Weil formula with a rho-function

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. ∎