  We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag
varieties using the moment map approach. We define the right Hecke action on
this cohomology ring, and then prove that the respective Bott-Samelson classes
in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental
class of the identity point, hence generalizing previously known results by
Brion, Knutson, Peterson, Tymoczko and others.
  We then focus on the equivariant oriented cohomology theory corresponding to
the 2-parameter Todd genus. We give a new interpretation of Deodhar's
construction of the parabolic Kazhdan-Lusztig basis. Based on it, we define the
parabolic Kazhdan-Lusztig (KL) Schubert classes independently of a reduced
word. We make a positivity conjecture, and a conjecture about the relationship
of such classes with smoothness of Schubert varieties. We then prove several
special cases.
