  Francois Rodier proved that it is possible to view smooth representations of
certain totally disconnected abelian groups (the underlying additive group of a
finite-dimensional p-adic vector space, for example) as sheaves on the
Pontryagin dual group. For nonabelian totally disconnected groups, the
appropriate dual space necessarily includes representations which are not
one-dimensional, and does not carry a group structure. The general definition
of the topology on the dual space is technically unwieldy, so we provide three
different characterizations of this topology for a large class of totally
disconnected groups (which includes, for example, p-adic unipotent groups),
each with a somewhat different flavor. We then use these results to demonstrate
some formal similarities between smooth representations and sheaves on the dual
space, including a concrete description of the Bernstein center of the category
of smooth representations.
