Strong continuity of unitary induction

Fix $F\in C_c(G,H;V)$ and let $K$ be its compact quotient support. Choose a compact identity neighborhood $C\subset G$ and put $Q=K\cup CK$, a compact subset of $G\mathrm{/}H$. For $g\in C$ both $F$ and ${\mathrm{\Pi }}_{\rho }(g)F$ vanish off $Q$.

Choose a compact lift $K_0\subset G$ of $Q$ using [F3] and [A1]. On $C\times K_0$, joint continuity of $(g,x)\mapsto D_g(xH{)}^{1\mathrm{/}2}F(g^{-1}x)$ and compactness imply uniform convergence to $F(x)$ as $g\to e$. The fiber norm of the difference is right-$H$ invariant, so this gives uniform convergence on $Q$. Since ${\mu }_{\rho }(Q)<\mathrm{\infty }$, its $L^2$ norm is at most ${\mu }_{\rho }(Q{)}^{1\mathrm{/}2}$ times that uniform bound, and tends to zero.

For arbitrary $u$ in the completion and $ϵ>0$, choose $F\in C_c(G,H;V)$ with $\mathrm{\parallel }u-F{\mathrm{\parallel }}_2<ϵ$ by [F2]. Unitarity gives $\mathrm{\parallel }{\mathrm{\Pi }}_{\rho }(g)u-u{\mathrm{\parallel }}_2\le 2ϵ+\mathrm{\parallel }{\mathrm{\Pi }}_{\rho }(g)F-F{\mathrm{\parallel }}_2\mathrm{.}$ Step 2.1 makes the last term tend to zero; then let $ϵ\downarrow 0$. This proves strong continuity for every vector. ∎