双仿射Hecke代数上的bar运算
A bar operation on the double affine Hecke algebra
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中文总结 AI 辅助
本文为对称化Kac-Moody群的仿射Hecke代数构造bar运算，利用Steinberg簇的等变K-理论定义几何双模，引入完备化解决无穷展开问题，并在仿射型给出双仿射Hecke代数局部化上的显式公式，且当q=1时成为对合。
中文摘要 AI 辅助
我们为对称化Kac-Moody群所关联的仿射Hecke代数$\mathcal{H}_W^a$构造了一种bar型运算。我们的主要工具是一个几何双模，它是利用该Kac-Moody群的Steinberg簇的适当版本的等变$K$-理论定义的。利用该双模中对应于Kashiwara旗簇上的标准与余标准Hodge $D$-模的元素，我们通过关联$\mathcal{H}_W^a$在该双模上的左、右作用，定义了$\mathcal{H}_W^a$上的bar对合。一个核心问题是，由此得到的bar运算公式并不保持代数本身，并且自然会产生无穷展开。为解决此问题，我们引入了$\mathcal{H}_W^a$的某些完备化，在这些完备化上bar运算是良定义的。为此构造，我们证明了$\mathcal{H}_W^a$中乘积的若干组合有限性结果。在仿射型情形，我们进一步分析了代数的level-zero部分，并得到了Cherednik双仿射Hecke代数的适当局部化上bar运算的显式组合公式。在此情形下，bar运算本身并非对合，但我们也证明了当格参数$q$设为$1$时，该bar运算成为对合。
英文摘要
We construct a bar-type operation for the affine Hecke algebra $\Ha$ attached to a symmetrizable Kac--Moody group. Our main tool is a geometric bimodule defined using the equivariant $K$-theory of a suitable version of the Steinberg variety for this Kac--Moody group. Using elements of this bimodule corresponding to standard and costandard Hodge $D$-modules on the Kashiwara flag variety, we define a bar involution on $\mathcal{H}_{W}^a$ by relating the left and right actions of $\mathcal{H}_{W}^a$ on this bimodule. A central issue is that the resulting formulas for the bar operation do not preserve the algebra itself and naturally produce infinite expansions. To address this, we introduce certain completions of $\mathcal{H}_{W}^a$ on which the bar operation is well-defined. For this construction, we prove a number of combinatorial finiteness results for products in $\mathcal{H}_{W}^a$. In affine type, we further analyze the level-zero part of the algebra and obtain an explicit combinatorial formula for a bar operation on a suitable localization of Cherednik's double affine Hecke algebra. In this case the bar operation itself is not an involution but we also show that this bar operation becomes an involution when the lattice parameter $q$ is set to $1$.
