Let μ be any positive definite
measure on a locally compact group, and let (πμ,ℋμ) be the associated unitary
representation of G. Previous work of the authors’ showed that a cyclic vector exists
for πμ if G is second countable; there is now a simple proof of this result,
due to Hulanicki. Rather elementary conditions on the way μ is related to
the geometry of G are examined which are necessary, or sufficient, for the
existence of a cyclic vector. These conditions require μ to be “constant” on
cosets (or double cosets) of certain subgroups of G. A conjectured necessary
and sufficient conditions is presented. These results are adequate to decide
whether or not πu is cyclic for various nontrivial measures. As a special case
it is shown that the left regular representation of G is cyclic ⇔ G is first
countable.
