  We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra
in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first
show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases
produce dual bases, one of which can be interpreted as characteristic classes
of the intersection homology mixed Hodge modules. In equivariant hyperbolic
cohomology, we show that if the Schubert variety is smooth, then the class it
determines coincides with the class of the Kazhdan-Lusztig basis; this was
known as the Smoothness Conjecture. For Grassmannians, we prove that the
classes of the Kazhdan-Lusztig basis coincide with the classes determined by
Zelevinsky's small resolutions. These properties of the so-called KL-Schubert
basis show that it is the closest existing analogue to the Schubert basis for
hyperbolic cohomology; the latter is a very useful testbed for more general
elliptic cohomologies.
