Smooth right regular representation ($(\rho (g)f)(x)=f(xg)$)

The right action on compactly supported locally constant functions is a smooth representation. For a compact-open subgroup $K$, its fixed-vector space has basis the characteristic functions of cosets in $G/K$, since compact support on that discrete quotient is finite support. Consequently this representation is admissible exactly when $G$ is compact.