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.

中文翻译：

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半无限旗形流形的逆 K-Chevalley 公式，I：ADE 类型中的微小权重

我们在等变式中证明了一个显式的逆 Chevalley 公式ķ- 简单系带类型的半无限旗形流形理论。“逆 Chevalley 公式”是指等变标量与舒伯特类的乘积公式，表示为 $\mathbb {Z}\left [q^{\pm 1}\right ]$ -由等变线束扭曲的舒伯特类的线性组合。我们的公式适用于简单系带类型和等变标量的半无限旗形流形中的任意舒伯特类 $e^{\lambda }$ ， 在哪里 $\λ$ 是任意微小的权重。通过 Stembridge 的结果，我们的公式完全确定了除了类型之外的简单系带类型中任意重量的逆 Chevalley 公式 $E_8$ . 我们公式的组合由量子布鲁哈特图控制，证明基于双仿射赫克代数的极限。因此，我们的公式还提供了所有非对称的明确确定q- ADE 类型中用于微小权重的 Toda 运算符。