Unitary induction from a closed subgroup

Choose $\rho$ and ${\mu }_{\rho }$ by [F1] under [A1]. The covariance equations make the pointwise inner product a well-defined positive form by [F2,F3]. Its completion is a Hilbert space.

The formula ${\mathrm{\Pi }}_{\rho }(g)F(x)=D_g(xH{)}^{1\mathrm{/}2}F(g^{-1}x)$ preserves the dense covariant model and is a unitary representation by [F4]. The strong continuity lemma [F5] extends this property to every completed vector. This gives ${\mathrm{Ind}}_H^G\sigma$.

If $H=G$, then $G\mathrm{/}H$ is a singleton and every covariant section is determined by $v=F(e)$, with $F(x)=\sigma (x{)}^{-1}v$. Rescale $\rho$ by a positive constant so the quotient point has measure one; evaluation at $e$ is then an isometry, and the action becomes $v\mapsto \sigma (g)v$. If $H=\{e\}$, put $c=dh(\{e\})$ and choose the constant rho-function $\rho (x)=c$. The Weil formula [F6] then gives ${\mu }_{\rho }=dx$, so the model completes from $C_c(G;V)$ to $L^2(G;V)$. The action is $F(x)\mapsto F(g^{-1}x)$, namely ${\lambda }_G\otimes I_V$; for $V=\mathbb{C}$ this is the scalar left regular representation. If ${\mu }_{\rho }$ is invariant, then $d(g_{\ast }{\mu }_{\rho })\mathrm{/}d{\mu }_{\rho }=1$; the continuous density $D_g$ is therefore one everywhere by full support, so the action has no cocycle factor. These are the three stated reductions. ∎