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We prove a Chevalley formula to multiply the motivic Chern classes ofSchubert cells in a generalized flag manifold $G/P$ by the class of any linebundle $\mathcal{L}_\lambda$. Our formula is given in terms of the$\lambda$-chains of Lenart and Postnikov. Its proof relies on a change of basisformula in the affine Hecke algebra due to Ram, and on the Hecke algebra actionon torus-equivariant K-theory of the complete flag manifold $G/B$ via leftDemazure--Lusztig operators. We revisit some wall-crossing formulae for thestable envelopes in $T^*(G/B)$. We use our Chevalley formula, and theequivalence between motivic Chern classes of Schubert cells and K-theoreticstable envelopes in $T^*(G/B)$, to give formulae for the change ofpolarization, and for the change of slope for stable envelopes. We proveseveral additional applications, including Serre, star, and Dynkin, dualitiesof the Chevalley coefficients, new formulae for the Whittaker functions, andfor the Hall--Littlewood polynomials. We also discuss positivity properties ofspecial cases of the Chevalley coefficients, some of them conjectural.

Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes