Moduli of nondegenerate unipotent representations in characteristic zero

TL;DR

This paper introduces a new invariant called width for unipotent group representations in characteristic zero, showing that nondegenerate representations of certain dimensions form a quasi-projective variety and exploring their moduli.

Contribution

It defines the width invariant for unipotent groups and establishes conditions under which nondegenerate representations form a quasi-projective variety.

Findings

Width is bounded by the length of a composition series.

Nondegenerate representations of dimension ≤ width+1 form a quasi-projective variety.

Studied the gluing problem for pairs of nondegenerate representations.

Abstract

With this work we initiate a study of the 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 \textit{width}, as well as a certain nondegeneracy condition on representations, and we prove that nondegenrate representations of dimension $n \le w+1$ 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. Finally, we study the problem of gluing a pair of nondegenerate representations along a common subquotient.