| With this work we initiate a study of representations of algebraic groups from the modular point of view, extending classical representation theory to include representations which are not semisimple. Let G be an algebraic group over a field k of characteristic zero. We focus on the case that G is unipotent. In this case the diagonal of the stack Mn (G) of n-dimensional representations is positive dimensional, and its fiber dimensions can jump in families. We introduce a nondegeneracy condition which gives rise to an immersed substack Mndn (G) ⊂ Mn (G) whose diagonal is flat and which admits a coarse space Mndn (G). For general n our nondegeneracy condition is somewhat opaque; we are able, however, to define an invariant w of G, its width, which singles out a best case scenario for the construction of moduli, and to give, for n ≤ w + 1, a concrete criterion for nondegeneracy. With the help of this criterion we show that for n ≤ w + 1, Mndn (G) is quasi-projective.;Along the way we study the structure of flag representations: representations whose associated filtration is a full flag, as well as wide representations: those which are nondegenerate of dimension n ≤ w + 1. We associate to a flag representation a set of "canonical matrix entries" which record the associated graded representation, and we prove that the canonical matrix entries of a wide representation are nonzero. As a corollary, we find that the width is bounded by the length of a composition series for G. This enables us to compute the width in several examples.;Finally, flag representations have two canonically defined codimension-one subquotients; we study a certain compatibility condition obeyed by pairs of representations which arise in this way. This gives rise to a closed subspace Mcndn (G) of Mndn (G) x Mnd n-1G Mndn (G) and to a fibration of Mndn+1 (G) over Mcndn (G). |
