Modular categories associated with unipotent groups

Abstract

Let \(G\) be a unipotent algebraic group over an algebraically closed field \(\mathtt{k }\) of characteristic \(p>0\) and let \(l\ne p\) be another prime. Let \(e\) be a minimal idempotent in \(\mathcal{D }_G(G)\), the \(\overline{\mathbb{Q }}_l\)-linear triangulated braided monoidal category of \(G\)-equivariant (for the conjugation action) \(\overline{\mathbb{Q }}_l\)-complexes on \(G\) under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to \(G\) and \(e\) a modular category \(\mathcal{M }_{G,e}\). In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class \(\mathfrak{C }_p^{\pm }\).