We explain an extension of the (equivariant) MacPherson Chern class transformation to a bivariant theory with respect to suitable convolutions as conjectured by Ginzburg. This is based on the Lagrangian approach via characteristic cycles. As an application we show, that for flag manifolds G/B the two cohomological Weyl group actions constructed by Ginzburg and Aluffi-Mihalcea coincide. These Weyl group actions permute the (equivariant) Chern classes of the corresponding Schubert cells. This is joint work with P. Aluffi, L. Mihalcea and C. Su.