Higher Verlinde categories of reductive groups

Abstract

We define tensor categories ${\sf Ver}_{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}_p(G)$ originating from Gelfand-Kazhdan and the higher Verlinde categories ${\sf Ver}_{p^n}$ for ${\rm SL}_2$ defined by Benson-Etingof-Ostrik. The construction is based on the definition of ${\sf Ver}_{p^n}$ as an abelian envelope of a quotient of a category of tilting modules, but we also introduce an expanded construction which refines the ${\rm SL}_2$ case and gives new results. In particular, the union ${\sf Ver}_{p^\infty}(G)$ can be derived from the perfection of $G$; certain exact sequences in ${\sf Rep}G$ map to exact sequences in ${\sf Ver}_{p^n}(G)$; and the underlying abelian category of ${\sf Ver}_{p^n}$ can be expressed as a subcategory of ${\sf Rep}{\rm SL}_2$, or as a Serre quotient of a subcategory of ${\sf Rep}{\rm SL}_2$.