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We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flagvarieties using the moment map approach. We define the right Hecke action onthis cohomology ring, and then prove that the respective Bott-Samelson classesin $h_{T}(G/P)$ can be obtained by applying this action to the fundamentalclass of the identity point, hence generalizing previously known results byBrion, Knutson, Peterson, Tymoczko and others.  We then focus on the equivariant oriented cohomology theory corresponding tothe 2-parameter Todd genus. We give a new interpretation of Deodhar'sconstruction of the parabolic Kazhdan-Lusztig basis. Based on it, we define theparabolic Kazhdan-Lusztig (KL) Schubert classes independently of a reducedword. We make a positivity conjecture, and a conjecture about the relationshipof such classes with smoothness of Schubert varieties. We then prove severalspecial cases.

Parabolic Kazhdan-Lusztig basis, Schubert classes, and equivariant oriented cohomology