Mathematics Lecture Notes

Theorem8.1Peter-Weyl Theorem

Let $G$ be a compact group with normalized Haar measure $\mu$ ( $\mu(G)=1$ ). Let $\{\rho_\lambda\}_{\lambda\in\hat{G}}$ be the set of (equivalence classes of) irreducible unitary representations of $G$ , with $\rho_\lambda: G \to \mathrm{U}(V_\lambda)$ and $\dim V_\lambda = d_\lambda$ . Then:

(Orthogonality) The matrix coefficients $\rho_\lambda(g)_{ij}$ satisfy $\int_G \overline{\rho_\lambda(g)_{ij}}\,\rho_\mu(g)_{kl}\,d\mu(g) = \frac{1}{d_\lambda}\delta_{\lambda\mu}\delta_{ik}\delta_{jl}$ .
(Completeness) The functions $\{\sqrt{d_\lambda}\,\rho_\lambda(g)_{ij}\}$ form a complete orthonormal basis of $L^2(G)$ : $f(g) = \sum_{\lambda\in\hat{G}} d_\lambda\sum_{i,j} \hat{f}_\lambda^{ij}\,\rho_\lambda(g)_{ij}$ where $\hat{f}_\lambda^{ij} = \int_G f(g)\overline{\rho_\lambda(g)_{ij}}\,d\mu(g)$ .
(Regular representation) The left regular representation on $L^2(G)$ decomposes as $L^2(G) \cong \bigoplus_{\lambda\in\hat{G}} V_\lambda \otimes V_\lambda^*$ , with each irreducible representation appearing with multiplicity $d_\lambda$ .

Proof

Step 1: Orthogonality from Schur's lemma. Define $T = \int_G \rho_\mu(g)^{-1}E\rho_\lambda(g)\,d\mu(g)$ for any linear map $E: V_\lambda \to V_\mu$ . By left-invariance of Haar measure, $\rho_\mu(h)T = T\rho_\lambda(h)$ for all $h \in G$ , so $T$ intertwines $\rho_\lambda$ and $\rho_\mu$ .

If $\lambda \neq \mu$ (inequivalent irreducible representations), Schur's lemma gives $T = 0$ . Taking $E = e_{kj}$ (the matrix with $1$ in position $(k,j)$ , $0$ elsewhere): $\int_G \overline{\rho_\mu(g)_{ik}}\rho_\lambda(g)_{j\ell}\,d\mu = 0$ .

If $\lambda = \mu$ , Schur's lemma gives $T = cI$ . Taking the trace: $c\,d_\lambda = \mathrm{tr}(E)$ , so $c = \mathrm{tr}(E)/d_\lambda$ . With $E = e_{kj}$ : $\int_G \overline{\rho_\lambda(g)_{ik}}\rho_\lambda(g)_{j\ell}\,d\mu = \frac{1}{d_\lambda}\delta_{ij}\delta_{k\ell}$ .

Step 2: Completeness (sketch). Consider the convolution algebra $L^2(G)$ and the operator $T_f: L^2(G) \to L^2(G)$ defined by $(T_f h)(x) = \int_G f(y^{-1}x)h(y)\,d\mu(y)$ . For $f \in L^2(G)$ , $T_f$ is a compact operator (by the Hilbert-Schmidt theorem, since $G$ is compact).

The spectral theorem for compact operators gives eigenspaces, which by Schur's lemma must be irreducible representations. To show the matrix coefficients span $L^2(G)$ , suppose $f \perp$ all matrix coefficients. Then $\hat{f}_\lambda = 0$ for all $\lambda$ , which means $T_f = 0$ (since $T_f$ restricted to each $V_\lambda$ acts by $\hat{f}_\lambda$ ). But $T_f = 0$ implies $f * g = 0$ for all $g \in L^2(G)$ . Choosing $g$ to be an approximate identity (peaked at the identity element): $f * g \to f$ in $L^2$ , so $f = 0$ .

Step 3: Decomposition of the regular representation. The left regular representation $L_g f(x) = f(g^{-1}x)$ preserves the finite-dimensional subspace $W_\lambda = \mathrm{span}\{\rho_\lambda(\cdot)_{ij}\}_{i,j}$ . By Step 1, these subspaces are mutually orthogonal. The representation on $W_\lambda$ is equivalent to $d_\lambda$ copies of $V_\lambda$ (varying the column index $j$ with fixed row $i$ gives a copy of $V_\lambda$ ; there are $d_\lambda$ row indices). More precisely, $W_\lambda \cong V_\lambda \otimes V_\lambda^*$ . By completeness, $L^2(G) = \overline{\bigoplus_\lambda W_\lambda}$ . $\square$

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ExamplePeter-Weyl for SO(3) and Spherical Harmonics

For $G = \mathrm{SO}(3)$ : irreducible representations are $D^\ell$ of dimension $2\ell+1$ ( $\ell = 0,1,2,\ldots$ ). The matrix elements $D^\ell_{mm'}(R)$ are the Wigner $D$ -matrices. Peter-Weyl gives: any $f \in L^2(\mathrm{SO}(3))$ expands in $D$ -matrices. Restricting to functions on $S^2 \cong \mathrm{SO}(3)/\mathrm{SO}(2)$ (functions depending only on the coset): only $m'=0$ survives, and $D^\ell_{m0}(\theta,\phi) \propto Y_\ell^m(\theta,\phi)$ . This recovers the spherical harmonic expansion on $L^2(S^2)$ .

RemarkNoncompact Groups and Plancherel Theorem

For noncompact groups (e.g., the Lorentz group $\mathrm{SO}(3,1)$ , the Heisenberg group), the Peter-Weyl theorem fails: irreducible representations are generally infinite-dimensional and the discrete sum becomes a continuous integral. The analog is the Plancherel theorem: $\|f\|_2^2 = \int_{\hat{G}} \|\hat{f}(\pi)\|_\mathrm{HS}^2\,d\mu_P(\pi)$ where $\mu_P$ is the Plancherel measure on the unitary dual $\hat{G}$ . For $G = \mathbb{R}$ , this reduces to Parseval's theorem for the Fourier transform.