On some representations of the Drinfelʹd double

For H a finite-dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero the induced representations from H and H* to the Drinfelʹd double D(H) are studied. The product of two such representations is a sum of copies of the regular representation of D(H). The action of certain irreducible central characters of H* on the simple modules of H is considered. The modules that receive trivial action from each such irreducible central character are precisely the constituents of the tensor powers of the adjoint representation of H.