Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 1 / 1 / b / Solution

An $\ell$-group here means a locally profinite group, so identity neighbourhoods contain compact-open subgroups. The right action is a group representation because $\rho (g)\rho (h)f(x)=f(xgh)=\rho (gh)f(x)$.

Let $f$ be a compactly supported locally constant function with support $S$. For every $x\in S$, local constancy supplies a compact-open subgroup $K_x$ with $f$ constant on $x{K}_x$, and this coset lies in $S$. Compactness gives a finite cover $S={\bigcup }_{j=1}^r{x}_j{K}_j$. Put $K={\bigcap }_j{K}_j$. Each coset in this cover is right $K$-invariant, so $S$ is right $K$-invariant. Within the support, $xk$ and $x$ lie in the same chosen $x_j{K}_j$ and have the same function value; outside the support, right $K$-invariance of $S$ ensures that both values remain zero. Hence $\rho (k)f=f$ for all $k\in K$. The zero function is fixed by every subgroup. This proves that the smooth right regular representation is smooth.

For a fixed compact-open subgroup $K$, right-invariant functions are functions on the discrete coset space $G/K$. Compact support becomes finite support there, since the image of a compact set in a discrete space is finite. Conversely every finitely supported function on $G/K$ lifts to a compactly supported locally constant function on $G$. Thuswhere the superscript indicates finite support. Its basis is the set of characteristic functions $1_{gK}$, and its dimension is finite exactly when $[G:K]$ is finite.

If $G$ is compact, the discrete quotient $G/K$ is compact and therefore finite for every such $K$; hence the representation is admissible. Conversely, admissibility gives finite index for any one compact-open subgroup. Then $G$ is a finite union of compact cosets, so is compact. Therefore the representation is admissible if and only if $G$ is compact.