Motivic classes of classifying stacks of some semi-direct products

Abstract: Let k be a field, let G be a finite group and let T be a split k-torus on which G acts multiplicatively, and for every m greater than 1 denote by T[m] the m-torsion subgroup of T. Under a suitable assumption on m, we show that the motivic class of the classifying stack of the semi-direct product of T[m] and G in K_0(Stacks_k) is trivial. As a consequence, we prove that the motivic class of BW is trivial for a large class of complex reflection groups W.