Modular categories, orbit method and character sheaves on unipotent groups

Abstract

Let $G$ be a unipotent group over a field of characteristic $p > 0$. The theory of character sheaves on $G$ was initiated by V. Drinfeld and developed jointly with D. Boyarchenko. They also introduced the notion of $\mathbb{L}$-packets of character sheaves. Each $\mathbb{L}$-packet can be described in terms of a modular category. Now suppose that the nilpotence class of $G$ is less than $p$. Then the $\mathbb{L}$-packets are in bijection with the set $\mathfrak{g}^*/G$ of coadjoint orbits, where $\mathfrak{g}$ is the Lie ring scheme obtained from $G$ using the Lazard correspondence and $\mathfrak{g}^*$ is the Serre dual of $G$. If $\Omega$ is a coadjoint orbit, then the corresponding modular category can be identified with the category of $G$-equivariant local systems on $\Omega$. This in turn is equivalent to the category of finite dimensional representations of a finite group. However, the associativity, braiding and ribbon constraints are nontrivial. Drinfeld gave a conjectural description of these constraints in 2006. In this article, we prove the formula describing the ribbon structure when $\dim(\Omega)$ is even.