$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ Let us remind ourselves of the finite group story. We are investigating group algebra homomorphisms $\rho:\CC[G]\to \End V$ for finite-dimensional $V$. In the Lie group case, the algebra on the LHS is $L^2(G,\CC)$ with the algebra operation $\ast$ via$(f\ast h)(x) = \int_G f(g^{-1}x)h(g)dg.$The embedding is $g\mapsto \delta(xg^{-1})$, where the RHS is a distribution to be precise, but we won't be here. The ==**left regular representation**== has the interpretation of left-shifting now; $\psi\mapsto \psi\circ L_{g^{-1}}$. Yes, in addition to being the left-multiplication map under this algebra, this is the representation you remember, $\rho(g): h\mapsto gh$. Yes, we can flip signs or transpose and get other conventions, just work with me here. This regular representation is extremely important in both the finite and Lie case. >[!idea] Even in the finite case, we should regard $\CC[G]\equiv L^2(G)$. Then, we obtain a unitary structure. The key idea here is that under the $\braket{\bullet,\bullet}$ endowed by $\CC[G]$ or $L^2(G)$, any Hermitian form $[\bullet,\bullet]$ can be upgraded to an averaged Hermitian form $\{\bullet,\bullet\}$ which is $G$-invariant, by embedding $\eta: V\curvearrowright L^2(G,V)\equiv L^2(G)\otimes V$ via $\eta(v)(x) = \rho(x)(v)$ and then considering the tensor product $\braket{\bullet,\bullet}\otimes [\bullet,\bullet]$ We obtain that sub-representations admit orthogonal complements, and thus all finite-dimensional representations are completely reducible. >[!idea] This $\eta$ map is a morally correct way to think about any finite-dimensional representation. >There are several moving pieces here. $L^2(G)\otimes V$ is the natural ==**induction**== extending $V$ as a $\CC$-module to $L^2(G)$. Okay, now what. In both the finite group story and the Lie group story, we now define ==**characters**== $\chi_V\in L^2(G)$ via $\chi_V(g) = \Tr(\rho(g))$. This is immediately sensible for finite-dimensional representations. Drawing analogy from Fourier analysis, the finite case, or formally evaluating the integral over $L^2(G)$, we obtain $\chi_{L^2(G)} = \delta(x)$. The ==**matrix coefficients**== are defined by considering the doubly $G$-invariant map $G\to V^*\otimes V$ given by $\rho$, and flip this to an isometric $\xi_V\in\Hom_G(V\otimes V^*, L^2(G))$ which extends to an isometric doubly $G$-invariant map $\bigoplus_{V\in \Irrep(G)} V\otimes V^*\hookrightarrow L^2(G)$. The characters are thus also orthonormal inside the conjugation-invariant functions $L^2(G)^G$ (as $\chi_V = \xi_V(\id_V)$ and $\id_V$ is permeable), also known as the ==**class functions**==. In the finite case, we immediately get that **the number of irreps is $\leq$ the number of conjugacy classes lt;\infty$**. >[!idea] Deep thing >In the infinite-dimensional case, there are a continuum of conjugacy classes, but a countable number of *rational* conjugacy classes. Look at the $U(1)$ case; any fixed $q\in \ZZ$ yields a conjugacy class $\{\frac{p}{q}: (p,q) = 1\}$, and the only representations are $z\mapsto z^q$. Conjugacy classes do not naturally biject to irreps, but the cardinality arguments should carry over. > >More specifically, the space $L^2(G)$ is ==**separable**==; $G$ can be given a finite atlas, on which each chart is a Borel measurable compact subset of $\RR^{\dim G}$. Thus, it admits a countable orthonormal basis. This is why things like $\sum_{V\in \Irrep(G)}$ are valid. The really good thing about characters is that they're additive under $\bigoplus$; the matrix coefficients don't have the correct type and the best you can do is a big tensor product. Thus, we already know by complete reducibility that $\chi_W = \sum_{V\in \Irrep(G)} \braket{\chi_W | \chi_V}\chi_V$; you don't need Peter-Weyl to see this. If only we could apply this to the regular representation, we would obtain Peter-Weyl, because then $L^2(G) = \bigoplus_{V\in \Irrep(G)} V^{\dim V}$ as desired.