On equivariant oriented cohomology of Bott-Samelson varieties

For any Bott-Samelson resolution $q_I:\widehat{X_I}\to G\mathrm{/}B$ of the flag variety $G\mathrm{/}B$, and any torus equivariant oriented cohomology $h_T$, we compute the restriction formula of certain basis ${\eta }_L$ of $h_T(\widehat{X_I})$ determined by the projective bundle formula. As an application, we show that $h_T(\widehat{X_I})$ embeds into the equivariant oriented cohomology of $T$-fixed points, and the image can be characterized by using the Goresky-Kottwitz-MacPherson (GKM) description. Furthermore, we compute the push-forward of the basis ${\eta }_L$ onto $h_T(G\mathrm{/}B)$, and their restriction formula.