I was pretty shocked by this one: if $V$ is the additive group of a finite-dimensional vector space over a $p$-adic field, the category of smooth (sometimes called algebraic or discrete) complex representations of $V$ is equivalent to the category of sheaves of complex vector spaces on the Pontryagin dual of $V$.
More generally, this is true of any Hausdorff, locally compact, and totally disconnected abelian group which is a filtered union of its compact open subgroups.
