Smooth right regular representation
= Smooth right regular representation
{title2=$(\rho(g)f)(x)=f(xg)$}
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.