Support theory for Drinfeld doubles of some infinitesimal group schemes

TL;DR

This paper develops a support theory for the Drinfeld doubles of Frobenius kernels in algebraic groups, classifying thick ideals via cohomological support and establishing a unified support framework.

Contribution

It introduces a support theory for Drinfeld doubles of Frobenius kernels, including classification of thick ideals and a $ ext{pi}$-point style rank variety, extending to groups with a quasi-logarithm.

Findings

Thick ideals are classified by cohomological support.

The Balmer spectrum of the stable category is computed.

Support theories satisfy the tensor product property.

Abstract

Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $\pi$-point style rank variety for the Drinfeld double, identify $\pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.