Isotypic projections are mutually orthogonal equivariant projections

If $H=\{0\}$ then $P_{\sigma }=0$, $H_{\sigma }=\{0\}$ and every assertion is immediate, so assume $H\mathrm{\ne }\{0\}$ throughout; nothing below uses more than $H\mathrm{\ne }\{0\}$ where a vector is exhibited.

Linearity and boundedness. Let $v,v^{\mathrm{\prime }}\in H$, $a,b\in \mathbb{C}$ and $y\in H$. By [F2], linearity of each $\pi (k)$ and termwise integration [F5], $⟨P_{\sigma }(av+b{v}^{\mathrm{\prime }}),y⟩=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨\pi (k)(av+b{v}^{\mathrm{\prime }}),y⟩d\mu (k)=a⟨P_{\sigma }v,y⟩+b⟨P_{\sigma }{v}^{\mathrm{\prime }},y⟩$; a vector is determined by its pairings, so $P_{\sigma }(av+b{v}^{\mathrm{\prime }})=a{P}_{\sigma }v+b{P}_{\sigma }{v}^{\mathrm{\prime }}$. Moreover $\mathrm{\mid }⟨P_{\sigma }v,y⟩\mathrm{\mid }\le d_{\sigma }{\int }_K\mathrm{\mid }{\chi }_{\sigma }(k)\mathrm{\mid }\mathrm{\mid }⟨\pi (k)v,y⟩\mathrm{\mid }d\mu (k)\le d_{\sigma }\left({\int }_K\mathrm{\mid }{\chi }_{\sigma }\mathrm{\mid }d\mu \right)\mathrm{\parallel }v\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$ by Cauchy--Schwarz [F10]; applying this with $y=P_{\sigma }v$ when $P_{\sigma }v\mathrm{\ne }0$ gives $\mathrm{\parallel }{P}_{\sigma }v\mathrm{\parallel }\le d_{\sigma }\left({\int }_K\mathrm{\mid }{\chi }_{\sigma }\mathrm{\mid }d\mu \right)\mathrm{\parallel }v\mathrm{\parallel }$. Thus $P_{\sigma }$ is a bounded linear operator.

Self-adjointness. By [F2] and unitarity of $\pi (k)$, $⟨P_{\sigma }v,y⟩=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨v,\pi (k{)}^{-1}y⟩d\mu (k)$; substituting $k\mapsto k^{-1}$, which preserves $\mu$ by [F4], and using $\overline{{\chi }_{\sigma }(k^{-1})}={\chi }_{\sigma }(k)$ from [F3] turns this into $d_{\sigma }{\int }_K{\chi }_{\sigma }(k)⟨v,\pi (k)y⟩d\mu (k)=\overline{d_{\sigma }{\int }_K\stackrel{‾}{{\chi }_{\sigma }(k)}⟨\pi (k)y,v⟩d\mu (k)}=\overline{⟨P_{\sigma }y,v⟩}=⟨v,P_{\sigma }y⟩$, the conjugation inside the scalar integral being justified by [F5]. Hence $P_{\sigma }$ is self-adjoint.

Equivariance. Let $g\in K$. For $v,y\in H$, [F2] and the homomorphism property give $⟨P_{\sigma }\pi (g)v,y⟩=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨\pi (kg)v,y⟩d\mu (k)$; right translation $u\mapsto u{g}^{-1}$ is measure preserving by [F4], so substituting $k=u{g}^{-1}$ this equals $d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(u{g}^{-1})}⟨\pi (u)v,y⟩d\mu (u)$, and since ${\chi }_{\sigma }$ is a class function the identity ${\chi }_{\sigma }(u{g}^{-1})={\chi }_{\sigma }(g^{-1}u)$ of [F3] gives $d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(g^{-1}u)}⟨\pi (u)v,y⟩d\mu (u)$; left translation $u\mapsto gu$ is measure preserving by [F4], so substituting $u=gk$ gives $d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨\pi (gk)v,y⟩d\mu (k)=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨\pi (k)v,\pi (g{)}^{-1}y⟩d\mu (k)=⟨P_{\sigma }v,\pi (g{)}^{-1}y⟩=⟨\pi (g)P_{\sigma }v,y⟩$, the second-to-last equality by unitarity of $\pi (g)$. As $y$ is arbitrary, $P_{\sigma }\pi (g)=\pi (g)P_{\sigma }$.

Fixing a single $\sigma$-copy. Let $M\subseteq H$ be a $\sigma$-copy with unitary intertwiner $U:V_{\sigma }\to M$, and let $x\in M$, $y\in H$. Writing $p_M$ for the orthogonal projection onto the closed subspace $M$ [F9], the invariance of $M$ gives $\pi (k)x\in M$, hence $⟨\pi (k)x,y⟩=⟨\pi (k)x,p_{M}y⟩$. For $x=Ua$ and $p_{M}y=Ub$ with $a,b\in V_{\sigma }$, intertwining and unitarity of $U$ give $⟨\pi (k)Ua,Ub⟩=⟨\sigma (k)a,b⟩$, and the character expansion of [F6] together with Schur orthogonality (ii) yields $⟨P_{\sigma }x,y⟩=d_{\sigma }{\sum }_i{\int }_K⟨\sigma (k)a,b⟩\overline{⟨\sigma (k)e_i,e_i⟩}d\mu (k)=d_{\sigma }{\sum }_i{d}_{\sigma }^{-1}⟨a,e_i⟩\overline{⟨b,e_i⟩}={\sum }_i⟨a,e_i⟩\overline{⟨b,e_i⟩}=⟨a,b⟩=⟨Ua,Ub⟩=⟨x,p_{M}y⟩=⟨x,y⟩$. Hence $P_{\sigma }x=x$ for every $x$ in any $\sigma$-copy.

Killing inequivalent copies. Let $\tau$ be an irreducible strongly continuous unitary representation of $K$ inequivalent to $\sigma$, let $M\subseteq H$ be a $\tau$-copy with unitary intertwiner $U:V_{\tau }\to M$, and let $x=Ua\in M$, $y\in H$ with $p_{M}y=Ub$. Then $\pi (k)x\in M$, so $⟨\pi (k)x,y⟩=⟨\tau (k)a,b⟩$, and [F2], [F6] with Schur orthogonality (i) applied to the inequivalent irreducibles $\tau$ and $\sigma$ give $⟨P_{\sigma }x,y⟩=d_{\sigma }{\sum }_i{\int }_K⟨\tau (k)a,b⟩\overline{⟨\sigma (k)e_i,e_i⟩}d\mu (k)=0$. Hence $P_{\sigma }x=0$ for every $x$ in any $\tau$-copy.

The range lies in the isotypic subspace. Fix $v\in H$ and, for each $i$, define $A_i(w):={\int }_K⟨w,\sigma (k)e_i⟩\pi (k)vd\mu (k)$ for $w\in V_{\sigma }$. The integrand $g_{i,w}(k):=⟨w,\sigma (k)e_i⟩\pi (k)v$ is continuous and satisfies $\mathrm{\parallel }{g}_{i,w}(k)\mathrm{\parallel }\le \mathrm{\parallel }w\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ for every $k$, because $\sigma (k)e_i$ has norm $1$ and $\pi (k)v$ has norm $\mathrm{\parallel }v\mathrm{\parallel }$, so $g_{i,w}$ is Bochner integrable and $\mathrm{\parallel }{A}_i(w)\mathrm{\parallel }\le \mathrm{\parallel }w\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ by [F8]; moreover, since the bounded linear functional $x\mapsto ⟨x,y⟩$ commutes with the Bochner integral, $⟨A_i(w),y⟩={\int }_K⟨w,\sigma (k)e_i⟩⟨\pi (k)v,y⟩d\mu (k)$ for every $y\in H$. The map $A_i$ is linear: the scalar integrand in this pairing is linear in $w$ and the scalar integral is linear [F5], so $⟨A_i(aw+b{w}^{\mathrm{\prime }}),y⟩=a⟨A_i(w),y⟩+b⟨A_i(w^{\mathrm{\prime }}),y⟩$ for all $y$, and a vector is determined by its pairings. For $h\in K$ and $y\in H$, unitarity of $\pi (h)$ together with the pairing formula gives $⟨\pi (h)A_i(w),y⟩=⟨A_i(w),\pi (h{)}^{-1}y⟩={\int }_K⟨w,\sigma (k)e_i⟩⟨\pi (hk)v,y⟩d\mu (k)$, and the substitution $k\mapsto h^{-1}k$, measure preserving by [F4], turns this into ${\int }_K⟨w,\sigma (h^{-1}k)e_i⟩⟨\pi (k)v,y⟩d\mu (k)={\int }_K⟨\sigma (h)w,\sigma (k)e_i⟩⟨\pi (k)v,y⟩d\mu (k)=⟨A_i(\sigma (h)w),y⟩$; hence $\pi (h)A_i=A_i\sigma (h)$ and each $A_i$ is a bounded intertwiner. If $A_i\mathrm{\ne }0$ then $\ker A_i$ is a $\sigma$-invariant subspace, because $A_i\sigma (h)=\pi (h)A_i$ for all $h$, and $\ker A_i\mathrm{\ne }{V}_{\sigma }$; by irreducibility $\ker A_i=\{0\}$, so $A_i$ is injective, and its image $M_i=A_i(V_{\sigma })$ is finite dimensional (hence closed [F10]) and $\pi (K)$-invariant, because $\pi (h)M_i=A_i\sigma (h)V_{\sigma }\subseteq M_i$; a closed invariant subspace $N\subseteq M_i$ pulls back under the injective intertwiner $A_i$ to the $\sigma$-invariant subspace $A_i^{-1}(N)$ of the irreducible $V_{\sigma }$, which is $\{0\}$ or $V_{\sigma }$, so $\pi {\mathrm{\mid }}_{M_i}$ is irreducible and the nonzero bounded intertwiner $A_i$ makes $\pi {\mathrm{\mid }}_{M_i}$ unitarily equivalent to $\sigma$ by Schur's lemma [F7]; thus $M_i$ is a $\sigma$-copy and $A_i(e_i)\in H_{\sigma }$, while if $A_i=0$ then $A_i(e_i)=0\in H_{\sigma }$. Finally, for every $y$, summing the pairing formula over $i$ and using linearity of the scalar integral [F5] and the trace formula ${\chi }_{\sigma }(k)={\sum }_i⟨\sigma (k)e_i,e_i⟩$ of [F3] together with conjugate symmetry $⟨e_i,\sigma (k)e_i⟩=\overline{⟨\sigma (k)e_i,e_i⟩}$ gives $⟨d_{\sigma }{\sum }_i{A}_i(e_i),y⟩=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}⟨\pi (k)v,y⟩d\mu (k)=⟨P_{\sigma }v,y⟩$ by [F2]; hence $P_{\sigma }v=d_{\sigma }{\sum }_i{A}_i(e_i)$ lies in $H_{\sigma }$.

Fixing the isotypic subspace and identifying the range. By step 1.5, $P_{\sigma }$ fixes every vector lying in a $\sigma$-copy; by linearity (step 1.2) it fixes the linear span of all $\sigma$-copies, and since it is bounded, hence continuous, it fixes the closure of that span: $P_{\sigma }x=x$ for every $x\in H_{\sigma }$. Combined with step 1.7, for every $v\in H$ the vector $P_{\sigma }v$ lies in $H_{\sigma }$ and is therefore fixed, so $P_{\sigma }^2=P_{\sigma }$. Hence $\mathrm{range}(P_{\sigma })\subseteq H_{\sigma }$ by step 1.7 and $H_{\sigma }\subseteq \mathrm{range}(P_{\sigma })$ because $x=P_{\sigma }x$ for $x\in H_{\sigma }$; the range is exactly $H_{\sigma }$.

Distinct inequivalent types. Let $\tau$ be irreducible and inequivalent to $\sigma$ with isotypic projection $P_{\tau }$ and isotypic subspace $H_{\tau }$. Repeating steps 1.6, 1.7 and 2.1 with $\tau$ in place of $\sigma$ shows $P_{\sigma }$ vanishes on every $\tau$-copy, hence by linearity and continuity on $H_{\tau }$, and that $\mathrm{range}(P_{\tau })=H_{\tau }$; therefore $P_{\sigma }{P}_{\tau }=0$. Exchanging the roles of $\sigma$ and $\tau$ gives $P_{\tau }{P}_{\sigma }=0$. Finally, for $v,w\in H$, self-adjointness (step 1.3) and idempotence (step 2.1) give $⟨P_{\sigma }v,P_{\tau }w⟩=⟨P_{\sigma }^{2}v,P_{\tau }w⟩=⟨P_{\sigma }v,P_{\sigma }{P}_{\tau }w⟩=0$, so the ranges of $P_{\sigma }$ and $P_{\tau }$ are orthogonal.

Collecting steps 1.2, 1.3, 1.4, 1.5, 2.1 and 3.1: $P_{\sigma }$ is a bounded self-adjoint idempotent commuting with $\pi (K)$, fixes every $\sigma$-copy, has range exactly the $\sigma$-isotypic subspace $H_{\sigma }$, and the projections of inequivalent irreducibles multiply to zero with orthogonal ranges. At no point is any sum over the unitary dual asserted, and no density or completeness statement is used.