Harmonic analysis on SU(2)

Example

Let $G = \mathrm{SU}(2)$ (SU(2)) with Haar measure $dg$ of total mass $1$. By the representation theory of SU(2), the irreducible representations are the spaces $V_n$ for $n = 0, 1, 2, \dots$. Here $V_n$ is the space of homogeneous polynomials of degree $n$ in two variables, and $\dim V_n = n + 1$.

The Peter–Weyl theorem gives the orthogonal decomposition $$L^2(\mathrm{SU}(2)) = \widehat{\bigoplus_{n \ge 0}} \mathcal{M}_n,$$ where $\mathcal{M}_n$ is the span of the matrix coefficients of $V_n$, of dimension $(n + 1)^2$. The Fourier series of $f \in L^2(G)$ is $\sum_{n \ge 0} (n + 1) \tr(\hat f(n) \pi_n(g))$, and the Plancherel formula reads $\int_G \abs{f}^2 \, dg = \sum_{n \ge 0} (n + 1) \norm{\hat f(n)}_{\mathrm{HS}}^2$.

Class functions give a concrete check. Every $g \in G$ is conjugate to $t_\theta = \mathrm{diag}(e^{i \theta}, e^{-i \theta})$ for a unique $\theta \in [0, \pi]$. For a continuous class function $f$, the Weyl integration formula for $\mathrm{SU}(2)$ is $$\int_G f(g) \, dg = \frac{2}{\pi} \int_0^\pi f(t_\theta) \sin^2 \theta \, d\theta.$$ The characters are $\chi_n(t_\theta) = \sin((n+1)\theta) / \sin \theta$, so $$\int_G \chi_n \overline{\chi_m} \, dg = \frac{2}{\pi} \int_0^\pi \sin((n+1)\theta) \sin((m+1)\theta) \, d\theta = \delta_{nm}.$$ Thus the characters form an orthonormal basis of the class functions in $L^2(G)$, and the Fourier series of a class function $f$ is $\sum_n \ip{f}{\chi_n} \chi_n$. Under $f(t_\theta) \sin \theta \leftrightarrow f$, this is exactly the classical sine series on $[0, \pi]$.