Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Abstract

We study the equivariant oriented cohomology ring hT (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 hT (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. 3.6. An example 13 3.7. A generalized Deodhar's resolution 15 4. Deodhar's parabolic Hecke modules revisited 17 4.1. The case of a multiplicative formal group law 17 4.2. Modules over the Hecke algebra

Geometric properties of the Kazhdan-Lusztig Schubert basis

“…In this section, we recall the definition of the Kazhdan-Lusztig Schubert (KL-Schubert) classes, following [LZZ19].…”

“…Aiming for the definition of Schubert classes, in [LZ17,LZZ19], the authors consider the so-called hyperbolic cohomology, denoted by h. A Riemann-Roch type map is defined from K-theory to the hyperbolic cohomology theory, which induces an action of the Hecke algebra (considered on the K-theory side) on the hyperbolic cohomology of G/B. In this way, the action of the Kazhdan-Lusztig basis defines classes KL w in h T (G/B), called KL-Schubert classes.…”

“…In this way, the action of the Kazhdan-Lusztig basis defines classes KL w in h T (G/B), called KL-Schubert classes. In [LZ17,LZZ19], there is a conjecture stating that, if the Schubert variety X(w) is smooth, then its fundamental class coincides with the class KL w . It is proved in some special cases in [LZ17,LZZ19].…”

“…In [LZ17,LZZ19], there is a conjecture stating that, if the Schubert variety X(w) is smooth, then its fundamental class coincides with the class KL w . It is proved in some special cases in [LZ17,LZZ19]. The second main result of this paper is to prove this conjecture in full generality, see Theorem 23.…”

Loop Models and K-Theory

“…Since no elliptic weights are known for loop models, it is unclear how to generalize our work in this direction. However, one can extend a little less by going over to the singular elliptic cohomology considered in [22], where loop models should still play a role.…”