We describe the action of the Weyl group of a semisimple linear group G on cohomological and K-theoretic invariants of the generalized flag variety G/B. We study the automorphism \(s_i\) , induced by the reflection in a simple root, on the equivariant K-theory ring \(K_T(G/B)\) using divided-difference operators. Using the localization theorem for torus actions and the Borel presentation for the equivariant K-theory ring, we derive a formula for this automorphism. Moreover, we expand this formula in the basis consisting of the classes of structure sheaves of Schubert varieties. We provide an explicit formula (using properties of Weyl groups) to approximate this expansion, specifically the part corresponding to Schubert varieties of fixed dimension, which in the case of G being a special linear group is more precise. Finally, we discuss the above-mentioned formula in the basis of motivic Chern classes of Schubert varieties.
