  The Balmer spectrum of a monoidal triangulated category is an important
geometric construction which is closely related to the problem of classifying
thick tensor ideals. We prove that the forgetful functor from the Drinfeld
center of a finite tensor category ${\mathbf{C}}$ to ${\mathbf{C}}$ extends to
a monoidal triangulated functor between their corresponding stable categories,
and induces a continuous map between their Balmer spectra. We give conditions
under which it is injective, surjective, or a homeomorphism. We apply this
general theory to prove that Balmer spectra associated to finite-dimensional
cosemisimple quasitriangular Hopf algebras (in particular, group algebras in
characteristic dividing the order of the group) coincide with the Balmer
spectra associated to their Drinfeld doubles, and that the thick ideals of both
categories are in bijection. An analogous theorem is proven for certain
Benson--Witherspoon smash coproduct Hopf algebras, which are not
quasitriangular in general.
