# Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} \subseteq \mathbb{C}$) form an LCA group themselves under pointwise multiplication, and additionally the relationship between $G$ and $\widehat{G}$ is a duality (the Pontryagin duality).

The notion of characters is similarly defined for Banach algebras, where they are non-trivial multiplicative linear functionals. If we consider the group algebra $L^1(G)$ under a Haar measure with convolution for the product, then the use of the group structure in convolution allows us put characters $\mathcal{M}(L^1(G))$ on $L^1(G)$ (here $\mathcal{M}$ stands for the maximal ideal space of $L^1(G)$) and characters $\widehat{G}$ on $G$ in direct correspondence. In particular, to every character $\psi \in \mathcal{M}(L^1(G))$ there is some $\chi_\psi \in \widehat{G}$ such that $$\psi(f) = \int_G f(x) \ \overline{\chi_\psi (x)} \ d\mu(x).$$ Then the Gelfand transform, which transforms an object in a Banach algebra into a decaying continuous function on the space of characters of the algebra, is precisely the Fourier transform on $L^1(G)$, which decomposes an integrable function into its effect on characters of $G$.

It seems as though there are many directions that one can take this. If commutative Lie groups are all that one is concerned with, these can be exhausted rather quickly to form classical Fourier transforms (the transform on $\mathbb{R}$, discrete Fourier transform, and Fourier series). One can then proceed to talk about noncommutative harmonic analysis where the trace of representations (from where we derive characters) is no longer a group, or you can investigate the connection with the Laplacian and just look at the Laplacian on a general manifold.

When looking at this whole thing however, one gets a little tingling feeling that the characters going from $G \to S^1$ feels like homotopy. Of course we desperately need the group structure on $S^1$, and this is precisely what higher dimensional structures like $S^n$ lack. On the other hand, $S^2$ can be identified with the Riemann sphere. Although this isn't a group and doesn't have some immediate "structure", it does have ties (so to speak) with $\mathbb{C}$ and $\mathbb{C}^\times$ that makes it more significant than arbitrary $S^n$.

My question is this: Is there some generalization of a character that maps in some (possibly vague) way to $S^2$, that generalizes some of the things about harmonic analysis summarized here?

• What about $S^3$? This carries the structure of a Lie group; the unit quaternions. Nov 4, 2014 at 7:59
• Related : this comment Feb 28, 2016 at 20:44

See Representation theory non commutative harmonic analysis from A.A. Kirillov page 71, the idea is that the sphere is a quotient of the group of rotations on three dimensions and you can do harmonic analysis ther. Explicitly if you consider the action of $SO(3)$ on the three dimensional euclidean space and you look for the orbit of any unitary vector, its stabilizer is a copy of $SO(2)$ and you get the sphere as an homogeneous space.