  We address the problem of defining Schubert classes independently of a
reduced word in equivariant elliptic cohomology, 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.
