Smooth representation of a totally disconnected group

Definition

Let $G$ be a locally profinite group. A complex representation $(\pi,V)$ of $G$ is smooth if every vector has an open stabilizer:

V=\bigcup_{K\leq G\text{ open}}V^K, \qquad V^K=\{v\in V:\pi(k)v=v\text{ for all }k\in K\}.

Equivalently, every orbit map $g\mapsto\pi(g)v$ is locally constant. No topology on $V$ is part of this algebraic notion of smoothness.

Smooth vectors

If a continuous representation is initially given on a topological vector space, its smooth part is the union of its open-subgroup fixed spaces. This is different from differentiable smooth vectors for a real Lie group.

Hecke action

For a compact open subgroup $K\leq G$ , the fixed space $V^K$ carries an action of the Hecke algebra $\mathcal H(G,K)$ . Much of nonarchimedean representation theory studies a smooth representation through these fixed spaces as $K$ varies.

References

Joseph Bernstein and Andrei Zelevinsky, “Induced representations of reductive $p$ -adic groups I,” Annales scientifiques de l’École Normale Supérieure 10 (1977), 441–472. Numdam.
Jayce R. Getz, An Introduction to Automorphic Representations, §§5 and 10.9. Author notes.