  The present paper links the representation theory of Lie groupoids and
infinite-dimensional Lie groups. We show that smooth representations of Lie
groupoids give rise to smooth representations of associated Lie groups. The
groups envisaged here are the bisection group and a group of groupoid self
maps. Then representations of the Lie groupoids give rise to representations of
the infinite-dimensional Lie groups on spaces of (compactly supported) bundle
sections. Endowing the spaces of bundle sections with a fine Whitney type
topology, the fine very strong topology, we even obtain continuous and smooth
representations. It is known that in the topological category, this
correspondence can be reversed for certain topological groupoids. We extend
this result to the smooth category under weaker assumptions on the groupoids.
