A representation of a Lie groupoid G ⇉ ⇉ M consists of a Lie groupoid action on a vector bundle π π : E → → M , such that the action is fibrewise linear, i.e. each bijection π π − − 1 ( x ) → → π π − − 1 ( y ) is a linear isomorphism. Equivalently, a representation of G on E can be described as a Lie groupoid morphism from G to the general linear groupoid G L ( E ) .