We address the problem of defining Schubert classes independently of
a reduced word in equivariant cohomology theory corresponding to a hyperbolic formal group law, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important
conjectures about them: a positivity conjecture, and the agreement with the topologically defined Schubert classes in the smooth case. We prove some special cases of these conjectures.