Laplacian on $S^3$ are given by harmonic homogeneous polynomials all with respect to the Euclidean $\mathbb{R}^4$, aren't they? So why does he talk about just homogeneous polynomials without the harmonic restriction?
I already consulted Bröcker and Hall and cannot find anything helpful.
Obviously, I don't understand the connection between Laplace eigenspaces and irreps and definitely miss something. Maybe it's a silly question. But unfortunately, I don't get the point. So I would appreciate some help.
Thank you! 🙂
EDIT: I worked up a little and come to the conclusion that the Peter-Weyl theorem is the key point here. It seems that the functions given by the matrix coefficents
\begin{align}
g \mapsto f_{v,w}(g) = \left< \pi_k(g)v,w\right>
\end{align}
on the irreps $\pi_k$ of SU(2), namely homogeneous polynomials in two complex variables of degree $k$, must give the spherical harmonics I'm seeking for. Can anybody confirm this?
Best Answer
Ah, yes, the words here create confusion.
As it turns out, the eigenfunctions for the rotation-invariant Laplacian on ("round") spheres are the restrictions to the spheres of Euclidean-harmonic (yes, note the modifier) polynomials on the ambient Euclidean space.
Further, as it turns out, the Euclidean-harmonic polynomials of a given degree are an irreducible repn for orthogonal and/or unitary groups acting on the physical space.
Yes, many chances for cognitive dissonance.
No, it is not obvious that ambient-harmonic implies or is related to eigenfunction properties on the sphere. (The facts here are pretty standard. I do have notes about this, for example. No mysteries here, folks.)
EDIT: the most typical presentation refers to orthogonal groups action on spaces $\mathbb R^n$, etc. But, also, unitary groups act nicely on $\mathbb C^n \supset S^{2n-1}$, and on harmonic complex polynomials, etc., in a similar fashion. In part, even though $SU(n)$ is quite a lot smaller than $SO(2n)$, it is still transitive on $S^{2n-1}$, and so on.