  For a finite abelian group action on a linear category, we study the dual
action given by the character group acting on the category of equivariant
objects. We prove that the groups of equivariant autoequivalences on these two
categories are isomorphic. In the triangulated situation, this isomorphism
implies that the classifications of stable tilting objects for these two
categories are in a natural bijection. We apply these results to stable tilting
complexes on weighted projective lines of tubular type.
