The normalized irreducible matrix coefficient family

Assume the Axiom of Choice. Let $K$ be a compact Hausdorff group with normalized Haar probability $\mu$ and unitary dual $\widehat{K}$ (The unitary dual of a compact group). For each class $\pi \in \widehat{K}$ fix a representative, still written $\pi$, on a finite-dimensional carrier $H_{\pi }$ with $d_{\pi }:={\dim }_{\mathbb{C}}{H}_{\pi }\ge 1$ (Irreducible unitary representations of compact groups are finite dimensional, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis), and fix an orthonormal basis $e_1^{\pi },\dots ,e_{d_{\pi }}^{\pi }$ (Every finite-dimensional real or complex inner product space has an orthonormal basis); both choices are licensed by AC (The Axiom of Choice). The normalized irreducible matrix coefficient family is $\mathcal{B}=(u_{ij}^{\pi }{)}_{\pi \in \widehat{K},1\le i,j\le d_{\pi }},u_{ij}^{\pi }(k):=\sqrt{d_{\pi }}⟨\pi (k)e_i^{\pi },e_j^{\pi }⟩,$ the matrix coefficients being those of Matrix coefficient of a unitary representation.

The normalization is the one that makes $\mathcal{B}$ orthonormal. For a class $\pi$ and indices $i,j,k,l$, Schur orthogonality in the convention $⟨\pi (k)v,w⟩$ with its $1\mathrm{/}{d}_{\pi }$ constant (Schur orthogonality for general compact groups) gives ${\int }_K{u}_{ij}^{\pi }(k)\overline{u_{kl}^{\pi }(k)}d\mu (k)=d_{\pi }\cdot \frac{1}{d_{\pi }}⟨e_i^{\pi },e_k^{\pi }⟩\overline{⟨e_j^{\pi },e_l^{\pi }⟩}={\delta }_{ik}{\delta }_{jl},$ and for two inequivalent classes the same theorem gives inner product $0$ between any two of their coefficients; thus the normalization $\sqrt{d_{\pi }}$ is exactly the factor that converts the $1\mathrm{/}{d}_{\pi }$ Schur constant into the unit of the family. No completeness claim is made here; it is the content of the theorem that the closed span of $\mathcal{B}$ is $L^2(K)$.

Choice invariance. Different choices of representatives and orthonormal bases produce the same family up to a unitary change of coordinates in each block and a relabeling of its indices. Explicitly, let $e_j^{\mathrm{\prime }\pi }={\sum }_k{U}_{kj}{e}_k^{\pi }$ be another orthonormal basis of the same carrier, with $U$ unitary. Then for all $i,j$ $u_{ij}^{\mathrm{\prime }\pi }(k)=\sqrt{d_{\pi }}⟨\pi (k){\sum }_m{U}_{mi}{e}_m^{\pi },{\sum }_n{U}_{nj}{e}_n^{\pi }⟩={\sum }_{m,n}{U}_{mi}\overline{U_{nj}}{u}_{mn}^{\pi }(k),$ so the block $(u_{ij}^{\mathrm{\prime }\pi }{)}_{i,j}$ is obtained from $(u_{ij}^{\pi }{)}_{i,j}$ by the unitary change of coordinates $U$; replacing the representative of $\pi$ by a unitarily equivalent one acts by a further fixed unitary in that block. Every statement about $\mathcal{B}$ made in this development is invariant under these changes.