Hello,


I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.


Let F be a p-adic field. Then (from what I understand) $C\_c^{\infty}(F)$ is the set of functions $f: F \to \mathbb{C}$ such that $f$ is locally constant and $f$ has compact support. Is the locally constant part the same as smooth or continuous (the $\infty$)? What exactly does smooth mean in this case?


Thanks,
Tom



Locally constant functions with compact support = smooth ?
A function $f : F \to \mathbb{C}$ is said to be smooth provided that $f$ is invariant under translation by some open subgroup of $F$ (hence by some compact open subgroup). Notice that when $f$ has compact support, this is equivalent to requiring that $f$ be locally constant. I don't have time to cook up a counterexample at the moment (nor do I have one readily available, which I suppose is a problem), but without the compact support assumption this is definitely a stronger condition than locally constant.


More generally, if $\pi : F \to GL(V)$ is a complex representation of the additive (topological) group of $F$, a vector $v \in V$ is called smooth provided that there exists an open subgroup of $F$ which fixes $v$. Taking $V$ to be the space of all functions $f : F \to \mathbb{C}$ with the action of $F$ given by translations, one arrives at the definition above.


This is relevant when, for example, one studies induced representations of $p$-adic groups.


