Support theory for Drinfeld doubles of some infinitesimal group schemes

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