Isotypic Haar projections specialize to finite character sums

$F$ is a compact Hausdorff topological group. Every subset of $F$ is open and closed in the discrete topology, and the product topology on $F\times F$ is discrete because its points are the basic open sets $\{g\}\times \{h\}$; hence the inversion $F\to F$ and the multiplication $F\times F\to F$ are continuous, being functions on discrete domains [F1]. Thus $F$ is a topological group; it is Hausdorff because distinct $g,h$ are separated by the disjoint open sets $\{g\}$ and $\{h\}$; and it is compact: given an open cover, enumerate $F=\{g_1,\dots ,g_n\}$ and choose a covering member $U_i\ni g_i$ for each $i$, which is a finite choice, and $U_1,\dots ,U_n$ is a finite subcover. [F1] 2.1 The normalized Haar probability $\mu$ exists on $F$ by [F2]. Every singleton is open, hence Borel [F3]. For every $g\in F$ the set $\{g\}=g\{e\}$ is a left translate of $\{e\}$, so left invariance gives $\mu (\{g\})=\mu (\{e\})$; writing $F=\{g_1,\dots ,g_n\}$ as a disjoint union of singletons and using finite additivity and $\mu (F)=1$ gives $1=\mu (F)={\sum }_{i=1}^n\mu (\{g_i\})=n\mu (\{e\})$, so $\mu (\{g\})=1\mathrm{/}\mathrm{\mid }F\mathrm{\mid }$ for every $g\in F$. [F2, F3, step 1.1] 3.1 Fix $v\in H$. The function $f_v(k):=\overline{{\chi }_{\sigma }(k)}\pi (k)v$ is constant on each singleton, with value $x_g:=\overline{{\chi }_{\sigma }(g)}\pi (g)v$ on $\{g\}$; its nonzero fibres are therefore unions of those singletons $g$ for which $x_g$ takes one fixed nonzero value, so $f_v={\sum }_j{y}_j{\mathbf{1}}_{B_j}$ for finitely many pairwise disjoint Borel sets $B_j$ and distinct nonzero $y_j\in H$ [F5]. Each $\mu (B_j)\le \mu (F)=1$ is finite, so $f_v$ is an integrable simple function and hence Bochner integrable, with Bochner integral equal to its simple integral; regrouping the singletons into the fibres and using finite additivity of $\mu$ gives ${\int }_F{f}_{v}d\mu ={\sum }_j\mu (B_j)y_j={\sum }_{g\in F}\mu (\{g\})x_g=\frac{1}{\mathrm{\mid }F\mathrm{\mid }}{\sum }_{g\in F}\overline{{\chi }_{\sigma }(g)}\pi (g)v$. [F3, F5, step 2.1] 4.1 Multiplying the identity of step 3.1 by $d_{\sigma }$ and comparing with the definition $P_{\sigma }v=d_{\sigma }{\int }_F{f}_{v}d\mu$ of [F4] yields $P_{\sigma }v=\frac{d_{\sigma }}{\mathrm{\mid }F\mathrm{\mid }}{\sum }_{g\in F}\overline{{\chi }_{\sigma }(g)}\pi (g)v$ for every $v\in H$, which is the displayed operator identity; the finite sum is independent of the enumeration by [F8]. [F4, F8, step 3.1] 5.1 The representation $\sigma$ and $\pi$ are strongly continuous, since every function on the discrete space $F$ is continuous, and $F$ is a compact Hausdorff group by step 1.1; so the A-page theorem [F6] applies and shows that this operator $P_{\sigma }$ is a bounded linear self-adjoint idempotent commuting with $\pi (F)$ whose range is exactly $H_{\sigma }$, and that for every irreducible $\tau$ inequivalent to $\sigma$ the corresponding projections satisfy $P_{\sigma }{P}_{\tau }=0=P_{\tau }{P}_{\sigma }$ and have orthogonal ranges. [F6, step 1.1, step 4.1] 6.1 Special cases of the formula of step 4.1. For the trivial representation ${\sigma }_0$ on $\mathbb{C}$ one has $d_{{\sigma }_0}=1$ and ${\chi }_{{\sigma }_0}\equiv 1$, so $P_{{\sigma }_0}=\frac{1}{\mathrm{\mid }F\mathrm{\mid }}{\sum }_{g\in F}\pi (g)$; the ${\sigma }_0$-copies are exactly the lines spanned by nonzero vectors fixed by $\pi (F)$ (a fixed vector spans a one-dimensional invariant subspace on which $\pi$ acts trivially, and conversely every ${\sigma }_0$-copy consists of fixed vectors); the fixed space is ${\bigcap }_{g\in F}\ker (\pi (g)-I)$, which is closed because each $\pi (g)-I$ is bounded, so its closed span of fixed lines is itself; by step 5.1 the range of $P_{{\sigma }_0}$ is exactly that fixed space. If $d_{\sigma }=1$, then on a $\sigma$-copy $\pi (g)$ acts as the scalar ${\chi }_{\sigma }(g)$ with $\mathrm{\mid }{\chi }_{\sigma }(g)\mathrm{\mid }=1$, so the formula gives $P_{\sigma }x=\frac{1}{\mathrm{\mid }F\mathrm{\mid }}{\sum }_{g\in F}\overline{{\chi }_{\sigma }(g)}{\chi }_{\sigma }(g)x=x$ for every $x$ in that copy, in agreement with the fixing property of step 5.1. For the one-element group $F=\{e\}$ irreducibility forces $d_{\sigma }=1$, because for $d_{\sigma }\ge 2$ a line in the finite-dimensional space $V_{\sigma }$ is a proper nontrivial closed $\sigma (e)$-invariant subspace, since $\sigma (e)=I$ and every finite-dimensional subspace is closed; then ${\chi }_{\sigma }(e)=1$, $\pi (e)=I$ and the formula gives $P_{\sigma }=I$, while $v=0$ gives $P_{\sigma }v=0$. [F4, F6, F7, step 5.1] 7.1 Explicit two-element group. Let $F=\mathbb{Z}\mathrm{/}2\mathbb{Z}=\{e,t\}$ and $H={\mathbb{C}}^2$ with orthonormal basis $e_1,e_2$, let $\pi (e)=I$, $\pi (t)=\mathrm{diag}(1,-1)$, and let ${\sigma }_0$ be the trivial representation and ${\sigma }_1$ the sign representation ${\sigma }_1(t)=-1$ on $\mathbb{C}$, both irreducible of degree one by [F7] and inequivalent because their characters differ at $t$. Applying the formula of step 4.1 ($\mathrm{\mid }F\mathrm{\mid }=2$, characters ${\chi }_{{\sigma }_0}(e)={\chi }_{{\sigma }_0}(t)=1$ and ${\chi }_{{\sigma }_1}(e)=1,{\chi }_{{\sigma }_1}(t)=-1$) gives $P_{{\sigma }_0}=\frac{1}{2}(I+\pi (t))=\mathrm{diag}(1,0)$ and $P_{{\sigma }_1}=\frac{1}{2}(I-\pi (t))=\mathrm{diag}(0,1)$; both matrices are self-adjoint idempotents, they are mutually orthogonal, and their ranges $\mathbb{C}{e}_1$ and $\mathbb{C}{e}_2$ are the trivial and sign copies inside $H$, so the ranges are orthogonal and span $H$.