  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.
