We explain in the context of complete flag varieties X = G/B the inductive calculation of equivariant motivic characteristic classes of Schubert cells via suitable Demazure-Lusztig operators, fitting with convolution actions of corresponding Hecke-algebras and Weyl groups. Applications include solutions of: (1) a conjecture of Bump, Nakasuji and Naruse. Here a specialization of the equivariant K-theory of G/B was identified, as a Hecke module, to the Iwahori-fixed part of the the principal series representation of a p-adic Langlands dual group. Under this identification, certain classes related to motivic Chern classes of Schubert cells, were sent to the standard basis elements of the principal series representation, and the fixed point classes were sent to the Casselman basis elements. (2) a positivity conjecture of Aluffi-Mihalcea for the non-equivariant MacPherson Chern classes of Schubert cells (3) a positivity conjecture about the Euler characteristic of generic triple intersections of Schubert cells. This is joint work with P. Aluffi, L. Mihalcea and C. Su, and for (3) with C. Simpson and B. Wang.