Geometric properties of the Kazhdan-Lusztig Schubert basis

TL;DR

This paper explores the geometric and algebraic properties of the Kazhdan-Lusztig basis within K-theory and hyperbolic cohomology of flag varieties, revealing dualities, smoothness relations, and connections to resolutions.

Contribution

It demonstrates dual bases in K-theory, confirms the Smoothness Conjecture in hyperbolic cohomology, and links Kazhdan-Lusztig classes to Zelevinsky's resolutions for Grassmannians.

Findings

Dual bases in K-theory from Kazhdan-Lusztig bases.

Verification of the Smoothness Conjecture in hyperbolic cohomology.

Kazhdan-Lusztig classes match Zelevinsky's resolutions for Grassmannians.

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.