Geometric properties of the Kazhdan-Lusztig Schubert basis

Abstract

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.

arXiv comment24 pages, 4 figures. Comments are welcome

References in corpus (7)

• Motivic Chern classes and K-theoretic stable envelopes
• Formal Hecke algebras and algebraic oriented cohomology theories
• Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds
• Inductive construction of stable envelopes
• A coproduct structure on the formal affine Demazure algebra
• Elliptic classes of Schubert varieties via Bott-Samelson resolution
• Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory