Geodesic
Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type
Forum of Mathematics, Sigma, Tome 9 (2021)

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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 $\mathbb {Z}\left [q^{\pm 1}\right ]$-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. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type. 
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     author = {Takafumi Kouno and Satoshi Naito and Daniel Orr and Daisuke Sagaki},
     title = {Inverse {K-Chevalley} formulas for semi-infinite flag manifolds, {I:} minuscule weights in {ADE} type},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2021},
     doi = {10.1017/fms.2021.45},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.45/}
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Takafumi Kouno; Satoshi Naito; Daniel Orr; Daisuke Sagaki. Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.45

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