Action of Weyl group on equivariant K-theory of flag varieties

TL;DR

This paper investigates how the Weyl group acts on the equivariant K-theory of flag varieties, providing explicit formulas and expansions in various bases, with applications to special linear groups.

Contribution

It introduces explicit formulas for Weyl group automorphisms on equivariant K-theory, including expansions in Schubert basis and motivic Chern classes, advancing computational methods.

Findings

Derived formulas for Weyl group automorphisms on K-theory

Expanded formulas in Schubert basis and motivic Chern classes

Provided effective approximations for special linear groups

Abstract

We describe the action of the Weyl group of a semi simple linear group $G$ on cohomological and K-theoretic invariants of the generalized flag variety $G/B$. We study the automorphism $s_i$, induced by the reflection in the simple root, on the equivariant $K$-theory ring $K_T(G/B)$ using divided difference operators. Using the localization theorem for torus action and Borel presentation for the equivariant K-theory ring, we calculate the formula for this automorphism. Moreover, we expand this formula in the basis consisting of structure sheaves classes of Schubert varieties. We provide effective formula (applying properties of Weyl groups) for the approximation of this expansion, more specifically for the part corresponding to Schubert varieties with the fixed dimension, which in the case of $G$ being a special linear group is more exact. Finally, we discuss the above-mentioned formula in the basis of motivic Chern classes of Schubert varieties.