AI 中文总结
本文通过不动点局部化技术，将仿射格拉斯曼流形的等变$K$-理论环描述为仿射爆发的函数环，推广并证明了Bezukavnikov猜想，并给出无穷拓扑生成元。
AI 中文摘要
我们重新审视等变拓扑$K$-理论中的不动点局部化技术，将其在等变基的不同点上的纤维呈现为相应不动点方案的上同调。然后，我们利用这一框架对仿射格拉斯曼流形$\mathcal{G}r_G$的等变$K$-理论环进行有效描述。为此，我们详细研究了相关的不动点，将已知结果推广到任意$(x, \zeta) \in T \times \mathbb{G}_{m}^{\mathrm{rot}}$的作用。我们得到了复化拓扑$K$-理论环$K^{\mathrm{top}, 0}_{T\times \mathbb{G}_{m}^{\mathrm{rot}}}(\mathcal{G}r_G; \mathbb{C})$作为一族仿射爆发的函数环的描述，从而推广并证明了Roman Bezrukavnikov的一个猜想。虽然这个环相当大，但我们提供了一个优选的无穷拓扑生成元集合。
英文摘要
We revisit fixed-point localization techniques in equivariant topological $K$-theory, presenting its fibers over varying points of the equivariant base as cohomology of the corresponding fixed-point schemes. We then employ this framework for an effective description of the equivariant $K$-theory ring of the affine Grassmannian $\mathcal{G}r_G$. We do so by a detailed study of the relevant fixed points, extending known results to the action of any $(x, ζ) \in T \times \mathbb{G}_{m}^{\mathrm{rot}}$. We obtain a description of the complexified topological $K$-theory ring $K^{\mathrm{top}, 0}_{T\times \mathbb{G}_{m}^{\mathrm{rot}}}(\mathcal{G}r_G; \mathbb{C})$ as the ring of functions on a family of affine blowups, extending and proving a conjecture of Roman Bezrukavnikov. While this ring is fairly big, we provide a preferred infinite set of topological generators.
Comments70 pages, comments welcome
