Moduli of unipotent representations II: Wide representations and the width

With this work and its prequel [Ann. Inst. Fourier (Grenoble) 62 (2012), no. 3, 1123-1187] we initiate a study of the finite dimensional representations of a unipotent group over a field of characteristic zero from the modular point of view. Let G be such a group. The stack of all representations of a fixed finite dimension n is badly behaved. We introduce an invariant, w, of G, its width, as well as a certain nondegeneracy condition on representations, and we prove that nondegenerate representations of dimension n ≤ w + 1 form a quasi-projective variety. Our definition of the width is opaque; as a first attempt to elucidate its behavior, we prove that it is bounded by the length of a composition series.