Induced Unitary Representations of Locally Compact Groups

The quotient map is open because $p^{-1}(p(O))=OH$ is open whenever $O\subseteq G$ is open. To separate distinct cosets $xH,yH$, note $y^{-1}x\mathrm{\notin }H$. Closedness gives an open neighborhood $W$ of $y^{-1}x$ disjoint from $H$. Continuity of $(v,u)\mapsto v^{-1}{y}^{-1}xu$ gives identity neighborhoods $V,U$ with $V^{-1}{y}^{-1}xU\subseteq W$; hence $p(xU)$ and $p(yV)$ are disjoint. Thus $X$ is Hausdorff. If $U$ is a relatively compact open neighborhood of $x$, then $p(U)$ is open and its closure lies in the compact, hence closed, set $p(\bar{U})$, so $X$ is locally compact.

For compact $Q\subseteq X$, cover $Q$ by sets $p(U_q)$ where each $U_q$ is relatively compact and open. A finite subcover exists, and the union $K$ of the corresponding finitely many compact closures satisfies $Q\subseteq p(K)$.

For each $f\in C_c(G)$, the function $x\mapsto {\int }_{H}f(xh)dh$ is continuous locally on $G$: around any $x_0$ choose a compact neighborhood $K$; the kernel $(x,h)\mapsto f(xh)$ on $K\times H$ is supported in the compact set $\{(x,h)\in K\times H:xh\in \mathrm{supp}f\}$, so [F2] gives continuity there. Left invariance of $dh$ makes this function right $H$-invariant, and openness of $p$ makes its descended function $T_{H}f$ continuous. Its support lies in the compact set $p(\mathrm{supp}f)$, so $T_{H}f\in C_c(X)$.

Let $\varphi \in C_c(X)$ and $Q=\mathrm{supp}\varphi$. By [A1], choose a lift $x_q$ of each $q\in Q$. For each lift apply [F1] to the singleton and a relatively compact open neighborhood to obtain a nonnegative $u_q\in C_c(G)$ with $u_q(x_q)=1$. A nonzero left Haar measure has full support: its support is a nonempty closed set invariant under every left translation, hence is all of $H$. Thus $u_q(x_q\cdot )$ has positive integral on $H$, so $T_H{u}_q(q)>0$. By continuity from step 2.2, $T_H{u}_q$ stays positive on a neighborhood of $q$. A finite subcover of $Q$ gives $u={\sum }_i{u}_{q_i}$ with $T_{H}u>0$ on an open neighborhood $W$ of $Q$. The function $\psi =\varphi \mathrm{/}(T_{H}u)$ on $W$, extended by zero, is in $C_c(X)$ because its support is contained in the compact set $Q\subset W$. Set $f=(\psi \circ p)u$. Then $f\in C_c(G)$ and $T_{H}f=\psi T_{H}u=\varphi$, also when $\varphi =0$ (use $f=0$). Thus $T_H:C_c(G)\to C_c(X)$ is onto; compact lifts were proved in step 2.1. ∎

Choose a relatively compact symmetric open identity neighborhood $U$ in $G$ and let $L={\bigcup }_{n\ge 1}{U}^n$. Then $L$ is an open subgroup and is $\sigma$-compact, since $L={\bigcup }_n(\bar{U}{)}^n$. Its orbits on $X$ are open and disjoint; each is a continuous image of $L$, hence $\sigma$-compact. As an open subspace of the LCH space $X$, each orbit is regular and Lindelöf. By [F1] and [F3], every orbit is paracompact, and their topological sum $X$ is paracompact.

Cover $X$ by relatively compact open sets. By [F4] choose a locally finite partition of unity $({\psi }_i)$ subordinate to this cover; each $\mathrm{supp}{\psi }_i$ is compact. Use [F5] to choose ${\chi }_i\in C_c(X)$ with ${\chi }_i=1$ on $\mathrm{supp}{\psi }_i$, and [F2] to choose a nonnegative $u_i\in C_c(G)$ with $T_H{u}_i={\chi }_i$. Define $b_i=({\psi }_i\circ p)u_i$. It is continuous, nonnegative and compactly supported, and $T_H{b}_i={\psi }_i{\chi }_i={\psi }_i$.

Set $\beta ={\sum }_i{b}_i$. Since $({\psi }_i)$ is locally finite and $p$ is continuous, the sum is locally finite on $G$, hence continuous and nonnegative. Fiber integration gives $T_H\beta ={\sum }_i{\psi }_i=1$. For compact $Q\subseteq X$, only finitely many $\mathrm{supp}{\psi }_i$ meet $Q$; the support of $\beta$ over $Q$ is contained in the finite union of the compact sets $\mathrm{supp}{u}_i\cap p^{-1}(Q)$. Thus it is compact. AC supplies the choices, and the construction applies to non-$\sigma$-compact $X$ because it uses the open $L$-orbits from step 1.1. ∎

For $f,g\in C_c(G)$, the kernel $(x,h)\mapsto f(x)g(xh)\rho (x)$ has compact support in $G\times H$: its support lies in $\mathrm{supp}f\times ((\mathrm{supp}f{)}^{-1}\mathrm{supp}g\cap H)$. Thus [F3] permits interchanging the two integrations. Right-translation change of variables in $G$, [F1], and inversion in $H$ using [F6] give ${\int }_{G}f(x)(T_{H}g)(xH)\rho (x)dx={\int }_G(T_{H}f)(xH)g(x)\rho (x)dx\mathrm{.}$ Explicitly, the inner integral at $h$ becomes ${\mathrm{\Delta }}_H(h{)}^{-1}{\int }_{G}f(y{h}^{-1})g(y)\rho (y)dy$; integrating this in $h$ and applying [F6] gives ${\int }_{H}f(yh)dh$.

Define $\mathrm{\Lambda }(T_{H}f)={\int }_{G}f\rho dx$. If $T_{H}f=0$, let $Q=p(\mathrm{supp}f)$ and choose $g\in C_c(G)$ with $T_{H}g=1$ on $Q$, as supplied by the compact-set lift construction in [F2]. The identity in step 1.1 gives ${\int }_{G}f\rho dx={\int }_G(T_{H}f)(xH)g(x)\rho (x)dx=0$. Thus $\mathrm{\Lambda }$ is well defined. If $\varphi \ge 0$, choose a nonnegative lift $f$ with $T_{H}f=\varphi$ using [F2]; then $\mathrm{\Lambda }(\varphi )=\int f\rho dx\ge 0$.

By [F4] and [F7], $\mathrm{\Lambda }$ is represented by a Radon measure ${\mu }_{\rho }$, and [F5] makes it unique. The defining identity for $\mathrm{\Lambda }$ is the displayed Weil formula. For any nonempty open $O\subseteq X$, [F8] gives a nonzero nonnegative $\varphi \in C_c(X)$ supported in $O$. Choose the nonnegative lift $f$ from [F2]. Since $T_{H}f=\varphi$ is nonzero, $f$ is positive at some point and hence on a nonempty open subset of $G$. A nonzero left Haar measure has full support: its support is nonempty, closed, and invariant under every left translation, so it is all of $G$. The positive continuous weight $\rho$ therefore gives ${\int }_{G}f\rho dx>0$. The Weil identity implies ${\mu }_{\rho }(O)>0$, proving full support. ∎

Define $\rho (x)={\int }_H\beta (xh){\mathrm{\Delta }}_G(h)\mathrm{/}{\mathrm{\Delta }}_H(h)dh$. For each $x$, the integrand is supported on the compact fiber intersection $x^{-1}\mathrm{supp}\beta \cap H$, so its integral is finite. It is positive because $\beta \ge 0$, the weight is positive, and ${\int }_H\beta (xh)dh=1$.

Near $x_0$ choose a compact neighborhood $K$. The set $S=\mathrm{supp}\beta \cap p^{-1}(p(K))$ is compact by [F2], and all $h$ for which $xh\in \mathrm{supp}\beta$ with $x\in K$ lie in the compact set $K^{-1}S\cap H$. The integrand is jointly continuous with this common compact support; [F4] gives continuity of its integral. Thus $\rho$ is positive and continuous.

For $h_0\in H$, substitute $k=h_{0}h$; left invariance of $dh$ and the homomorphism laws give $\rho (x{h}_0)={\mathrm{\Delta }}_H(h_0){\mathrm{\Delta }}_G(h_0{)}^{-1}\rho (x)$. Hence $\rho$ is a rho-function.

Apply [F5] to obtain ${\mu }_{\rho }$ and the Weil formula. The ratio $D_g(xH)=\rho (g^{-1}x)\mathrm{/}\rho (x)$ is independent of the representative by [F3] and is positive continuous. For $\varphi =T_{H}f$, Weil and left invariance give ${\int }_{G\mathrm{/}H}\varphi (gq)d{\mu }_{\rho }(q)={\int }_{G}f(gx)\rho (x)dx={\int }_{G}f(y)\rho (g^{-1}y)dy={\int }_{G\mathrm{/}H}\varphi (q)D_g(q)d{\mu }_{\rho }(q)\mathrm{.}$ By [F6] this holds for every $\varphi \in C_c(G\mathrm{/}H)$, and [F7] identifies $g_{\ast }{\mu }_{\rho }=D_g{\mu }_{\rho }$. Positivity of $D_g$ gives equivalence of measures; the ratio descends continuously jointly in $(g,q)$ through the open quotient map. Thus ${\mu }_{\rho }$ is strongly quasi-invariant. Full support is part of [F5]. ∎

Assume $\nu$ is a nonzero invariant Radon measure on $X=G\mathrm{/}H$. Define $L(f)={\int }_X{T}_{H}fd\nu$ for real $f\in C_c(G)$. This functional is positive. If it were zero, surjectivity [F1] would make every $C_c(X)$ integral against $\nu$ zero, and [F2] would force $\nu =0$. Thus $L$ is nonzero.

Conversely suppose the modular functions agree on $H$. Then $\rho =1$ satisfies the covariance in [F5]. Let ${\mu }_1$ be the Weil measure. For $a\in G$ and $\varphi =T_{H}f$, its translate satisfies ${\int }_X\varphi (a^{-1}q)d{\mu }_1(q)={\int }_{G}f(a^{-1}x)dx={\int }_{G}f(x)dx={\int }_X\varphi (q)d{\mu }_1(q),$ by left invariance.

For $a\in G$, let $L_{a}f(x)=f(a^{-1}x)$. Then $T_H(L_{a}f)(xH)=T_{H}f(a^{-1}xH)$, so invariance of $\nu$ gives $L(L_{a}f)=L(f)$. By [F2], $L$ is represented by a Radon measure $\lambda$ on $G$; it is left invariant and nonzero, hence a left Haar measure.

By [F3], $\lambda =cdx$ for $c>0$. For $h\in H$, right translation gives $T_H(R_{h}f)={\mathrm{\Delta }}_H(h{)}^{-1}{T}_{H}f$ by [F4] applied in $H$. Hence $L(R_{h}f)={\mathrm{\Delta }}_H(h{)}^{-1}L(f)$. Since $\lambda =cdx$, [F4] applied in $G$ also gives $L(R_{h}f)={\mathrm{\Delta }}_G(h{)}^{-1}L(f)$. Choose $f$ with $L(f)>0$; equality forces ${\mathrm{\Delta }}_G(h)={\mathrm{\Delta }}_H(h)$.

Surjectivity [F1] gives this equality for every $C_c(X)$ test function. The Radon uniqueness in [F2] shows $a_{\ast }{\mu }_1={\mu }_1$; the Weil measure is nonzero. This proves sufficiency and the equivalence. ∎

Define $D_g(xH)=\rho (g^{-1}x)\mathrm{/}\rho (x)$. Replacing $x$ by $xh$ multiplies numerator and denominator by the same factor from [F1], so the ratio is well-defined and positive. The continuous function $(g,x)\mapsto \rho (g^{-1}x)\mathrm{/}\rho (x)$ is constant on fibers in the second coordinate; [F5] makes its descent through $G\times G\to G\times G\mathrm{/}H$ continuous.

For $\varphi =T_{H}f$, Weil’s formula and left invariance give ${\int }_{G\mathrm{/}H}\varphi (gq)d{\mu }_{\rho }(q)={\int }_{G}f(gx)\rho (x)dx={\int }_{G}f(y)\rho (g^{-1}y)dy={\int }_{G\mathrm{/}H}\varphi (q)D_g(q)d{\mu }_{\rho }(q)\mathrm{.}$ The positive continuous density is locally bounded, so it defines a Radon measure relative to the Radon measure ${\mu }_{\rho }$. By [F3] the equality holds on every $C_c(G\mathrm{/}H)$ function; [F4] identifies $g_{\ast }{\mu }_{\rho }=D_g{\mu }_{\rho }$. This proves the derivative formula.

For $q=xH$, the ratios telescope: $D_{g_1}(q)D_{g_2}(g_1^{-1}q)=\frac{\rho (g_1^{-1}x)}{\rho (x)}\frac{\rho (g_2^{-1}{g}_1^{-1}x)}{\rho (g_1^{-1}x)}=D_{g_1{g}_2}(q)\mathrm{.}$ Together with step 1.1, this proves the stated cocycle identity and continuity. ∎

For two covariant sections, [F1] gives $\mathrm{\parallel }{F}_1(xh)-F_2(xh)\mathrm{\parallel }=\mathrm{\parallel }{F}_1(x)-F_2(x)\mathrm{\parallel }$. Thus their difference norm descends continuously to $G\mathrm{/}H$. Compact support modulo $H$ and Radon finiteness on compact sets make its square integrable, so the stated norm is well-defined.

Let $(F_n)$ be Cauchy in this norm. Choose a subsequence $(F_{n_k})$ such that ${\sum }_k\mathrm{\parallel }{F}_{n_{k+1}}-F_{n_k}{\mathrm{\parallel }}_2<\mathrm{\infty }$. The partial sums $S_m(q)={\sum }_{k<m}\mathrm{\parallel }{F}_{n_{k+1}}(x)-F_{n_k}(x)\mathrm{\parallel }$, with $q=xH$, satisfy $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_2\le {\sum }_k\mathrm{\parallel }{F}_{n_{k+1}}-F_{n_k}{\mathrm{\parallel }}_2$ by Minkowski. By [F3] and monotone convergence, $S={\lim }_m{S}_m$ is finite almost everywhere and has finite $L^2$ norm. Its infinite-value set is a measurable null set in $G\mathrm{/}H$. Outside its saturated preimage, the sections are pointwise Cauchy for every lift and converge to a covariant function $F$; set $F=0$ on the null fibers. On each compact neighborhood in $G$, the continuous approximants have jointly separable range, so their pointwise limit is locally strongly measurable. The $L^2$ norm of the tail is bounded by the tail of the same summable series, again by Minkowski and monotone convergence. Thus the embedded $L^2$ classes converge to $F$, which represents the original completion vector.

Equivalent Cauchy sequences have difference norm zero and so have representatives equal almost everywhere. Conversely, representatives equal almost everywhere have zero difference norm, hence define the same completion vector. This proves the asserted identification. ∎

For $h\in H$, covariance and unitarity give $⟨F_1(xh),F_2(xh)⟩=⟨\sigma (h{)}^{-1}{F}_1(x),\sigma (h{)}^{-1}{F}_2(x)⟩=⟨F_1(x),F_2(x)⟩\mathrm{.}$ Thus the scalar is independent of the representative. Its continuous lift to $G$ descends continuously because the quotient map is open; its support lies in the intersection of the compact quotient supports.

Radon finiteness on that compact support makes the integral finite. For $F_1=F_2=F$, the integral is nonnegative. If it were zero but $F$ were nonzero at some $x$, continuity would make $\mathrm{\parallel }F{\mathrm{\parallel }}^2$ positive on a nonempty open subset of $G\mathrm{/}H$, which has positive measure by full support [F4], a contradiction. Hence the norm is positive definite; integrating the pointwise sesquilinear form gives the asserted inner product. ∎

For fixed $x$, the integrand defining ${\xi }_{f,v}$ is supported on the compact set $x^{-1}\mathrm{supp}f\cap H$, so the Bochner integral exists. Replacing $x$ by $x{h}_0$ and substituting $k=h_{0}h$ gives ${\xi }_{f,v}(x{h}_0)=\sigma (h_0{)}^{-1}{\xi }_{f,v}(x)$. For a relatively compact neighborhood $N$ of a fixed $x_0$, every contributing $h$ lies in the compact set $K={\bar{N}}^{-1}\mathrm{supp}f\cap H$. The integrand $(x,h)\mapsto f(xh)\sigma (h)v$ is jointly continuous on a compact neighborhood times $K$, so its uniform variation in $h$ tends to zero as $x\to x_0$; the integral therefore varies continuously. Its quotient support lies in $p(\mathrm{supp}f)$, which is compact. Thus ${\xi }_{f,v}\in C_c(G,H;V)$.

Now let $F$ be a locally strongly measurable covariant section with finite quotient $L^2$ norm. By [F5] choose $\psi \in C_c(G\mathrm{/}H)$ close in scalar $L^2$ to $q\mapsto \mathrm{\parallel }F(q)\mathrm{\parallel }$; outside $Q=\mathrm{supp}\psi$ the $L^2$ tail of $F$ is therefore small. Put $F_Q={\mathbf{1}}_{Q}F$ and choose a cutoff $\chi \in C_c(G\mathrm{/}H)$ with $\chi =1$ on $Q$ by [F3], then choose a nonnegative lift $u_0\in C_c(G)$ with $T_H{u}_0=\chi$ by [F3]. The measurable map $U=u_0{F}_Q$ is supported in a compact subset of $G$. For $\varphi \in C_c(G\mathrm{/}H)$, apply the Weil formula [F8] to $f(y)=(\varphi \circ p)(y)\mathrm{\mid }{u}_0(y){\mathrm{\mid }}^2$. It identifies the finite Radon measures $B\mapsto {\int }_{p^{-1}(B)}\mathrm{\mid }{u}_0(y){\mathrm{\mid }}^2\rho (y)dy$ and $B\mapsto {\int }_B{T}_H(\mathrm{\mid }{u}_0{\mathrm{\mid }}^2)(q)d{\mu }_{\rho }(q)$, first on $C_c(G\mathrm{/}H)$ and then on Borel sets by [F8]. Integrating $q\mapsto \mathrm{\parallel }{F}_Q(q){\mathrm{\parallel }}^2$ gives ${\int }_G\mathrm{\parallel }U(x){\mathrm{\parallel }}^2\rho (x)dx={\int }_Q\mathrm{\parallel }F(q){\mathrm{\parallel }}^2{T}_H(\mathrm{\mid }{u}_0{\mathrm{\mid }}^2)(q)d{\mu }_{\rho }(q)<\mathrm{\infty }\mathrm{.}$ On its compact support $\rho dx$ is finite, so [F6] makes $U$ Bochner square integrable.

Let $F\in C_c(G,H;V)$ and $K={\mathrm{supp}}_{G\mathrm{/}H}F$. Choose a cutoff $\chi \in C_c(G\mathrm{/}H)$ with $\chi =1$ on $K$ by [F3], then a nonnegative lift $u_0\in C_c(G)$ with $T_H{u}_0=\chi$ by [F3]. The map $u_{0}F$ is continuous and compactly supported on $G$. Cover its compact support by finitely many open sets on which $F$ varies by less than $ϵ$ in norm; [F4] supplies a subordinate partition ${\theta }_j$. For chosen $v_j$ from each patch, $u_{0}F$ is uniformly within $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}ϵ$ of ${\sum }_j(u_0{\theta }_j)v_j$. Averaging the latter gives a finite sum of generators. The averaging error is bounded uniformly on the compact quotient support because, after choosing a compact lift $C$ of that support, all relevant $h$ lie in the fixed compact set $C^{-1}\mathrm{supp}{u}_0\cap H$, of finite Haar measure. Since $A(u_{0}F)=T_H{u}_{0}F=F$ on $K$ and both vanish off $K$, the generators approximate $F$ uniformly.

Strong measurability approximates $U$ by finite-valued simple maps; scalar $C_c$ density [F5] approximates their coefficients in $L^2(G,\rho dx)$. Multiplying by one fixed compactly supported cutoff equal to one on $\mathrm{supp}U$ makes all approximants supported in a common compact $C$. The averaging operator $A(W)(x)={\int }_H\sigma (h)W(xh)dh$ is bounded on continuous maps supported in $C$. If $C=\mathrm{\varnothing }$ then $W=0$ and the bound is immediate. Otherwise choose a compact lift $C_0$ of $p(C)$ and let $m=dh(C_0^{-1}C\cap H)<\mathrm{\infty }$. For each $q\in p(C)$ choose a representative $x\in C_0$. Cauchy–Schwarz gives $\mathrm{\parallel }A(W)(x){\mathrm{\parallel }}^2\le m{T}_H(\mathrm{\parallel }W{\mathrm{\parallel }}^2)(q)$. Integrating over $G\mathrm{/}H$ and applying [F8] to $\mathrm{\parallel }W{\mathrm{\parallel }}^2\in C_c(G)$ yields $\mathrm{\parallel }A(W){\mathrm{\parallel }}_2^2\le m{\int }_G\mathrm{\parallel }W(y){\mathrm{\parallel }}^2\rho (y)dy=m\mathrm{\parallel }W{\mathrm{\parallel }}_{L^2(G,\rho dy)}^2\mathrm{.}$ Therefore averages of the finite-sum $C_c(G,V)$ approximants converge to $A(U)=F_Q$. Each average is a finite sum of the generators in step 1.1. Letting the discarded tail tend to zero proves density in the full measurable $L^2$ space. ∎

Since $D_g$ is a function on $G\mathrm{/}H$, it is right $H$-invariant. Thus $F(g^{-1}xh)=\sigma (h{)}^{-1}F(g^{-1}x)$ proves covariance of ${\mathrm{\Pi }}_{\rho }(g)F$. Its quotient support is the translate by $g$ of the compact support of $F$.

Applying twice gives ${\mathrm{\Pi }}_{\rho }(g_1){\mathrm{\Pi }}_{\rho }(g_2)F(x)=(D_{g_1}(xH)D_{g_2}(g_1^{-1}xH){)}^{1\mathrm{/}2}F(g_2^{-1}{g}_1^{-1}x)={\mathrm{\Pi }}_{\rho }(g_1{g}_2)F(x)$ by the cocycle identity [F1]. Also ${\mathrm{\Pi }}_{\rho }(e)=I$, so ${\mathrm{\Pi }}_{\rho }(g^{-1})$ is the inverse.

The cocycle identity with $g_1=g^{-1},g_2=g$ gives $D_g(grH)D_{g^{-1}}(rH)=1$. The change-of-measure formula $d((g^{-1}{)}_{\ast }{\mu }_{\rho })\mathrm{/}d{\mu }_{\rho }=D_{g^{-1}}$ then yields $\mathrm{\parallel }{\mathrm{\Pi }}_{\rho }(g)F{\mathrm{\parallel }}_2^2=\int D_g(q)\mathrm{\parallel }F(g^{-1}q){\mathrm{\parallel }}^{2}d{\mu }_{\rho }(q)=\int D_g(gr)D_{g^{-1}}(r)\mathrm{\parallel }F(r){\mathrm{\parallel }}^{2}d{\mu }_{\rho }(r)=\mathrm{\parallel }F{\mathrm{\parallel }}_2^2\mathrm{.}$ Thus the operator is an isometry on the dense continuous model, and its inverse from step 2.1 makes its extension unitary on the completion. ∎