Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

TL;DR

This paper derives an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds for simply-laced types, connecting quantum Bruhat graph combinatorics with double affine Hecke algebra techniques.

Contribution

It provides the first explicit inverse Chevalley formula for arbitrary Schubert classes and minuscule weights in simply-laced semi-infinite flag manifolds, extending prior results.

Findings

Explicit inverse Chevalley formula for simply-laced types.

Determination of nonsymmetric q-Toda operators for minuscule weights.

Connection between quantum Bruhat graph and double affine Hecke algebra.

Abstract

We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^{\lambda}$, where $\lambda$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply-laced type, except for type $E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. As such, our formula also provides an explicit determination of all nonsymmetric $q$-Toda operators for minuscule weights in ADE type.