A foliation on a manifold $M$ can be seen as a sub bundle of the tangent bundle $\mathcal{F}\subseteq TM$ that is closed under Lie bracket of vector fields.
One can think of foliation as a Lie algebroid. There is an existing representation theory of Lie algebroids (there is something more, that of, representation unto homotopy of Lie algebroids).
It is understood that representation theory of Lie algebroids is not nice enough for multiple reasons.
As not all Lie algebroids are foliations, one reasonable question is, does the representation theory becomes interesting if we focus to representations of foliations?
How about asking for some more conditions taking into account the structure of foliation?
Or, is there already an existing representation theory of foliations?
