We classify Lagrangian subcategories of the representation category of a
twisted quantum double of a finite group. In view of results of 0704.0195v2
this gives a complete description of all braided tensor equivalent pairs of
twisted quantum doubles of finite groups. We also establish a canonical
bijection between Lagrangian subcategories of the representation category of a
twisted quantum double of a finite group G and module categories over the
category of twisted G-graded vector spaces such that the dual tensor category
is pointed. This can be viewed as a quantum version of V. Drinfeld's
characterization of homogeneous spaces of a Poisson-Lie group in terms of
Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we
obtain that two group-theoretical fusion categories are weakly Morita
equivalent if and only if their centers are equivalent as braided tensor
categories.
