For any Bott-Samelson resolution of the flag variety, and any torus equivariant oriented cohomology, we compute the restriction formula of certain basis η_L of equivariant oriented cohomology of Bott-Samelson variety determined by the projective bundle formula. As an application, we show that the equivariant oriented cohomology of Bott-Samelson variety 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 η_L onto equivariant oriented cohomology of flag variety, and their restriction formula.

The second author would like to thank Leonardo Mihalcea and Rebecca Goldin for helpful conversations.