Certain (good) functors on vector spaces, for example volume elements, and densities, lead to 1-dimensional representations of Lie groupoids. Invariant sections of such representations are transverse volume elements and transverse densities for the groupoid.
These representations, both using volume elements, and densities, can be used to construct the modular class of the Lie groupoid.

These constructions will be explained, as well as the conclusions that:
- densities are orientable in a manner compatible with the representation, and so are better suited (than volume elements) to define modular classes;
- a Lie groupoid has an invariant positive transverse density exactly when the modular class vanishes;
- it has a transverse volume element under an additional orientability condition (of representations).

This is based on joint work with Marius Crainic