A型量子Satake：一般与泛情形
Quantum Satake in Type A: The General and Generic Case
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中文总结 AI 辅助
本文在泛化条件下证明了A_{n-1}型（n≥4）的量子Satake等价，将代数重构及其q-变形从A₁、A₂推广到一般情形，并推广了相关文献结果。
中文摘要 AI 辅助
在第一作者的工作中，几何Satake等价被（非严格地）重新解释为两个代数定义的2-范畴之间的等价，一个由李代数的表示构成，另一个使用奇异Soergel双模构建。随后解释了如何在A型中对此等价进行q-变形，将特殊线性李代数替换为其量子群，并使用变形反射表示的奇异Soergel双模。代数重构及其q-变形仅在A₁和A₂型中得到证明。本文中，我们在泛化条件下证明n≥4的A_{n-1}型结果：具体而言，我们工作在特征零的域上，其中q不是单位根，并且已添加q的n次根。在此过程中，我们将文献中的某些结果（例如Soergel-Williamson范畴化定理、球面元素的Soergel猜想、各种对称性）推广到变形反射表示。
英文摘要
In work of the first author, the geometric Satake equivalence was (non-rigorously) reinterpreted as an equivalence between two algebraically-defined $2$-categories, one built from representations of a Lie algebra, and one built using singular Soergel bimodules. It was then explained how to $q$-deform this equivalence in type $A$, replacing the special linear Lie algebra with its quantum group, and using singular Soergel bimodules for a deformed reflection representation. Both the algebraic reformulation and its $q$-deformation were only proven in types $A_1$ and $A_2$. In this paper, we prove the result in type $A_{n-1}$ for $n \ge 4$, while working generically: more precisely, we work over a field of characteristic zero, where $q$ is not a root of unity, and having adjoined an $n$-th root of $q$. Along the way we generalize certain results in the literature (e.g. the Soergel-Williamson categorification theorem, the Soergel conjecture for spherical elements, various symmetries) to the deformed reflection representation.
