  Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two
orbits: an open orbit and a a closed orbit of codimension $1$. We give an
algebraic description of the category of $G$-equivariant vector bundles on $X$
under a mild technical hypothesis. We deduce simpler classifications in the
special cases of line bundles and vector bundles which are generically local
systems. We apply our results to the study of admissible representations of
semisimple Lie groups. Our main result gives a new set of constraints on the
associated cycles of unipotent representations.
