Higher Verlinde categories of reductive groups

Abstract

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

25 pages, 1 figure. v2: expanded Theorem 3(1) and Conjecture 3.8