Spencer delta-cohomology, restrictions, characteristics and involutive symbolic PDEs

We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context, though in a modified form. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating Spencer delta-cohomology of a symbolic system and its restriction to a non-characteristic subspace.